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A mechanistic damage model for ligaments
Accepted Manuscript A Mechanistic Damage Model for Ligaments Jeff M. Barrett, Jack P. Callaghan PII: DOI: Reference:
S0021-9290(17)30349-4 http://dx.doi.org/10.1016/j.jbiomech.2017.06.039 BM 8278
To appear in:
Journal of Biomechanics
Received Date: Revised Date: Accepted Date:
1 March 2017 3 June 2017 25 June 2017
Please cite this article as: J.M. Barrett, J.P. Callaghan, A Mechanistic Damage Model for Ligaments, Journal of Biomechanics (2017), doi: http://dx.doi.org/10.1016/j.jbiomech.2017.06.039
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A Mechanistic Damage Model for Ligaments
A Mechanistic Damage Model for Ligaments Barrett, Jeff M. University of Waterloo 200 University Avenue, Waterloo, Ontario N2L 3G1 [email protected] Callaghan, Jack P. 1 University of Waterloo 200 University Avenue, Waterloo, Ontario N2L 3G1 [email protected] ASME Membership Number: 101987826
Word Count: 3572
1
Corresponding author information can be added as a footnote.
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A Mechanistic Damage Model for Ligaments
Abstract Purpose: The accuracy of biomechanical models is predicated on the realism by which they represent their biomechanical tissues. Unfortunately, most models use phenomenological ligament models that neglect the behaviour in the failure region. Therefore, the purpose of this investigation was to test whether a mechanistic model of ligamentous tissue portrays behaviour representative of actual ligament failure tests. Model: The model tracks the time-evolution of a population of collagen fibres in a theoretical ligament. Each collagen fibre is treated as an independent linear cables with constant stiffness. Model equations were derived by assuming these fibres act as a continuum and applying a conservation law akin to Huxley’s muscle model. A breaking function models the rate of collagen fibre breakage at a given displacement, and was chosen to be a linear function for this preliminary analysis. Methods: The model was fitted to experimental average curves for the cervical anterior longitudinal ligament. In addition, the model was cyclically loaded to test whether the tissue model behaves similarly Results: The model agreed very well with experiment with an RMS error of 14.23 N and an R 2 of 0.995. Cyclic loading exhibited a reduction in force similar to experimental data. Discussion and Conclusion: The proposed model showcases behaviour reminiscent of actual ligaments being strained to failure and undergoing cyclic load. Future work could incorporate viscous effects, or validate the model further by testing it in various loading conditions. Characterizing the breaking function more accurately would also lead to better results.
Keywords: Tissue model, ligament, damage, failure, toe-region, mechanistic model
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A Mechanistic Damage Model for Ligaments
1.0 Introduction
The currently accepted notion of tissue injury pins an applied load against the tissue’s tolerance: when an applied load exceeds the tissue tolerance, there is a corresponding injury (Kumar, 2001). The situation is complicated by the weakening of tissues with repeated loading attributed to micro-trauma within the tissue. These situations represent events where the overall applied load is below the tissue’s ultimate tolerance, but are enough to impart some damage. As the tissue’s tolerance decreases, a subsequent load may cause an injury at a smaller applied load. To complicate matters further, with a large enough resting period tissues can recover or even exceed the original tissue tolerance. Despite the welltested nature of this theory (c.f. Brinckmann et al., 1987; Callaghan and McGill, 2001; Gooyers and Callaghan, 2015; Hansson et al., 1987), there has yet to be a rigorous quantitative description of it. Ligaments exhibit well-known viscoelastic effects: namely creep, stress-relaxation and hysteresis. In addition, their force-deflection relationship is highly nonlinear, featuring a prominent toe region often described mathematically using exponential springs (Lucas et al., 2009; Troyer and Puttlitz, 2011), and a complicated failure region which has, until recently, been largely ignored (DeWit and Cronin, 2012). Mechanistically, the toe region arises from the uncrimping of collagen and elastin fibres in the ligament as it is stretched, whereas the failure region results from the progressive failure of these fibres (Frisén et al., 1969a, 1969b). The current phenomenological approach treats each ligament as a single constituent responsible for exhibiting all of these behaviours rather than a population of fibres, greatly abstracting the model from the underlying mechanism. Therefore, the principle aim of this work was to present a mechanistic model detailing the behaviour of ligaments that is consistent with the currently accepted rationales for both the toe and failure regions of the force-deflection curve.
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A Mechanistic Damage Model for Ligaments
2.0 Model Derivation
The proposed model is derived from analyzing the time-evolution of a population of collagen fibres. Each fibre has its own displacement,
, from its neutral position (Figure 1) such that a positive
value
indicates that the fibre is being stretched, and when negative the fibre is crimped. The entire population of fibres is described as a distribution in their displacement, denoted , for small a magnitude of
. For example, the product
, can be thought of as roughly the number of fibres in the population displaced by
from their neutral position. [Figure 1 Here]
The goal of the derivation is to obtain a differential equation describing how
will evolve
over time. To do so, consider the rate of change of the total fibres stretched between a length of
and ,
such that this interval always contains the same fibres. Rather than consider this interval as stationary, as is classically done in deriving an advection equation, it will change as the ligament deforms so that it always contains the same fibres. It makes no difference on the end result, but the latter case is simpler to conceptualize and does not require considering the flux behaviour at the boundaries of the interval. The rate of change of fibres in this interval will be the difference between the creation of new fibres and the destruction of those stretched to failure. For simplicity, the creation of new fibres is ignored in this derivation since it occurs over a much longer timescale than breakage, although it would provide an avenue for the exploration of ligament healing and adaptation. The rate of breaking of fibres is governed by a “breaking function,”
, which describes the rate at which fibres will break if they
are stretched by an amount . Written mathematically as a global conservation expression:
(1)
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A Mechanistic Damage Model for Ligaments
The left hand side of Equation 1 can be simplified by using the Leibniz Rule for differentiation under the integral sign, assuming that all of the fibres in the ligament deform at the same rate, or that , and invoking the Fundamental Theorem of Calculus. Doing so homogenizes the intervals of all integrals in expression:
(2a)
(2b)
The final step in the derivation formally uses the Fundamental Lemma of the Calculus of Variations to infer that the integrand of these integrals is identically zero. This yields the linear-advection equation:
(3)
Thus far, this derivation has made no assumptions on how the fibres generate force. However, if they are assumed to be independent with force-deflection behaviour can be obtained from
, then the force in the ligament
through the integral:
(4) For simplicity, we take
, where
is the stiffness of each individual collagen fibre
multiplied by the cross-sectional area of the ligament, the “effective collagen stiffness”. This is justified as many investigations into the force-deflection behaviour of individual collagen fibrils have observed a linear relationship (Eppell et al., 2006; Kato et al., 1989; Sasaki and Odajima, 1996; Shen et al., 2008; Van Der Rijt et al., 2006). The integral term in this equation is similar to the moments defined by Ma and
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A Mechanistic Damage Model for Ligaments
Zahalak (1989) with the exception that this integral is taken over the positive real numbers as opposed to the entire real line.
