96
Int. J. Experimental and Computational Biomechanics, Vol. 1, No. 1, 2009
Variable stiffness rheological model for interrelating creep and stress relaxation in ligaments M.M. Reda Taha* and S. Neidigk Department of Civil Engineering, University of New Mexico, Albuquerque, NM, USA E-mail:
[email protected] E-mail:
[email protected] *Corresponding author
A. Noureldin Departments of Electrical and Computer Engineering, Queen’s University/Royal Military College of Canada, Kingston, ON, Canada E-mail:
[email protected] Abstract: Creep and stress relaxation are two fundamental operational principles of joints that are significant for joint laxity. Modelling and relating creep and stress relaxation of ligaments is important if synthetic grafts (e.g., artificial ligaments) are to be developed and used for reconstructive surgery. This article discusses the use of rheological models to simulate creep and stress relaxation of ligaments. Modelling is performed using theory of linear viscoelasticity. Using principles of system identification, the parameters of constant and variable stiffness rheological models are determined considering experimentally measured stress relaxation of the medial collateral ligaments (MCL). The models are then tested to predict experimentally measured creep of the MCL. The proposed method proves the need to consider collagen fibre recruitment to interrelate creep and stress relaxation of ligaments. The results show that a rheological model with variable stiffness is capable of predicting creep from experimentally measured stress relaxation with a reasonable accuracy. Keywords: creep; stress-relaxation; fibre recruitment; system identification. Reference to this paper should be made as follows: Reda Taha, M.M., Neidigk, S. and Noureldin, A. (2009) ‘Variable stiffness rheological model for interrelating creep and stress relaxation in ligaments’, Int. J. Experimental and Computational Biomechanics, Vol. 1, No. 1, pp.96–113. Biographical notes: Mahmoud Reda Taha is an Associate Professor and Regents’ Lecturer, Department of Civil Engineering, University of New Mexico, Albuquerque, NM 87131 USA. Stephen Neidigk is an Undergraduate Research Assistant, Department of Civil Engineering, University of New Mexico, Albuquerque, NM 87131 USA. Aboelmagd Noureldin is an Associate Professor, Department of Electrical and Computer Engineering, Royal Military College of Canada, PO Box 17000, Station Forces, Kingston, Ontario K7K 7B4, Canada.
Copyright © 2009 Inderscience Enterprises Ltd.
Variable stiffness rheological model
1
97
Introduction
1.1 Background It is evident that ligaments behave in a viscoelastic manner and experience creep, stress relaxation and hysteresis under constant stress, displacement and cyclic loading respectively (Fung, 1993). Creep represents the non-linear increase in deformation over time when the stress is constant. Researchers showed that ligaments also creep when subjected to cyclically repetitive stress (DeHaven, 1980; Woo et al., 1981; Thornton et al., 1997). The significance of creep and stress relaxation on laxity of joints after reconstructive surgery has also been a point of concern for a few studies (Snyder-Mackler et al., 1997). Laxity of joints due to viscoelastic behaviour of ligaments needs to be accurately modelled if synthetic grafts are to be used for reconstructive surgery. Moreover, successful design of artificial ligaments that mimic the behaviour of natural ligaments requires fundamental understanding of the response of all ligament constituents to the stress environment, mainly the collagen. This requires modelling techniques that can capture the response of the ligaments to different stress environments including creep and stress relaxation. Considering Boltzmann’s hypothesis for superposition within the framework of linear viscoelasticity, the strain of the ligament ε(t) can be related to its creep compliance J(t – τ) under a constant stress σ(τ) as t
∫
ε (t ) = J (t − τ ) 0
dσ (τ ) dτ dτ
(1)
Fung (1993) introduced the theory of quasi-linear viscoelasticity (QLV) where a reduced relaxation function G (t ) is formulated to be dependent on the ligament strain such that G (ε , t ) = G (t ) σ (ε (t ))
(2)
Where G (ε , t ) is a generalised relaxation function and G (t ) is the reduced relaxation function determined using equation (3) t
∫
σ (t ) = G (t − τ ) 0
dσ (ε (τ )) d ε (τ ) dτ dε dτ
(3)
If the stress relaxation function is directly related to strain, the non-linear viscoelastic behaviour of ligaments can be described by non-linear superposition (Fung, 1993) t
∫
σ (t ) = G (ε (τ ), t − τ ) 0
d ε (τ ) dτ dτ
(4)
G (ε (τ ), t − τ ) is the non-linear stress relaxation function. The above framework shows that it is possible to relate creep and stress relaxation of biological materials using principles of non-linear viscoelasticity. In this article, we discuss the ability to establish this connection using linear viscoelastic rheological models.
