Journal of Biomechanics 48 (2015) 4297–4302
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Quasi-linear viscoelastic properties of the human medial patello-femoral ligament G. Criscenti a,b, C. De Maria b,n, E. Sebastiani a,d, M. Tei a,d, G. Placella a,d, A. Speziali a,d, G. Vozzi b,c, G. Cerulli a,d a
Istituto di Ricerca Traslazionale per l’Apparato Locomotore – Nicola Cerulli – LPMRI, via A. Einstein 12, 52100 Arezzo, Italy Research Center “E. Piaggio”, University of Pisa, Largo Lucio Lazzarino 1, 56126 Pisa, Italy c Department of Ingegneria dell’Informazione, University of Pisa, via G. Caruso 16, 56126 Pisa, Italy d Department of Orthopedic surgery, University of Perugia, via S. Andrea delle Fratte 1, 06134 Perugia, Italy b
art ic l e i nf o
a b s t r a c t
Article history: Accepted 26 October 2015
The evaluation of viscoelastic properties of human medial patello-femoral ligament is fundamental to understand its physiological function and contribution as stabilizer for the selection of the methods of repair and reconstruction and for the development of scaffolds with adequate mechanical properties. In this work, 12 human specimens were tested to evaluate the time- and history-dependent non linear viscoelastic properties of human medial patello-femoral ligament using the quasi-linear viscoelastic (QLV) theory formulated by Fung et al. (1972) and modified by Abramowitch and Woo (2004). The five constant of the QLV theory, used to describe the instantaneous elastic response and the reduced relaxation function on stress relaxation experiments, were successfully evaluated. It was found that the constant A was 1.21 70.96 MPa and the dimensionless constant B was 26.03 74.16. The magnitude of viscous response, the constant C, was 0.11 70.02 and the initial and late relaxation time constants τ1 and τ2 were 6.32 71.76 s and 903.47 7504.73 s respectively. The total stress relaxation was 32.7 74.7%. To validate our results, the obtained constants were used to evaluate peak stresses from a cyclic stress relaxation test on three different specimens. The theoretically predicted values fit the experimental ones demonstrating that the QLV theory could be used to evaluate the viscoelastic properties of the human medial patello-femoral ligament. & 2015 Elsevier Ltd. All rights reserved.
Keywords: Medial patello-femoral ligament (MPFL) Ligament biomechanics Ligament mechanical properties Quasi-linear viscoelastic theory QLV theory Tissue Engineering
1. Introduction The medial patello-femoral ligament (MPFL) is the primary static stabilizer of the patello-femoral joint and is commonly injured during a first time lateral patellar dislocation (Ahmad et al, 2000; Desio et al., 1998; Greiwe et al., 2010; Nomura 1999; Ostermeier et al., 2006; Panagiotopoulos et al., 2006). The ligament natural healing process is often inadequate to restore its structural, mechanical and biological properties and, in most cases, it results in the formation of scar tissue with inferior mechanical properties (Hammoudi and Temenoff, 2011). Two different strategies can be followed to reach a complete and long-term functional repair of the damaged tissue: reconstructive and regenerative approaches. Different points of view regarding reconstructive methods have been presented, focusing on the graft choice and tension, knee flexion angle and fixation n
Corresponding author: Tel.: þ 39 50 2217056; fax: þ 39 50 2217051. E-mail address: [email protected] (C. De Maria).
http://dx.doi.org/10.1016/j.jbiomech.2015.10.042 0021-9290/& 2015 Elsevier Ltd. All rights reserved.
methods. Many of these methods have been compared, studying the stiffness of the reconstructed ligament and the elongation of the graft after cyclic tests, but results are controversial (Mountney et al., 2005; Lenschow et al., 2013). The regenerative approach represents a new strategy based on Tissue Engineering (TE) that aims at fabricating an immunological tolerant tissue substitute to permanently restore the functionality of the damaged one, without the need for supplementary therapies (Sachlos and Czernuszka, 2003). Nevertheless, important challenges must be solved to obtain complete ligament repair that will lead to a clinically effective and commercially successful application (Rodrigues et al., 2013): it is well known that the fate of cells seeded onto a scaffold is extremely dependent on the mechanical properties of this substrate (Engler et al., 2006; Tse and Engler, 2011). Thus, determining the viscoelastic properties of the MPFL is a critical point to understand its physiological function and its contribution as patellar stabilizer in order to select the optimal replacement graft and to define the correct mechanical parameters for the design and fabrication of scaffolds.