2.1 A Simplified Case with an Analytic Solution
If the breaking function is taken to be a linear function of the collagen distribution, with a proportionality constant related to the amount a fibre has been stretched (i.e.
),
then the analytic solution for the model can be obtained by using the Method of Characteristics:
(5a)
Where
is the initial distribution of collagen displacements at the initial time, and
displacement as a function of time (note that this solution has assumed that
is the
when
).
Immediately, it becomes apparent that the analytic solution is the product of two functions:
(5b)
In Equation 5b, the standard solution term is the solution one would get if damage were ignored. Conversely, the cumulative damage term is a discounting term which effectively counts the number of collagen fibres that have broken. For this preliminary work, we considered a relatively simple breaking function, taking the form of a scaled Heaviside-step function:
(6)
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A Mechanistic Damage Model for Ligaments
Intuitively, the solution to equation to Equation 3 translates the initial distribution of collagen fibres across the -axis with velocity
. If any section of the collagen distribution overlaps with where
the breaking function is non-zero, then the distribution decays in a first-order fashion. Additionally, the form of Equation 5 reveals a more mathematically stringent way of describing the damage to ligamentous tissue. It is the product of two terms: the translated initial distribution,
, the solution one
would get from ignoring the breaking collagen fibres; and a second, more complicated term involving an integral of the breaking function. This second term is effectively counting the number of collagen fibres that have broken and decrementing the initial distribution accordingly.
3.0 Methods
3.1 Obtaining Model Parameters An analytic solution (Equation 5a) and corresponding force (Equation 4) were fit to average force-deflection data for the Anterior-Longitudinal Ligament (ALL) of the cervical spine obtained and reported in Mattucci (2011), including the failure region. These average curves were obtained by loading cadaveric specimens (
) to failure while measuring elongation and force with strain gauges and
linear variable displacement transformers (LVDT) (Omegadyne Inc. Model LC412-500), at a low strainrate (
) and averaged across all specimens (Mattucci, 2011). The scipy package for Python (van der Walt et al., 2011) was used to carry-out the non-linear
least squares procedure, using the minimize function. Integrals in Equations 4 and 5 were evaluated using trapezoidal integration, with the -axis discretized into 1024-cells over a range of -5 to 10 mm. Average force-deflection curves from Mattucci (2011) were up-sampled to have 128 evenly-spaced data-points using 3rd-order spline interpolation; this was done to make the time-integration of Equation 5 more accurate. The initial distribution of collagen fibres was assumed to obey a normal-distribution with mean
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A Mechanistic Damage Model for Ligaments
and standard deviation
(Equation 7). The model itself included 5 free-parameters:
and
.
(7) Here,
represents the total number of collagen fibres in the ligament. The value of
into the effective collagen stiffness, , since in calculating force (Equation 4),
is absorbed
become redundant.
3.2 Cyclic Loading and Tissue Tolerance The response of the model to cyclic loading was also tested using the parameters obtained from the fitting procedure. The model was cyclically loaded to a displacement of 3 mm at 0.5 Hz in order to test if micro-trauma to the ligament model reflect of true ligament damage. A unit-less tissue tolerance (capacity) was defined as the ratio of total remaining collagen fibres over the initial total (Equation 8a, due to the absorption of
, the denominator is set to 1), while an approximation of the true tissue
tolerance was made by multiplying the current capacity by the maximum force generated in the failure trial (Equation 8b).
(8a) (8b) Where
is the maximum force measured in the quasi-static failure test done by Mattucci
(2011) for the ALL of the cervical spine: 342.0 N.
4.0 Results 4.1 Fitted Parameters The fitted model agreed with experiment very well (Figure 1; Parameters in Table), with a total RMS error of only 14.23 N (1.67 N in the toe and linear regions, and 15.70 N in the failure region), and a
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A Mechanistic Damage Model for Ligaments
Pearson-Product Moment Correlation Coefficient of 0.994 (0.999 in the toe and linear regions, and 0.995 in the failure region). In general, the model fit the toe and linear regions of the force-deflection curve almost exactly, and captured the failure region reasonably well, but exhibited a decay in force too quickly. [Table 1 Here] [Figure 2 Here]
4.2 Cyclic Loading and Tissue Tolerance Qualitatively, the model’s response to cyclic loading is also reminiscent of sub-failure cyclic loading. With more cycles, the ligament fails to produce an equivalent force to the previous cycle (Figure 3). Additionally, as these micro-traumas accumulate, there is a notable softening of the ligament (Figure 4). As expected, the cumulative damage inflicted on the ligament manifests itself with a decreased tissue tolerance with every cycle (Figure 5). [Figure 3 Here] [Figure 4 Here] [Figure 5 Here]
5.0 Discussion Biomechanical models have been extraordinarily useful in exploring injury pathways in both acute and chronic scenarios (Dickerson et al., 2007; Marras and Granata, 1997; McGill and Norman, 1986), in predicting results from surgical interventions (Ahn, 2005; Delp et al., 1990; Kumaresan et al., 1999), and, with the advent of more widely available software, are becoming more commonplace in ergonomic assessments and designing interventions (Duffy et al., 2007). Yet their success is predicated on the realism of their representation of the mechanical behaviour of biological tissues, most commonly taking the form of a phenomenological model representing the experimental force-deflection curve (Fung,
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A Mechanistic Damage Model for Ligaments
1967; Lucas et al., 2006; Troyer and Puttlitz, 2010). Such models range from representing biological tissues as simple linear-springs to more complicated viscoelastic rheological models which are still undergoing improvement (Haut and Little, 1972; Jamison et al., 1968; Troyer et al., 2012), but in many cases neglect the prevailing mechanisms which govern the shape of the force-deflection curve. The proposed model generalizes the pioneering work of Viidik (1968), who showed that a discrete population of spring elements could be used to give a piecewise linear approximation to the toe region of ligaments. Later, Liao and Belkoff (1999) showed that a similar formulation could be used to model the failure of ligaments by allowing each fibre to fail as a brittle material at a set failure strain, while also showing that treating the collagen fibres as a continuum improved the resolution of the toeregion. The model equations presented here are a natural extension of Liao and Belkoff (1999), and are obtained by modelling the time-evolution of the distribution rather than each fibre individually, analogous to Huxley’s famous muscle model (Huxley, 1957). The model predicts diminished force-producing capacity with repeated submaximal loads, which is consistent with experimental results (Möller et al., 1992; Solomonow et al., 2001; Woo et al., 1981). Similarly, the model predicts a considerable amount of hysteresis, which is also consistent with experimental data (Weisman et al., 1979). These phenomena have previously been associated with ligament viscoelasticity, a component of ligament force-production completely ignored in this formulation. This suggests that the stress-relaxation and hysteresis phenomena are partly attributable to tissue damage in addition to viscoelastic responses. This prediction is particularly noteworthy from an injury prevention standpoint, where repeated micro-trauma over a large period of time is hypothesized to be the culprit behind chronic injuries (Kumar, 2001; Norman et al., 1998). Therefore, a natural application of this plastic ligament model would be in characterizing the risk of injury from cumulative loading, something no previous model has been able to do. Once viscoelastic components are incorporated into this model, it would be possible to separate apparent viscoelastic components from actual tissue damage.