98
M.M. Reda Taha et al.
1.2 Rheological models for ligament viscoelasticity Many analytical models have been introduced to model creep and stress relaxation of ligaments (Thornton et al., 1997; Hingorani et al., 2004; Lakes and Vanderby, 1999). These models were based on standard mathematical formulations of viscoelastic materials describing linear, quasi-linear and non-linear viscoelasticity. A brief description of these linear, quasi-linear and non-linear viscoelasticity models is provided elsewhere (Woo et al., 1993). While the mechanical response of connective tissues including ligaments is known to follow non-linear viscoelasticity, ligaments are expected to behave in a linear viscoelastic manner when subjected to limited oscillation of small amplitude about an equilibrium state (Fung, 1993). Rheological models are mechanical models that consist of a group of connected springs and dashpots connected in series or parallel (Findley et al., 1976). Rheological models have been developed over the last few decades to describe linear viscoelastic behaviour of biological materials. The Maxwell, Kelvin-Voigt and Kelvin models, shown in Figure 1, are among those widely used models and are described by equations (5), (6) and (7) respectively: du F 1 ⎛ dF ⎞ = + ⎜ ⎟ dt η μ ⎝ dt ⎠
(5)
⎛ du ⎞ F = μ u +η ⎜ ⎟ ⎝ dt ⎠
(6)
⎡ ⎛ dF ⎞ ⎛ du ⎞ ⎤ F + τ1 ⎜ ⎟ = μ 0 ⎢u + τ 2 ⎜ ⎟ ⎥ ⎝ dt ⎠ ⎝ dt ⎠ ⎦ ⎣
(7)
as
τ1 =
η1 η ⎛ μ ⎞ and τ 2 = 1 ⎜1 + 0 ⎟ μ0 ⎝ μ 1 ⎠ μ1
(8)
F is the force and u is the displacement. μ, η are the spring and damping coefficient of the dashpot for the Maxwell and Kelvin-Voigt models while μ0, μ1 and η1 are the spring and damping coefficients of the Kelvin model. It is well established that Kelvin model (also known as the standard linear solid model) is more capable of describing the viscoelastic behaviour under creep and relaxation than the Maxwell and Kelvin-Voigt models (Findley et al., 1976). Creep compliance and stress relaxation of the Kelvin model can be identified by subjecting the model to a unit displacement function u(t) or unit force function F(t) as J (t ) = u (t )
μ0τ 2 τ1
G (t ) = F (t )
τ1 μ0τ 2
(9) (10)
Variable stiffness rheological model Figure 1
99
Mechanical rheological models for describing ligament viscoelasticity, (a) Maxwell model (b) Kelvin-Voigt model (c) Kelvin model
(a)
(b)
(c)
1.3 Interrelating creep and stress relaxation Equations (9) and (10) imply that creep and stress relaxation of the ligaments can be interrelated using the linear viscoelasticity represented by Kelvin model. However, Thornton et al. (1997) showed that creep and stress relaxation of ligaments cannot be interrelated when considering linear viscoelasticity. This was attributed to the fact that the rate of creep is different from the rate of stress relaxation due to changes of the level of fibre recruitment in the ligament during the creep process (Thornton et al., 1997). Fibre recruitment represents the change in the number of collagen fibres that are mobilized to resist the load. The effect of fibre crimp on ligament stiffness and strain was earlier proposed by Matyas et al. (1988). Therefore, failure to interrelate creep and stress relaxation was attributed to the fact that these two viscoelastic phenomena occur by fundamentally different mechanisms (Fung, 1993; Woo et al., 1981). Using a non-linear model that incorporates fibre recruitment, Thornton et al. (2001) showed that fibre recruitment helps the ligament to reduce creep. Although the model was capable of demonstrating the fact that creep of progressively recruited fibres could be used to predict creep from stress relaxation measurements, the model requires estimating collagen fibre elastic modulus at failure, which is usually unknown and cannot not be attained from either creep or stress relaxation experiments. Using the same set of experimental data, Lakes et al. (1999) showed that creep and stress relaxation can be also interrelated if described within the framework of non-linear viscoelasticity. While the methods by Lakes et al. (1999) showed good agreement with experimental data, modelling complexity might hinder its use in ligament modelling process for mechanical analysis. Other models based on probability distribution of collagen fibres waviness (Sverdlik and Lanir, 2002) or fuzzy evaluation of collagen fibre recruitment (Ali et al., 2005) were successfully capable of modelling the time-dependent effect of ligaments due to cyclic loading but did not address the problem of interrelating creep and stress relaxation. Here, we present a method to interrelate creep and stress relaxation using rheological models. We illustrate that the modified Kelvin model with variable stiffness has the ability to relate creep and stress relaxation of ligaments with a reasonable accuracy. The use of rheological models provides a modular format used by researchers in finite element models adding a significant value to successful modelling.