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A Mechanistic Damage Model for Ligaments
5.1 Relationship to Huxley’s Muscle Model The advection equation in Equation 3 is reminiscent of the underlying equation in Huxley’s muscle model. Indeed, with the inclusion of a healing term, the two models become indistinguishable. The realization that the underlying equations for a population of collagen fibres are the same as those for a population of myosin-actin cross-bridges is interesting for philosophical reasons, and brings with it some level of unification in the study of soft tissue. Seeing as the same derivation could be carried out for tendon tissue, annular layers of the intervertebral disc, or smooth muscular tissue, it may be worthwhile to consider Equation 3 as a Soft Tissue Law which describes how a population of soft-tissue constituents will evolve over time. Indeed, it follows almost immediately from applying a conservation law to the distribution of tissue constituents, an idea which may extend beyond the study of ligaments: to tendons or the passive component of muscle tissues.
6.2 The Indestructible Ligament Simplification In some applications it may be detrimental for a model to include ligamentous failure; namely models which use inverse dynamics in scenarios where ligaments are not expected to fail. In this case, the assumption that fibres do not break, which we call the Indestructible Ligament Simplification (ILS), makes Equation 4 tractable and an explicit force-deflection relationship can be achieved. For instance, in assuming that the initial collagen fibre population obeys a normal distribution, the ILS force-deflection function takes the form of Equation 9.
(9)
Where
is the effective stiffness of the collagen fibres,
deviation of the initial population of collagen fibres,
and
are the mean and standard
is the current elongation of the ligament, and
is the error-function. This expression models the toe and linear regions almost exactly, and
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A Mechanistic Damage Model for Ligaments
represents an improvement over the exponential springs, power laws, or piecewise linear approximations used in conventional ligament models (Chandrashekar et al., 2008; Fung, 1981; Provenzano et al., 2001; Troyer and Puttlitz, 2010). Though exponential springs and power laws provocatively capture the toe region, they fail to reproduce the linear region of a ligament’s force-deflection curve and are very sensitive to choices in the “slack length” of the ligament. A piecewise linear response, like that of Chandreshekar et al. (2008), is closer to the true force-deflection behaviour of the ligament, but suffers from its simplicity and poor representation in the toe region. Another approach has been to use a quadratic equation in the toe region and a linear function in the linear region (Li et al., 1999). The ILS is a compromise in terms of complexity while also being derived in a way that respects the established uncrimping mechanism.
6.3 The Breaking Function The breaking function used in this preliminary analysis is almost certainly oversimplified and future work should aim to describe it more completely. Admittedly, the Heaviside breaking function seems to cause the ligament to fail too quickly, and this problem is accentuated when ramp or stepwise quadratic functions were employed. Additionally, these functions assume independence on the rate of deformation, which is a strong assumption to make given the well-documented strain-rate dependent behaviour of biological tissues (Bass et al., 2007; Lucas et al., 2008; Mattucci et al., 2013). To complicate matters further, the first-order response from the breaking function, with parameters derived from one strain-rate, cause the ligament to fail too slowly at higher strain rates. The true breaking function may be a more complicated nonlinear function, requiring extensive data to define. Nevertheless, this novel formulation of ligament mechanics opens up a new avenue for future investigations to explore and quantify damage to soft tissues, a component of tissue mechanics not previously considered in other soft tissue models.
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A Mechanistic Damage Model for Ligaments
A consequence of including the breaking function in this model formulation is that it gives the proposed ligament model a plastic behaviour, which until now, has been a tremendously difficult challenge in the modelling of biological tissues.
6.5 Limitations and Future Directions Viscoelasticity is currently absent from this model description, subjecting the results presented here to an implicit quasi-static assumption. Thus far, attempts to add viscoelasticity by replacing the springs in the description with Voigt models (spring and damper in parallel) have not been promising. An arrangement like that of Viidik (1973) may be useful in future work, reconciling this mechanistic approach with a more traditional phenomenological one. The current model is one-dimensional in nature, although it can be generalized to threedimensions if needed using some methods from continuum mechanics in a way identical to what has been done with the Huxley muscle model (Oomens et al., 2003), or other collagenous tissues (Lanir, 1979). One limitation is that the model currently has no input from experiment on the fibre level, and Future research should endeavour to reconcile the effective collagen stiffness with ligamentous cross-sectional area, packing density of collagen fibres, and stiffness data from individual collagen fibres. In addition, some assumptions in the model derivation need to be addressed. For one, the assumption that each collagen fibre generates force independently neglects the cross-linking of collagen fibres. Secondly, it was assumed that each fibre produces force linearly with displacement: an assumption that has some support from the literature (Kato et al., 1989; Sasaki and Odajima, 1996). However, there are other studies which have advocated using a wormlike chain as a more accurate representation for the force production in individual collagen fibres (Graham et al., 2004; Shah et al., 1977; Sun et al., 2002). Using a wormlike chain may provide a more complete picture of ligament force-displacement properties, however, would have precluded any analytic discussion of the model. Lastly, the assumption that
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A Mechanistic Damage Model for Ligaments
ligament damage is attributable to fibre loss is inherent in the model formulation, however this may not be the only failure mode for ligaments: cleavage of cross-linkages between collagen fibres or the rearrangement of matrix constituents may also be considered damage (Bailey et al., 1974; Fratzl et al., 1998). These aspects of ligament behaviour are currently not included in the model. The computational complexity of solving a partial differential equation may also be a significant barrier to widespread use of the proposed framework. A distribution moment approach, like that of Ma and Zahalak (1988), is appealing as it would reduce the partial differential equation into a system of four coupled ordinary differential equations. Unfortunately, the formulation presented there, if applied to these equations, gives ligaments the ability to resist compressive loads as well as tensile ones. Such nonphysiological predictions precludes its application to this ligament model. Adapting the distribution moment approach successfully to the study of ligaments is contingent on the identification of an experimentally justified breaking function The model presented was developed for the cervical ALL; however the general framework extends well beyond the study of this ligament by adjusting the parameters , , ,
and
. The same
procedure, simple least-squares fitting, can be applied to solve for these variables for any extrinsic ligament.
6.0 Conclusions The proposed model of ligament force-deflection behaviour showcased responses reflective of true ligaments being strained to failure; featuring toe and failure regions that are consistent with the currently accepted mechanisms that govern their shape. Additionally, the predicted tissue-tolerance and applied load dynamics produced responses that are qualitatively consistent with the conceptual repetitive injury model. This detailed model of a ligament provides a rigorous definition for tissue tolerance and provides equations for the modelling of injury from repetitive trauma. When damage is ignored, the resulting model provides an analytic representation of the toe and linear regions which may be more
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A Mechanistic Damage Model for Ligaments
appropriate in larger-scale biomechanical models where ligament failure in not a desired component. In addition, the model exhibits tissue softening with repeated loading that is similar to actual ligaments. The softening of tissues is accompanied by an expected decrease in the tolerance of the tissue and therefore an increased susceptibility to injury. Lastly, the general model equations lend themselves well to formalizing the study of failure in ligaments through the characterization of the breaking function. The proposed model represents a step toward a more rigorous treatment of the theoretical side of injury prediction, and lends itself well to making quantitative predictions that can either be falsified, or used in practice to assess the risk of injury of an activity.