100
2
M.M. Reda Taha et al.
Methods
2.1 Laplace analysis of Kelvin model We start by transforming Kelvin model to the s-domain using Laplace transform to enable identification of model parameters using experimental data. This can be done by considering a rational transfer function F(s)/u(s) as
μ (η s) F (s) = μ0 + 1 1 μ1 + η1s u(s) Equivalently, with the definition of τ1E =
(11)
η1 η ⎛ μ ⎞ and τ 2 E = 1 ⎜1 + 0 ⎟ , the rational μ0 ⎝ μ 1 ⎠ μ1
transfer function u(s)/F(s) can be written as 1 ⎛ 1 + τ1E s ⎞ u(s) = ⎜ ⎟ F ( s) μ0 ⎝ 1 + τ 2 E s ⎠
(12)
A state variable x(t) is introduced to obtain a state space representation of the system so that its corresponding Laplace transform X(s) is related to both u(s) and F(s) as X ( s) 1 ⎛ 1 ⎞ = ⎜ ⎟ F ( s) μ0 ⎝ 1 + τ 2 E s ⎠
(13)
u ( s) = 1 + τ 1E s X (s)
(14)
The state space representation of the Kelvin model can be obtained by transforming equations (13) and (14) into time domain as dx 1 1 =− x(t ) + F (t ) τ 2E μ0τ 2 E dt
(15)
⎛ τ ⎞ τ u (t ) = ⎜1 − 1E ⎟ x ( t ) + 1E F (t ) τ μ 2E ⎠ 0τ 2 E ⎝
(16)
Creep can be modelled using the time variable displacement function u(t) that follows the creep shape function with a maximum displacement of u0 u (t ) = u0 J (t )
(17)
Then, creep compliance can be expressed as J (t ) =
Where
u (t ) μ0τ 2 E F0 τ1 E
(18)
Variable stiffness rheological model
τ1E =
η1 η ⎛ μ ⎞ and τ 2 E = 1 ⎜1 + 0 ⎟ μ0 ⎝ μ 1 ⎠ μ1
101 (19)
Considering equation (9) creep compliance J(t) can be computed as the displacement u(t) derived from the Kelvin model when subjected to a unit force F0 = 1, thus J (t ) = u (t )
μ0τ 2 E τ1 E
(20)
Where u(t) represents the displacement of the Kelvin model under unit force. Similarly, the stress relaxation G(t) can be computed as the force F(t) of the Kelvin model when the model is subjected to a unit displacement as G (t ) = F (t )
τ1E μ0τ 2 E
(21)
If the stress relaxation G(t) is measured experimentally, values of the Kelvin model parameters (μ0, μ1 and η1) can be determined from equation (21) using a system identification method. Model parameters represent specific ligament characteristics can then be used to predict creep of the ligament J(t) using equation (20). The predicted creep compliance is compared to experimentally measured creep compliance. The process of predicting creep compliance of ligaments from stress relaxation measurements using rheological models and system identification is shown in schematically in Figure 2. Figure 2
Schematic representation of the method used for predicting creep compliance of ligaments from experimentally measured stress relaxation using rheological models (see online version for colours)
102
M.M. Reda Taha et al.