7.0 Acknowledgements Jeff M. Barrett is supported by the National Science and Engineering Research Council of Canada (NSERC) through a PGS-D scholarship. Jack P. Callaghan is a Tier 1 Canada Research Chair in Spine Biomechanics and Injury Prevention.
8.0 Conflicts of Interest None to declare.
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Delp, S.L., Loan, J.P., Hoy, M.G., Zajac, F.E., 1990. An interactive Graphics-Based Model of the Lower Extremity to Study Orthopaedic Surgical Procedures. IEEE Trans. Biomed. Eng. 37, 757–767. doi:10.1109/10.102791 DeWit, J. a., Cronin, D.S., 2012. Cervical spine segment finite element model for traumatic injury prediction. J. Mech. Behav. Biomed. Mater. 10, 138–150. doi:10.1016/j.jmbbm.2012.02.015 Dickerson, C.R., Chaffin, D.B., Hughes, R.E., 2007. A mathematical musculoskeletal shoulder model for proactive ergonomic analysis. Comput. Methods Biomech. Biomed. Engin. 10, 389–400. doi:10.1080/10255840701592727 Duffy, V.G., Christy, J., Duffy, V.G., 2007. A methodology for assessing industrial workstations using optical motion capture integrated with digital human models A methodology for assessing industrial workstations using optical motion capture integrated with digital human models. Occup. Ergon. 11– 25. Eppell, S.., Smith, B.., Kahn, H., Ballarini, R., 2006. Nano measurements with micro-devices: mechanical properties of hydrated collagen fibrils. J. R. Soc. Interface 3, 117–121. doi:10.1098/rsif.2005.0100 Fratzl, P., Misof, K., Zizak, I., Rapp, G., Amenitsch, H., Bernstorff, S., 1998. Fibrillar Structure and Mechanical Properties of Collagen. J. Struct. Biol. 122, 119–122. doi:10.1006/jsbi.1998.3966 Frisén, M., Mägi, M., Sonnerup, I., Viidik, a, 1969a. Rheological analysis of soft collagenous tissue. Part I: theoretical considerations. J. Biomech. 2, 13–20. Frisén, M., Mägi, M., Sonnerup, L., Viidik, A., 1969b. Rheological analysis of soft collagenous tissue Part II: Experimental Evaluations and Verification. J. Biomech. 2, 21–28. doi:10.1016/00219290(69)90038-4 Fung, Y.C., 1981. Biomechanics: Mechanical Properties of Living Tissues, 2nd ed., Second. ed. Springer Science and Business Media. Fung, Y.C., 1967. Elasticity of soft tissues in simple elongation. Am. J. Physiol. 213, 1532–1544. Gooyers, C.E., Callaghan, J.P., 2015. Exploring interactions between force , repetition and posture on intervertebral disc height loss and bulging in isolated porcine cervical functional spinal units from sub-acute-failure magnitudes of cyclic compressive loading 1–8. Graham, J.S., Vomund, A.N., Phillips, C.L., Grandbois, M., 2004. Structural changes in human type I collagen fibrils investigated by force spectroscopy. Exp. Cell Res. 299, 335–342. doi:10.1016/j.yexcr.2004.05.022 Hansson, T.H., Keller, T.S., Spengler, D.M., 1987. Mechanical behavior of the human lumbar spine. II. Fatigue strength during dynamic compressive loading. J. Orthop. Res. 5, 479–487. doi:10.1002/jor.1100050403 Haut, R.C., Little, R.W., 1972. A constitutive equation for collagen fibers. J. Biomech. 5, 423–430. doi:10.1016/0021-9290(72)90001-2 Huxley, A.F., 1957. Muscle structure and theories of contraction. Prog Biophys Biophys Chem 7, 255– 318. Jamison, C.., Marangoni, R.., Glaser, A.., 1968. Viscoelastic properties of soft tissue by discrete model characterization. J. Biomech. 1, 33–46. doi:10.1016/0021-9290(68)90036-5 Kato, Y.P., Christiansen, D.L., Hahn, R.A., Shieh, S.J., Goldstein, J.D., Silver, F.H., 1989. Mechanical properties of collagen fibres: a comparison of reconstituted and rat tail tendon fibres. Biomaterials 10, 38–42. doi:10.1016/0142-9612(89)90007-0
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A Mechanistic Damage Model for Ligaments
Kumar, S., 2001. Theories of doi:10.1080/00140130120716
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Kumaresan, S., Yoganandan, N., Pintar, F. a., 1999. Finite element analysis of the cervical spine: a material property sensitivity study. Clin. Biomech. 14, 41–53. doi:10.1016/S0268-0033(98)00036-9 Lanir, Y., 1979. A structural theory for the homogeneous biaxial stress-strain relationships in flat collagenous tissues. J. Biomech. 12, 423–436. doi:10.1016/0021-9290(79)90027-7 Li, G., Gil, J., Kanamori, A., Woo, S.L.-Y., 1999. A Validated Three-Dimensional Computational Model of a Human Knee Joint. J. Biomech. Eng. 121, 657. doi:10.1115/1.2800871 Liao, H., Belkoff, S.M., 1999a. A failure model for ligaments. J. Biomech. 32, 183–188. doi:10.1016/S0021-9290(98)00169-9 Liao, H., Belkoff, S.M., 1999b. A failure model for ligaments. J. Biomech. 32, 183–188. doi:10.1016/S0021-9290(98)00169-9 Lucas, S., Bass, C., Salzar, R., Shender, B.S., Paskoff, G., 2006. High-rate viscoelastic properties of human cervical spinal intervertebral discs, in: Annual Meeting of the American Society of Biomechanics. American Society of Biomechanics. Lucas, S.R., Bass, C.R., Crandall, J.R., Kent, R.W., Shen, F.H., Salzar, R.S., 2009. Viscoelastic and failure properties of spine ligament collagen fascicles. Biomech. Model. Mechanobiol. 8, 487–498. doi:10.1007/s10237-009-0152-7 Lucas, S.R., Bass, C.R., Salzar, R.S., Oyen, M.L., Planchak, C., Ziemba, A., Shender, B.S., Paskoff, G., 2008. Viscoelastic properties of the cervical spinal ligaments under fast strain-rate deformations. Acta Biomater. 4, 117–125. doi:10.1016/j.actbio.2007.08.003 Ma, S., Zahalak, G.I., 1988. Activation dynamics for a distribution-moment model of skeletal muscle. Math. Comput. Model. 11, 778–782. doi:10.1016/0895-7177(88)90599-7 Marras, W.S., Granata, K.P., 1997. The development of an EMG-assisted model to assess spine loading during whole-body free-dynamic lifting. J. Electromyogr. Kinesiol. 7, 259–268. doi:10.1016/S10506411(97)00006-0 Mattucci, S.F.E., 2011. Strain rate dependent properties of younger human cervical spine ligaments. University of Waterloo. Mattucci, S.F.E., Moulton, J. a., Chandrashekar, N., Cronin, D.S., 2013. Strain rate dependent properties of human craniovertebral ligaments. J. Mech. Behav. Biomed. Mater. 