2.2 Modelling the effect of collagen fibre recruitment The Kelvin model described above does not account for the effect of changes in the level of recruitment of the collagen fibre in the ligaments during creep, which was observed experimentally (Thornton et al., 1997; Hingorani et al., 2004). Thornton et al. (2001) showed that fibre recruitment during creep could be considered as an increase of the effective cross-sectional area of the ligament with time. We argue here that fibre recruitment can be modelled using a modified Kelvin model with a time-varying stiffness. The variable stiffness has an initial resistance μv0 that represents the initial level of fibre recruitment that is observed during stress relaxation. When modelling creep, the resistance shall increase with time to represent the ability of increased fibre recruitment to resist creep. Figure 3
Schematic representation of the modified Kelvin model to predict ligament creep from experimentally measured stress relaxation showing the use of variable stiffness μv
We introduce the modified Kelvin model (shown in Figure 3) including the time variable stiffness μv(t). We suggest that the variable spring stiffness μv(t) to follow equation (22) t ⎛ − τ cr ⎜ μv (t ) = μv 0 + Δμv 1 − e ⎜ ⎝
⎞ ⎟ ⎟ ⎠
(22)
μv0 is the initial spring stiffness derived from stress relaxation, Δμv is the change of the spring stiffness due to fibre recruitment during creep and τcr is a time constant to represent the delay in the increase of spring stiffness with time. Numerical values for Δμv and τcr can be determined from experimentally measured creep compliance. If the variable stiffness parameters are identified, the modified Kelvin model is used to predict creep compliance from experimentally measured stress relaxation of ligaments. The three stress relaxation model parameters μv0, μ1 and η1 are determined using stress relaxation data and methods of system identification. The variable resistance μv will be determined based on equation (22). The modified Kelvin model can then be used to relate creep compliance to stress relaxation.
2.3 System identification The experimental data published by Thornton et al. (1997) of creep and stress relaxation of the medial collateral ligaments (MCL) is used as a testing data to validate the models.
Variable stiffness rheological model
103
Briefly, the specimens were 14 MCLs from normal skeletally mature (one-year-old) female New Zealand white rabbits. The specimens were tested in cyclic loading for 30 cycles at 1 HZ in two conditions: the first group (n = 7) was tested in cyclic creep at a constant stress of 14 MPa, the second group (n = 7) was tested in cyclic relaxation at a constant deformation for an initial stress of 14 MPa. This testing stress represents 5% of the ligament strength and was reported to be at the start of the end of the toe region and the start of the linear stress region. The creep specimens are denoted C1 to C7 while the stress relaxation specimens were denoted RE1 to RE7. Specimens with the similar numbers were taken from the two knee joints of the same animal and thus can be used for comparison. The normalised stress relaxation GN was determined for test specimens RE1 to RE7 as GN (t , t0 ) =
G (t , t0 ) G(t 0 ,t 0 )
(23)
System identification methods are applied to determine the model parameters that can reduce the mean square error (MSEG) between the experimentally measured and predicted stress relaxation n
MSEG =
∑[G
2 MN i (t ) − GPN i (t )]
(24)
i =1
GPN(t) is the normalised predicted stress relaxation and GMN(t) is the normalised experimentally measured stress relaxation. Here, we implement a non-linear least square curve fitting method for system identification after Huet et al. (2003). The proposed method is a trust region method that implements the interior-reflective Newton optimisation method (Coleman and Li, 1996) with the target of determining the Kelvin model parameters [ μ 0 μ1 η1 ] needed to minimise the objective function MSEG. The model parameters are identified through an iterative process that is based on determining the approximate solution using the method of preconditioned conjugate gradient (PCG). Further details on that method can be found elsewhere (Dennis, 1977; Huet et al., 2003). The identified Kelvin model parameters are then used to predict the normalised creep compliance JN of the MCL, which was compared to the measured normalised creep compliance, determined as J N (t ) =
J (t , t0 ) J(t 0 ,t 0 )
(25)
The ability of the model to predict creep compliance was quantified by determining the mean square creep error MSEJ from equation (26) n
MSEJ =
∑[ J
MN i (t ) −
J PN i (t )]2
(26)
i =1
JPN(t) is the normalised predicted creep compliance and JMN(t) is the normalised experimentally measured creep compliance.
104
M.M. Reda Taha et al.
For the modified Kelvin model, the system identification process was performed in two steps. First, the model parameters [ μ v 0 μ1 η1 ] are identified using experimental stress relaxation data so that the system achieves the minim MSEG (equation 24). Moreover, we identify the time where 90% of stress relaxation took place denoted here as τ90%. Second, the variable stiffness model parameters presented in equation (22): Δμv and τcr can then be determined using equations (27) and (28) Δμv = α
μv0
(27)
τ cr = β τ 90%
(28)
μv0 is the original value for the variable stiffness defined by system identification of the stress relaxation data. Our investigations showed that α = 2.0 and β = 0.17. However, we argue that these values are dependent on the stress magnitude and the creep and relaxation rates. The modified Kelvin model is then used to predict creep compliance of the creep specimens C1 to C7 using stress relaxation data RE1 to RE7 while predicting the variable stiffness parameters using equations (27) and (28). Figure 4 shows a schematic presentation for using the current modified Kelvin model to predict creep compliance using stress relaxation experimental data. Figure 4
Schematic representation of predicting creep compliance using stress relaxation data through system identification of the modified Kelvin model and estimating the variable stiffness using equations (27) and (28) (see online version for colours)
Variable stiffness rheological model
3
105
Results and discussions
The Kelvin model parameters (μ0, μ1 and η1) for the stress relaxation specimens RE1 to RE7 are presented in Table 1. The MSEG values that represent goodness to fit of the Kelvin model with experimentally measured stress relaxation data for the seven testing specimens RE1 to RE7 are presented in Table 2. A sample presentation of the transfer functions that is used for modelling the Kelvin model in the state space representation is shown in Figure 5. The system identification method was capable of determining the Kelvin model parameters to fit the experimentally measured stress relaxation as demonstrated in Figure 6 for three sample specimens (RE1, RE4 and RE7). Comparison between predicted and experimentally measured creep compliance using the Kelvin model for these three specimens (C1, C4 and C7) is presented in Figure 7. The MSEJ values for predicting creep compliance using the Kelvin model are presented in Table 3. Table 1
System parameters of the Kelvin model identified using experimentally measured stress relaxation data
Specimen
RE1
RE2
RE3
RE4
RE5
RE6
RE7
μ0
6115
4917
5443
4955
4539
4901
4572
μ1
2052
2757
2194
2262
2756
2339
3382
η1
254410
299510
278390
298790
320220
299490
331390
Table 2
MSEG for stress relaxation representing the goodness of the Kelvin model to fit of the experimentally measured stress relaxation data
Specimen –3
MSEG*10 Figure 5
RE1
RE2
RE3
RE4
RE5
RE6
RE7
2.8
3.4
3.6
4.4
3.8
4.5
5.0
Transfer functions used for simulating the mechanical response in the s-domain to establish the relation between creep and stress relaxation
106 Figure 6
M.M. Reda Taha et al. Experimentally measured versus predicted stress relaxation of the MCL using the Kelvin model, (a) RE1 (b) RE4 (c) RE7 (see online version for colours)
(a)
(b)
(c) Note: Stress relaxation predicted by the Kelvin model (solid line) is in good agreement with experimentally measured stress relaxation (dashed line).
Variable stiffness rheological model Figure 7
107
Experimentally measured versus predicted creep compliance of the MCL using Kelvin model, (a) C1 (b) C4 (c) C7 (see online version for colours)
(a)
(b)
(c) Note: Creep compliance predicted by the Kelvin model (solid line) is much higher than experimentally measured creep compliance (dashed line).
108
M.M. Reda Taha et al.
Table 3
MSEJ for creep compliance representing the goodness of the Kelvin model to predict the experimentally measured creep data
Specimen
C1
C2
C3
C4
C5
C6
C7
MSEJ
0.38
1.57
0.68
0.83
1.66
0.95
3.20
Model validation of the Kelvin model proved the suggested analogy and showed the ability of rheological models to provide an adequate tool for modelling viscoelastic behaviour of ligaments. However, it is obvious from Figure 7 that creep and stress relaxation cannot be interrelated through the classical Kelvin model. Ligament creep compliances predicted from stress relaxation using the Kelvin model were significantly higher than experimentally measured creep compliance. That finding meets those reported by Thornton et al. (2001) and Ali et al. (2005) showing creep and stress relaxation mechanisms to be fundamentally different due to the dissimilarity in the collagen fibre recruitment process in creep and stress relaxation. On the other hand, the modified Kelvin model parameters (μv0, μ1, η1 and τ90%) for the stress relaxation specimens RE1 to RE7 are presented in Table 4. The MSEG values that represent goodness to fit of the modified Kelvin model with experimentally measured stress relaxation data for the seven