23, 71–79. doi:10.1016/j.jmbbm.2013.04.005 McGill, S.M., Norman, R.W., 1986. Partitioning of the L4-L5 dynamic moment into disc, ligamentous, and muscular components during lifting. Spine (Phila. Pa. 1976). 11, 666–78. Möller, J., Nolte, L.P., Visarius, H., Willburger, R., Crisco, J.J., Panjabi, M.M., 1992. Viscoelasticity of the alar and transverse ligaments. Eur. Spine J. 1, 178–184. doi:10.1016/0021-9290(92)90580-T Norman, R., Wells, R., Neumann, P., Frank, J., Shannon, H., Kerr, M., Beaton, D.E., Bombardier, C., Ferrier, S., Hogg-Johnson, S., Mondloch, M., Peloso, P., Smith, J., Stansfeld, S. a., Tarasuk, V., Andrews, D.M., Dobbyn, M., Edmonstone, M. a., Ingelman, J.P., Jeans, B., McRobbie, H., Moore, a., Mylett, J., Outerbridge, G., Woo, H., 1998. A comparison of peak vs cumulative physical work exposure risk factors for the reporting of low back pain in the automotive industry. Clin. Biomech. 13, 561–573. doi:10.1016/S0268-0033(98)00020-5 Oomens, C., Maenhout, M., van Oijen, C., Drost, M., Baaijens, F., 2003. Finite element modelling of
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Provenzano, P., Lakes, R., Keenan, T., Vanderby, Jr., R., 2001. Nonlinear Ligament Viscoelasticity. Ann. Biomed. Eng. 29, 908–914. doi:10.1114/1.1408926 Sasaki, N., Odajima, S., 1996. Stress-strain curve and Young’s modulus of a collagen molecule as determined by the X-ray diffraction technique. J. Biomech. 29, 655–658. doi:10.1016/00219290(95)00110-7 Shah, J.S., Jayson, M.I., Hampson, W.G., 1977. Low tension studies of collagen fibres from ligaments of the human spine. Ann. Rheum. Dis. 36, 139–45. doi:10.1136/ard.36.2.139 Shen, Z.L., Dodge, M.R., Kahn, H., Ballarini, R., Eppell, S.J., 2008. Stress-Strain Experiments on Individual Collagen Fibrils. Biophys. J. 95, 3956–3963. doi:10.1529/biophysj.107.124602 Solomonow, M., Eversull, E., Zhou, B.H., Baratta, R. V, 2001. Neuromuscular Neutral Zones Associated With Viscoelastic Hysteresis During Cyclic Lumbar Flexion 26, 314–324. Sun, Y.L., Luo, Z.P., Fertala, A., An, K.N., 2002. Direct quantification of the flexibility of type I collagen monomer. Biochem. Biophys. Res. Commun. 295, 382–386. doi:10.1016/S0006-291X(02)00685-X Troyer, K.L., Puttlitz, C.M., 2011. Human cervical spine ligaments exhibit fully nonlinear viscoelastic behavior. Acta Biomater. 7, 700–709. doi:10.1016/j.actbio.2010.09.003 Troyer, K.L., Puttlitz, C.M., 2010. Quasi-Linear Viscoelastic Theory is Insufficient to Comprehensively Describe the Time-Dependent Behavior of Human Cervical Spine Ligaments, in: ASME 2010 Summer Bioengineering Conference, Parts A and B. ASME, p. 705. doi:10.1115/SBC2010-19057 Troyer, K.L., Puttlitz, C.M., Shetye, S.S., 2012. Experimental Characterization and Finite Element Implementation of Soft Tissue Nonlinear Viscoelasticity. J. Biomech. Eng. 134, 114501. doi:10.1115/1.4007630 Van Der Rijt, J.A.J., Van Der Werf, K.O., Bennink, M.L., Dijkstra, P.J., Feijen, J., 2006. Micromechanical testing of individual collagen fibrils. Macromol. Biosci. 6, 699–702. doi:10.1002/mabi.200600063 van der Walt, S., Colbert, S.C., Varoquaux, G., 2011. The NumPy Array: A Structure for Efficient Numerical Computation. Comput. Sci. Eng. 13, 22–30. doi:10.1109/MCSE.2011.37 Viidik, A., 1973. Functional Properties of Collagenous Tissues. pp. 127–215. doi:10.1016/B978-0-12363706-2.50010-6 Viidik, a., 1968. A Rheological Model for Uncalcified Parallel-Fibred Collagenous Tissue. J. Biomech. 1, 3–11. doi:10.1016/0021-9290(68)90032-8 Weisman, G., Pope, M.H., Ph, D., Robert, J., 1979. Cyclic loading in knee ligament injuries *. doi:10.1177/036354658000800105 Woo, S.L., Gomez, M. a, Akeson, W.H., 1981. The time and history-dependent viscoelastic properties of the canine medical collateral ligament. J. Biomech. Eng. 103, 293–298.
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A Mechanistic Damage Model for Ligaments
Table 1: Fitted Parameters derived from experimental data for the Anterior Longitudinal Ligament.
Figure 1: Collagen fibres as an arrangement of parallel collagen fibres, each with its own slacklength denoted .
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A Mechanistic Damage Model for Ligaments
Figure 2: Model force predictions compared to the quasi-static curves reported by Mattucci (2011) for the Anterior Longitudinal Ligament.
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A Mechanistic Damage Model for Ligaments
Figure 3: (Top) The displacement waveform driving the simulation; (middle) the resulting force waveform predicted from the model; (bottom) the corresponding tissue capacity.
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A Mechanistic Damage Model for Ligaments
Figure 4: Response of the model's force-deflection behaviour. Here, the softening with repeated loading, as well as the plastic behaviour, are exemplified.
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A Mechanistic Damage Model for Ligaments
Figure 5: Response of the ligament's approximate tissue tolerance to cyclic loading at a controlled displacement.
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S0021-9290(17)30349-4 http://dx.doi.org/10.1016/j.jbiomech.2017.06.039 BM 8278
To appear in:
Journal of Biomechanics
Received Date: Revised Date: Accepted Date:
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Please cite this article as: J.M. Barrett, J.P. Callaghan, A Mechanistic Damage Model for Ligaments, Journal of Biomechanics (2017), doi: http://dx.doi.org/10.1016/j.jbiomech.2017.06.039
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A Mechanistic Damage Model for Ligaments
A Mechanistic Damage Model for Ligaments Barrett, Jeff M. University of Waterloo 200 University Avenue, Waterloo, Ontario N2L 3G1 [email protected] Callaghan, Jack P. 1 University of Waterloo 200 University Avenue, Waterloo, Ontario N2L 3G1 [email protected] ASME Membership Number: 101987826
Word Count: 3572
1
Corresponding author information can be added as a footnote.
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A Mechanistic Damage Model for Ligaments
Abstract Purpose: The accuracy of biomechanical models is predicated on the realism by which they represent their biomechanical tissues. Unfortunately, most models use phenomenological ligament models that neglect the behaviour in the failure region. Therefore, the purpose of this investigation was to test whether a mechanistic model of ligamentous tissue portrays behaviour representative of actual ligament failure tests. Model: The model tracks the time-evolution of a population of collagen fibres in a theoretical ligament. Each collagen fibre is treated as an independent linear cables with constant stiffness. Model equations were derived by assuming these fibres act as a continuum and applying a conservation law akin to Huxley’s muscle model. A breaking function models the rate of collagen fibre breakage at a given displacement, and was chosen to be a linear function for this preliminary analysis. Methods: The model was fitted to experimental average curves for the cervical anterior longitudinal ligament. In addition, the model was cyclically loaded to test whether the tissue model behaves similarly Results: The model agreed very well with experiment with an RMS error of 14.23 N and an R 2 of 0.995. Cyclic loading exhibited a reduction in force similar to experimental data. Discussion and Conclusion: The proposed model showcases behaviour reminiscent of actual ligaments being strained to failure and undergoing cyclic load. Future work could incorporate viscous effects, or validate the model further by testing it in various loading conditions. Characterizing the breaking function more accurately would also lead to better results.
Keywords: Tissue model, ligament, damage, failure, toe-region, mechanistic model
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A Mechanistic Damage Model for Ligaments
1.0 Introduction
The currently accepted notion of tissue injury pins an applied load against the tissue’s tolerance: when an applied load exceeds the tissue tolerance, there is a corresponding injury (Kumar, 2001). The situation is complicated by the weakening of tissues with repeated loading attributed to micro-trauma within the tissue. These situations represent events where the overall applied load is below the tissue’s ultimate tolerance, but are enough to impart some damage. As the tissue’s tolerance decreases, a subsequent load may cause an injury at a smaller applied load. To complicate matters further, with a large enough resting period tissues can recover or even exceed the original tissue tolerance. Despite the welltested nature of this theory (c.f. Brinckmann et al., 1987; Callaghan and McGill, 2001; Gooyers and Callaghan, 2015; Hansson et al., 1987), there has yet to be a rigorous quantitative description of it. Ligaments exhibit well-known viscoelastic effects: namely creep, stress-relaxation and hysteresis. In addition, their force-deflection relationship is highly nonlinear, featuring a prominent toe region often described mathematically using exponential springs (Lucas et al., 2009; Troyer and Puttlitz, 2011), and a complicated failure region which has, until recently, been largely ignored (DeWit and Cronin, 2012). Mechanistically, the toe region arises from the uncrimping of collagen and elastin fibres in the ligament as it is stretched, whereas the failure region results from the progressive failure of these fibres (Frisén et al., 1969a, 1969b). The current phenomenological approach treats each ligament as a single constituent responsible for exhibiting all of these behaviours rather than a population of fibres, greatly abstracting the model from the underlying mechanism. Therefore, the principle aim of this work was to present a mechanistic model detailing the behaviour of ligaments that is consistent with the currently accepted rationales for both the toe and failure regions of the force-deflection curve.
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A Mechanistic Damage Model for Ligaments
2.0 Model Derivation
The proposed model is derived from analyzing the time-evolution of a population of collagen fibres. Each fibre has its own displacement,
, from its neutral position (Figure 1) such that a positive
value
indicates that the fibre is being stretched, and when negative the fibre is crimped. The entire population of fibres is described as a distribution in their displacement, denoted , for small a magnitude of
. For example, the product
, can be thought of as roughly the number of fibres in the population displaced by
from their neutral position. [Figure 1 Here]
The goal of the derivation is to obtain a differential equation describing how
will evolve
over time. To do so, consider the rate of change of the total fibres stretched between a length of
and ,
such that this interval always contains the same fibres. Rather than consider this interval as stationary, as is classically done in deriving an advection equation, it will change as the ligament deforms so that it always contains the same fibres. It makes no difference on the end result, but the latter case is simpler to conceptualize and does not require considering the flux behaviour at the boundaries of the interval. The rate of change of fibres in this interval will be the difference between the creation of new fibres and the destruction of those stretched to failure. For simplicity, the creation of new fibres is ignored in this derivation since it occurs over a much longer timescale than breakage, although it would provide an avenue for the exploration of ligament healing and adaptation. The rate of breaking of fibres is governed by a “breaking function,”
, which describes the rate at which fibres will break if they
are stretched by an amount . Written mathematically as a global conservation expression:
(1)
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A Mechanistic Damage Model for Ligaments
The left hand side of Equation 1 can be simplified by using the Leibniz Rule for differentiation under the integral sign, assuming that all of the fibres in the ligament deform at the same rate, or that , and invoking the Fundamental Theorem of Calculus. Doing so homogenizes the intervals of all integrals in expression:
(2a)
(2b)
The final step in the derivation formally uses the Fundamental Lemma of the Calculus of Variations to infer that the integrand of these integrals is identically zero. This yields the linear-advection equation:
(3)
Thus far, this derivation has made no assumptions on how the fibres generate force. However, if they are assumed to be independent with force-deflection behaviour can be obtained from
, then the force in the ligament
through the integral:
(4) For simplicity, we take
, where
is the stiffness of each individual collagen fibre
multiplied by the cross-sectional area of the ligament, the “effective collagen stiffness”. This is justified as many investigations into the force-deflection behaviour of individual collagen fibrils have observed a linear relationship (Eppell et al., 2006; Kato et al., 1989; Sasaki and Odajima, 1996; Shen et al., 2008; Van Der Rijt et al., 2006). The integral term in this equation is similar to the moments defined by Ma and
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A Mechanistic Damage Model for Ligaments
Zahalak (1989) with the exception that this integral is taken over the positive real numbers as opposed to the entire real line.
2.1 A Simplified Case with an Analytic Solution
If the breaking function is taken to be a linear function of the collagen distribution, with a proportionality constant related to the amount a fibre has been stretched (i.e.
),
then the analytic solution for the model can be obtained by using the Method of Characteristics:
(5a)
Where
is the initial distribution of collagen displacements at the initial time, and
displacement as a function of time (note that this solution has assumed that
is the
when
).
Immediately, it becomes apparent that the analytic solution is the product of two functions:
(5b)
In Equation 5b, the standard solution term is the solution one would get if damage were ignored. Conversely, the cumulative damage term is a discounting term which effectively counts the number of collagen fibres that have broken. For this preliminary work, we considered a relatively simple breaking function, taking the form of a scaled Heaviside-step function:
(6)
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A Mechanistic Damage Model for Ligaments
Intuitively, the solution to equation to Equation 3 translates the initial distribution of collagen fibres across the -axis with velocity
. If any section of the collagen distribution overlaps with where
the breaking function is non-zero, then the distribution decays in a first-order fashion. Additionally, the form of Equation 5 reveals a more mathematically stringent way of describing the damage to ligamentous tissue. It is the product of two terms: the translated initial distribution,
, the solution one
would get from ignoring the breaking collagen fibres; and a second, more complicated term involving an integral of the breaking function. This second term is effectively counting the number of collagen fibres that have broken and decrementing the initial distribution accordingly.