testing specimens RE1 to RE7 are similar to that of the Kelvin model represented in Table 2. The variable stiffness parameters Δμv and τcr predicted from stress relaxation data using equations (27) and (28) are presented in Table 5. Figure 8 shows the time function relationship used to describe the variable stiffness in the modified Kelvin model for specimens C1, C4 and C7. The variable stiffness parameters Δμv and τcr from Table 5 are then used to predict creep compliance for specimens C1 to C7. Comparison between the predicted and experimentally measured creep compliance using the modified Kelvin model with variable stiffness predicted using equations (22), (27) and (28) for specimens (C1, C4 and C7) is presented in Figure 9. The MSEJ for predicting creep compliances using the modified Kelvin model are presented in Table 6. Table 4 System parameters of the modified Kelvin model identified using stress relaxation data Specimen
RE1
RE2
RE3
RE4
RE5
RE6
RE7
μv0
6115
4917
5443
4955
4539
4901
4572
μ1
2052
2757
2194
2262
2756
2339
3382
η1
254410
299510
278390
298790
320220
299490
331390
607.6
456.8
589.5
625.0
484.6
614.3
450.9
τ90% Table 5 Specimen
Variable stiffness parameters identified using the modified Kelvin model in Table 4 and equations (27) and (28) RE1
RE2
RE3
RE4
RE5
RE6
RE7
Δμv
12230.0
9834.0
10886.0
9910.0
9078.0
9802.0
9144.0
τcr
103.3
77.7
100.2
106.3
82.4
104.4
76.7
Variable stiffness rheological model Figure 8
109
Time varying stiffness function showing the change in stiffness to simulate the change in the level of fibre recruitment during creep, (a) Specimen C1 (b) Specimen C4 (c) C7
(a)
(b)
(c)
110 Figure 9
M.M. Reda Taha et al. Experimentally measured versus predicted creep compliance of the MCL using the modified Kelvin model, (a) C1 (b) C4 (c) C7
(a)
(b)
(c)
Variable stiffness rheological model
111
MSEJ for creep compliance representing the goodness of the modified Kelvin model to predict the experimentally measured creep data
Table 6 Specimen –3
MSEJ*10
C1
C2
C3
C4
C5
C6
C7
0.57
0.58
0.83
0.34
6.5
0.62
0.84
It can be observed from Figure 9 that the modified Kelvin model showed promising results in interrelating creep and stress relaxation. When a variable stiffness was considered and computed using experimentally measured stress relaxation, ligament creep compliance predicted from stress relaxation was close to experimentally measured compliance. Ligament creep compliances predicted from stress relaxation using the modified Kelvin model (Figure 9) are in good agreement with experimentally measured creep compliances with a relatively low root mean square error of 0.34 × 10–3 to 6.5 × 10–3 (Table 6) representing excellent modelling accuracy. The suggested variation in the model stiffness was therefore able to simulate the experimentally observed increase in ligament stiffness as a result of increasing the number of collagen fibres mobilized to resist creep. We suggest that the variable stiffness model with variable stiffness function (equations (22), (27) and (28)) are capable of modelling the variable ligament stiffness resistance during creep with an initial stiffness similar to that observed during ligament stress relaxation. It is worth noting other time functions suggested by other researchers (e.g., Hannafin and Arnoczky, 1994; Suki et al., 1994; Thornton et al., 2001) to describe collagen fibre recruitment were not capable to produce satisfactory results with the proposed rheological model. Finally, it is important to note that the proposed modified Kelvin model is easy to implement in finite element simulation and thus represents a good addition to the ability to model time-dependent behaviour of soft-tissue specially ligaments. Accurate time-dependent modelling of ligaments is necessary if synthetic grafts are to be used in reconstruction surgeries with torn ligaments.
4
Conclusions
A new rheological model based on variable stiffness named the modified Kelvin model was developed and was able to successfully predict creep compliance from experimentally measured stress relaxation. The proposed model utilises the framework of linear viscoelasticity and has been validated using experimentally measured creep and stress relaxation data. The proposed model accounts for the fact that collagen fibre recruitment with time reduces creep effect in ligaments. Fibre recruitment process is simulated by incorporating a time-variable stiffness with its initial value similar to that occurring during stress relaxation. The proposed model confirms the importance of considering collagen fibre recruitment to interrelate creep and stress relaxation of ligaments.
112
M.M. Reda Taha et al.
Acknowledgements This work has been supported by the Ralph Powe Junior Faculty Enhancement Award to M.M. Reda Taha by the Oak Ridge Associated Universities (ORAU), USA. The authors are grateful to this support. The financial support by the Natural Science and Engineering Research Council of Canada (NSERC) to A. Noureldin is much appreciated.