3.0 Methods
3.1 Obtaining Model Parameters An analytic solution (Equation 5a) and corresponding force (Equation 4) were fit to average force-deflection data for the Anterior-Longitudinal Ligament (ALL) of the cervical spine obtained and reported in Mattucci (2011), including the failure region. These average curves were obtained by loading cadaveric specimens (
) to failure while measuring elongation and force with strain gauges and
linear variable displacement transformers (LVDT) (Omegadyne Inc. Model LC412-500), at a low strainrate (
) and averaged across all specimens (Mattucci, 2011). The scipy package for Python (van der Walt et al., 2011) was used to carry-out the non-linear
least squares procedure, using the minimize function. Integrals in Equations 4 and 5 were evaluated using trapezoidal integration, with the -axis discretized into 1024-cells over a range of -5 to 10 mm. Average force-deflection curves from Mattucci (2011) were up-sampled to have 128 evenly-spaced data-points using 3rd-order spline interpolation; this was done to make the time-integration of Equation 5 more accurate. The initial distribution of collagen fibres was assumed to obey a normal-distribution with mean
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A Mechanistic Damage Model for Ligaments
and standard deviation
(Equation 7). The model itself included 5 free-parameters:
and
.
(7) Here,
represents the total number of collagen fibres in the ligament. The value of
into the effective collagen stiffness, , since in calculating force (Equation 4),
is absorbed
become redundant.
3.2 Cyclic Loading and Tissue Tolerance The response of the model to cyclic loading was also tested using the parameters obtained from the fitting procedure. The model was cyclically loaded to a displacement of 3 mm at 0.5 Hz in order to test if micro-trauma to the ligament model reflect of true ligament damage. A unit-less tissue tolerance (capacity) was defined as the ratio of total remaining collagen fibres over the initial total (Equation 8a, due to the absorption of
, the denominator is set to 1), while an approximation of the true tissue
tolerance was made by multiplying the current capacity by the maximum force generated in the failure trial (Equation 8b).
(8a) (8b) Where
is the maximum force measured in the quasi-static failure test done by Mattucci
(2011) for the ALL of the cervical spine: 342.0 N.
4.0 Results 4.1 Fitted Parameters The fitted model agreed with experiment very well (Figure 1; Parameters in Table), with a total RMS error of only 14.23 N (1.67 N in the toe and linear regions, and 15.70 N in the failure region), and a
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A Mechanistic Damage Model for Ligaments
Pearson-Product Moment Correlation Coefficient of 0.994 (0.999 in the toe and linear regions, and 0.995 in the failure region). In general, the model fit the toe and linear regions of the force-deflection curve almost exactly, and captured the failure region reasonably well, but exhibited a decay in force too quickly. [Table 1 Here] [Figure 2 Here]
4.2 Cyclic Loading and Tissue Tolerance Qualitatively, the model’s response to cyclic loading is also reminiscent of sub-failure cyclic loading. With more cycles, the ligament fails to produce an equivalent force to the previous cycle (Figure 3). Additionally, as these micro-traumas accumulate, there is a notable softening of the ligament (Figure 4). As expected, the cumulative damage inflicted on the ligament manifests itself with a decreased tissue tolerance with every cycle (Figure 5). [Figure 3 Here] [Figure 4 Here] [Figure 5 Here]
5.0 Discussion Biomechanical models have been extraordinarily useful in exploring injury pathways in both acute and chronic scenarios (Dickerson et al., 2007; Marras and Granata, 1997; McGill and Norman, 1986), in predicting results from surgical interventions (Ahn, 2005; Delp et al., 1990; Kumaresan et al., 1999), and, with the advent of more widely available software, are becoming more commonplace in ergonomic assessments and designing interventions (Duffy et al., 2007). Yet their success is predicated on the realism of their representation of the mechanical behaviour of biological tissues, most commonly taking the form of a phenomenological model representing the experimental force-deflection curve (Fung,
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A Mechanistic Damage Model for Ligaments
1967; Lucas et al., 2006; Troyer and Puttlitz, 2010). Such models range from representing biological tissues as simple linear-springs to more complicated viscoelastic rheological models which are still undergoing improvement (Haut and Little, 1972; Jamison et al., 1968; Troyer et al., 2012), but in many cases neglect the prevailing mechanisms which govern the shape of the force-deflection curve. The proposed model generalizes the pioneering work of Viidik (1968), who showed that a discrete population of spring elements could be used to give a piecewise linear approximation to the toe region of ligaments. Later, Liao and Belkoff (1999) showed that a similar formulation could be used to model the failure of ligaments by allowing each fibre to fail as a brittle material at a set failure strain, while also showing that treating the collagen fibres as a continuum improved the resolution of the toeregion. The model equations presented here are a natural extension of Liao and Belkoff (1999), and are obtained by modelling the time-evolution of the distribution rather than each fibre individually, analogous to Huxley’s famous muscle model (Huxley, 1957). The model predicts diminished force-producing capacity with repeated submaximal loads, which is consistent with experimental results (Möller et al., 1992; Solomonow et al., 2001; Woo et al., 1981). Similarly, the model predicts a considerable amount of hysteresis, which is also consistent with experimental data (Weisman et al., 1979). These phenomena have previously been associated with ligament viscoelasticity, a component of ligament force-production completely ignored in this formulation. This suggests that the stress-relaxation and hysteresis phenomena are partly attributable to tissue damage in addition to viscoelastic responses. This prediction is particularly noteworthy from an injury prevention standpoint, where repeated micro-trauma over a large period of time is hypothesized to be the culprit behind chronic injuries (Kumar, 2001; Norman et al., 1998). Therefore, a natural application of this plastic ligament model would be in characterizing the risk of injury from cumulative loading, something no previous model has been able to do. Once viscoelastic components are incorporated into this model, it would be possible to separate apparent viscoelastic components from actual tissue damage.
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A Mechanistic Damage Model for Ligaments
5.1 Relationship to Huxley’s Muscle Model The advection equation in Equation 3 is reminiscent of the underlying equation in Huxley’s muscle model. Indeed, with the inclusion of a healing term, the two models become indistinguishable. The realization that the underlying equations for a population of collagen fibres are the same as those for a population of myosin-actin cross-bridges is interesting for philosophical reasons, and brings with it some level of unification in the study of soft tissue. Seeing as the same derivation could be carried out for tendon tissue, annular layers of the intervertebral disc, or smooth muscular tissue, it may be worthwhile to consider Equation 3 as a Soft Tissue Law which describes how a population of soft-tissue constituents will evolve over time. Indeed, it follows almost immediately from applying a conservation law to the distribution of tissue constituents, an idea which may extend beyond the study of ligaments: to tendons or the passive component of muscle tissues.
6.2 The Indestructible Ligament Simplification In some applications it may be detrimental for a model to include ligamentous failure; namely models which use inverse dynamics in scenarios where ligaments are not expected to fail. In this case, the assumption that fibres do not break, which we call the Indestructible Ligament Simplification (ILS), makes Equation 4 tractable and an explicit force-deflection relationship can be achieved. For instance, in assuming that the initial collagen fibre population obeys a normal distribution, the ILS force-deflection function takes the form of Equation 9.