References Ali, A., Reda Taha, M.M., Thornton, G.M., Shrive, N.G. and Frank, C.B. (2005) ‘Biomechanical study using fuzzy systems to quantify collagen fibre recruitment and predict creep of the rabbit medial collateral ligament’, ASME Journal of Biomechanical Engineering, Vol. 127, pp.484–493. Coleman, T.F. and Li, Y. (1996) ‘An interior, trust region approach for non-linear minimization subject to bounds’, SIAM Journal on Optimization, Vol. 6, pp.418–445. DeHaven, K. (1980) ‘Diagnosis of acute knee injuries with hemoarthrosis’, American Journal of Sports Medicine, Vol. 8, pp.9–14. Dennis, J.E., Jr. (1977) ‘Non-linear least-squares’, in D. Jacobs (Ed.): State of the Art in Numerical Analysis, pp.269–312, Academic Press. Findley, W.N., Lai, J.S. and Onaran, K. (1976) Creep and Relaxation of Nonlinear Viscoelastic Materials with Introduction to Linear Viscoelasticity, Dover Publications. Fung, Y.C. (1993) Biomechanics: Mechanical Properties of Living Tissues, 2nd ed., Springer Verlag, New York, USA. Hannafin, J.A. and Arnoczky, S.P. (1994) ‘Effect of cyclic and static tensile loading on water content and solute diffusion in canine flexor tendons: an in vitro study’, Journal of Orthopedic Research. Vol. 12, pp.350–356. Hingorani, R.V., Provenzano, P.P., Lakes, R.S., Escarcega, A., Jr. and Vanderby, R. (2004) ‘Non-linear viscoelasticity in rabbit medial collateral ligament’, Annals of Biomedical Engineering, Vol. 32, No. 2, pp.306–312. Huet, S., Bouvier, A., Poursat, M-A. and Jolivet, E. (2003) Statistical Tools for Nonlinear Regression: A Practical Guide with S-PLUS and R Examples, 2nd ed., Springer. Lakes, R.S. and Vanderby, R. (1999) ‘Interrelation of creep and relaxation: a modelling approach for ligaments’, ASME Journal of Biomechanical Engineering, Vol. 121, No. 6, pp.612–615. Matyas, J.R., Chowdhury, P. and Frank, C.B. (1988) ‘Crimp as an index of ligament strain’, Proceedings of the 22nd Annual Meeting of the Canadian Orthopedic Research Society, Ottawa, Ontario, June, p.113. Snyder-Mackler, L.G., Fitzgerald, K., Bartolozzi, A.R. and Ciccotti, M.G. (1997) ‘The relationship between passive joint laxity and functional outcome after anterior cruciate ligament injury’, American Journal of Sports Medicine, Vol. 25, pp.191–195. Suki, B., Barabasi, A.L. and Lutchen, K.R. (1994) ‘Lung tissue viscoelasticity: a mathematical framework and its molecular basis’, Journal of Applied Physiology, Vol. 76, No. 6, pp.2749–3759. Sverdlik, A. and Lanir, Y. (2002) ‘Time-dependent mechanical behavior of sheep digital tendons, including the effects of preconditioning’, ASME J. Biomechanical Engineering, Vol. 124, pp.78–84. Thornton, G.M., Frank, C.B. and Shrive, N.G. (2001) ‘Ligament creep behaviour can be predicted from stress relaxation by incorporating fibre recruitment’, Journal of Rheology, Vol. 45, pp.493–507.
Variable stiffness rheological model
113
Thornton, G.M., Oliynyk, A., Frank, C.B. and Shrive, N.G. (1997) ‘Ligament creep cannot be predicted from stress relaxation at low stress: a biomechanical study of the rabbit medial collateral ligament’, Journal of Orthopaedic Research, Vol. 15, pp.652–656. Woo, SL-Y., Gomez, M.A. and Akeson, W.H. (1981) ‘The time and history-dependent viscoelastic properties of the canine medial collateral ligament’, ASME Journal of Biomechanical Engineering, Vol. 103, pp.293–298. Woo, SL-Y., Johnson, G.A. and Smith, B.A. (1993) ‘Mathematical modeling of ligaments and tendons’, ASME Journal of Biomechanical Engineering, Vol. 115, pp.468–473.