(9)
Where
is the effective stiffness of the collagen fibres,
deviation of the initial population of collagen fibres,
and
are the mean and standard
is the current elongation of the ligament, and
is the error-function. This expression models the toe and linear regions almost exactly, and
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A Mechanistic Damage Model for Ligaments
represents an improvement over the exponential springs, power laws, or piecewise linear approximations used in conventional ligament models (Chandrashekar et al., 2008; Fung, 1981; Provenzano et al., 2001; Troyer and Puttlitz, 2010). Though exponential springs and power laws provocatively capture the toe region, they fail to reproduce the linear region of a ligament’s force-deflection curve and are very sensitive to choices in the “slack length” of the ligament. A piecewise linear response, like that of Chandreshekar et al. (2008), is closer to the true force-deflection behaviour of the ligament, but suffers from its simplicity and poor representation in the toe region. Another approach has been to use a quadratic equation in the toe region and a linear function in the linear region (Li et al., 1999). The ILS is a compromise in terms of complexity while also being derived in a way that respects the established uncrimping mechanism.
6.3 The Breaking Function The breaking function used in this preliminary analysis is almost certainly oversimplified and future work should aim to describe it more completely. Admittedly, the Heaviside breaking function seems to cause the ligament to fail too quickly, and this problem is accentuated when ramp or stepwise quadratic functions were employed. Additionally, these functions assume independence on the rate of deformation, which is a strong assumption to make given the well-documented strain-rate dependent behaviour of biological tissues (Bass et al., 2007; Lucas et al., 2008; Mattucci et al., 2013). To complicate matters further, the first-order response from the breaking function, with parameters derived from one strain-rate, cause the ligament to fail too slowly at higher strain rates. The true breaking function may be a more complicated nonlinear function, requiring extensive data to define. Nevertheless, this novel formulation of ligament mechanics opens up a new avenue for future investigations to explore and quantify damage to soft tissues, a component of tissue mechanics not previously considered in other soft tissue models.
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A Mechanistic Damage Model for Ligaments
A consequence of including the breaking function in this model formulation is that it gives the proposed ligament model a plastic behaviour, which until now, has been a tremendously difficult challenge in the modelling of biological tissues.
6.5 Limitations and Future Directions Viscoelasticity is currently absent from this model description, subjecting the results presented here to an implicit quasi-static assumption. Thus far, attempts to add viscoelasticity by replacing the springs in the description with Voigt models (spring and damper in parallel) have not been promising. An arrangement like that of Viidik (1973) may be useful in future work, reconciling this mechanistic approach with a more traditional phenomenological one. The current model is one-dimensional in nature, although it can be generalized to threedimensions if needed using some methods from continuum mechanics in a way identical to what has been done with the Huxley muscle model (Oomens et al., 2003), or other collagenous tissues (Lanir, 1979). One limitation is that the model currently has no input from experiment on the fibre level, and Future research should endeavour to reconcile the effective collagen stiffness with ligamentous cross-sectional area, packing density of collagen fibres, and stiffness data from individual collagen fibres. In addition, some assumptions in the model derivation need to be addressed. For one, the assumption that each collagen fibre generates force independently neglects the cross-linking of collagen fibres. Secondly, it was assumed that each fibre produces force linearly with displacement: an assumption that has some support from the literature (Kato et al., 1989; Sasaki and Odajima, 1996). However, there are other studies which have advocated using a wormlike chain as a more accurate representation for the force production in individual collagen fibres (Graham et al., 2004; Shah et al., 1977; Sun et al., 2002). Using a wormlike chain may provide a more complete picture of ligament force-displacement properties, however, would have precluded any analytic discussion of the model. Lastly, the assumption that
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A Mechanistic Damage Model for Ligaments
ligament damage is attributable to fibre loss is inherent in the model formulation, however this may not be the only failure mode for ligaments: cleavage of cross-linkages between collagen fibres or the rearrangement of matrix constituents may also be considered damage (Bailey et al., 1974; Fratzl et al., 1998). These aspects of ligament behaviour are currently not included in the model. The computational complexity of solving a partial differential equation may also be a significant barrier to widespread use of the proposed framework. A distribution moment approach, like that of Ma and Zahalak (1988), is appealing as it would reduce the partial differential equation into a system of four coupled ordinary differential equations. Unfortunately, the formulation presented there, if applied to these equations, gives ligaments the ability to resist compressive loads as well as tensile ones. Such nonphysiological predictions precludes its application to this ligament model. Adapting the distribution moment approach successfully to the study of ligaments is contingent on the identification of an experimentally justified breaking function The model presented was developed for the cervical ALL; however the general framework extends well beyond the study of this ligament by adjusting the parameters , , ,
and
. The same
procedure, simple least-squares fitting, can be applied to solve for these variables for any extrinsic ligament.
6.0 Conclusions The proposed model of ligament force-deflection behaviour showcased responses reflective of true ligaments being strained to failure; featuring toe and failure regions that are consistent with the currently accepted mechanisms that govern their shape. Additionally, the predicted tissue-tolerance and applied load dynamics produced responses that are qualitatively consistent with the conceptual repetitive injury model. This detailed model of a ligament provides a rigorous definition for tissue tolerance and provides equations for the modelling of injury from repetitive trauma. When damage is ignored, the resulting model provides an analytic representation of the toe and linear regions which may be more
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A Mechanistic Damage Model for Ligaments
appropriate in larger-scale biomechanical models where ligament failure in not a desired component. In addition, the model exhibits tissue softening with repeated loading that is similar to actual ligaments. The softening of tissues is accompanied by an expected decrease in the tolerance of the tissue and therefore an increased susceptibility to injury. Lastly, the general model equations lend themselves well to formalizing the study of failure in ligaments through the characterization of the breaking function. The proposed model represents a step toward a more rigorous treatment of the theoretical side of injury prediction, and lends itself well to making quantitative predictions that can either be falsified, or used in practice to assess the risk of injury of an activity.
7.0 Acknowledgements Jeff M. Barrett is supported by the National Science and Engineering Research Council of Canada (NSERC) through a PGS-D scholarship. Jack P. Callaghan is a Tier 1 Canada Research Chair in Spine Biomechanics and Injury Prevention.
8.0 Conflicts of Interest None to declare.
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Table 1: Fitted Parameters derived from experimental data for the Anterior Longitudinal Ligament.
Figure 1: Collagen fibres as an arrangement of parallel collagen fibres, each with its own slacklength denoted .
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Figure 2: Model force predictions compared to the quasi-static curves reported by Mattucci (2011) for the Anterior Longitudinal Ligament.
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Figure 3: (Top) The displacement waveform driving the simulation; (middle) the resulting force waveform predicted from the model; (bottom) the corresponding tissue capacity.
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Figure 4: Response of the model's force-deflection behaviour. Here, the softening with repeated loading, as well as the plastic behaviour, are exemplified.
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Figure 5: Response of the ligament's approximate tissue tolerance to cyclic loading at a controlled displacement.
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