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The viscoelastic behavior of biological tissues depends on the complex interaction among collagen, elastin, proteoglycans and water (Woo et al., 1999) and it was studied for different tissues as aortic valve tissues (Rousseau et al., 1983; Sauren et al., 1983), cartilage (Woo et al., 1980), skeletal muscle (Best et al., 1994), bone (Lakes et al., 1979; Lakes and Katz, 1979), ligament (Abramowitch and Woo, 2004; Haut and Little 1969; Provenzano et al., 2001; Woo et al., 1981) and tendon (Atkinson et al., 1999; Haut and Little 1972). Ligaments are viscoelastic and display time-dependent and load-history-dependent mechanical behavior (Fung et al., 1972) described by hysteresis, creep and stress relaxation. Hysteresis is the energy dissipation with continuous phases of loading and unloading; creep is the increase of length over time under a constant load while stress relaxation is a decrease in the load when the ligament is held at a fixed elongation (Woo et al., 1999). The quasi-linear viscoelastic (QLV) theory formulated by Fung and modified by Abramowitch and Woo (2004) is the most used approach to describe the viscoelastic behavior of biological tissues (Carew et al., 1999; Elliott et al., 2003; Fung et al., 1972; Fung 1993; Funk et al., 2000; Kwan et al., 1993; Sauren et al., 1983; Thomopoulos et al., 2003; Woo et al., 1981), especially ligaments and tendons as the porcine anterior cruciate ligament (Kwan et al., 1993), human patellar tendon (Johnson et al., 1994) and the canine medial collateral ligament (MCL) (Abramowitch et al., 2004). The theory is based on the assumption of an instantaneous change of strain in the load application phase, although this is experimentally impossible to be performed. For this reason, many researchers tried to use high strain rates assuming that no relaxation occurred during the loading phase. With this approach, the parameters estimation was affected by high variability because experimental errors related to high strain rate (vibration of the testing machine, errors in reaching the final position, such as the overshoot) (Funk et al., 2000), obtaining estimated constants that were not comparable with other ones obtained from other biological tissues and in other laboratories. The modified quasi-linear viscoelastic theory (mQLV) allows to evaluate the reduced relaxation function and elastic response performing experiments with slow strain rate (Abramowitch and Woo, 2004). In the present work, we used this approach, avoiding errors related to the fast movements of the testing machines. Thus, the objective of this study was to determine the five constant of the modified QLV theory related to the MPFL, used to describe the instantaneous elastic response and the reduced relaxation function in stress relaxation experiments, and thus provide a complete characterization of its viscoelastic behavior. These values can be the target point for the selection of the optimal reconstructive and regenerative procedures to restore MPFL function. 2. Materials and methods 2.1. Modified quasi-linear viscoelastic (mQLV) theory The QLV theory formulated by Fung et al. (1972) and modified by Abramowitch and Woo (2004) was used to describe the time- and history-dependent viscoelastic and nonlinear mechanical properties of the MPFL. The Fung’s theory assumes that the general stress relaxation behavior of soft-tissue is expressed by the convolution integral of the reduced relaxation function G(t) and the elastic response σe(ε) (Eq. (1)): Z t ∂σ e ðεÞ ∂ε σ ðt Þ ¼ G ðt τ Þ ∂τ ð1Þ ∂ε ∂ τ 1 The instantaneous elastic response represents the maximum stress in response to an instantaneous step input of strain (Abramowitch and Woo, 2004) and is described by the exponential function (Eq. (2)): σ e ðϵÞ ¼ A eBϵ 1 ð2Þ where A and B are the material linear and nonlinear constants respectively. In
particular, the constant B represents the rate of change of the slope of the stress– strain curve and the product AB is the initial slope of the curve (Woo et al., 1981). The reduced relaxation function represents the time dependent stress response of the tissue normalized by the stress present at the time of the strain step input (Fung et al., 1972) and it was proposed by Fung to describe the sensitivity of soft tissues to strain rate. This equation is derived from the standard linear solid model that was chosen because it presents a finite plateau for a long relaxation time. In order to describe a constant spectrum of relaxation, the form of the reduced relaxation function is (Abramowitch and Woo, 2004) (Eq. (3)):
Gðt Þ ¼
h i 1þ C E1 τt2 E1 τt1
ð3Þ
1 þ Clnðττ21 Þ
R 1 z where E1 ðyÞ ¼ y e z dz is the exponential integral, and C, τ1 and τ2 are material constants with τ1 { τ2 . The dimensionless constant C represents the magnitude of viscous effects and is related to the percentage of relaxation (Abramowitch et al., 2004) while the time constants τ1 and τ2 describe the initial and the late relaxation of the slope of the stress-relaxation curve at early and late time periods, respectively (Sauren and Rousseau, 1983). According to the methodology proposed by Abramowitch and Woo (2004) slow strain rates during the loading phase of a stress relaxation test can be used. In particular, the stress resulting from the loading phase of the stress relaxation test with a constant strain rate γ over the time 0o t o t 0 is (Eq. (4)):
σ ðt Þ ¼
ABγ 1þ C ln τ2 =τ1
Z
t
tτ t τ 1 þ C E1 E1 eBγτ ∂τ
τ2
0
τ1
ð4Þ
The stress resulting from the relaxation phase (from t 0 to t ¼ 1) of the stress relaxation test is (Eq. (5)):
σ ðt Þ ¼
ABγ 1þ C ln τ2 =τ1
Z
1
t0
t τ t τ 1 þ C E1 E1 eBγτ ∂τ
τ2
τ1
ð5Þ
where A, B, C, τ1 , τ2 are the five constants of the QLV theory that need to be determined. This method showed a unique solution that was minimally sensitive to systematic and random experimental noise (Abramowitch et al., 2004). 2.2. Constant estimation and validation The five constants of the QLV theory were determined from the experimental data. Stress–strain data obtained from the loading portion of the stress relaxation experiments were used to fit Eq. (2) in order to determine constant A (Abramowitch et al., 2004). Constants B, C, τ1 and τ2 were obtained substituting constant A in Eqs. (4) and (5) and simultaneously fitting the stress–time data related to the loading and relaxation portions of the stress relaxation experiment (Abramowitch and Woo, 2004; Abramowitch et al., 2004). A modified Levenberg–Marquardt algorithm was used for non-linear optimization process (Abramowitch et al., 2004). To validate the results, the obtained constants were used to evaluate peak stresses from separate independent cyclic stress relaxation tests. In particular, the constants obtained from each specimen with the stress relaxation experiments were separately used in Eq. (1) to define the stress response of each sample and theoretical peak stresses at each cycle were determined (Woo et al., 1981). Subsequently, the theoretical peak stresses and experimental ones obtained during cyclic stress-relaxation test for each specimen were compared. 2.3. Preparation of the specimens A total of 15 human cadaveric knees from 5 women and 9 men with a mean age of 7678 years (range 68–84 years) were used in this experiment. None of these showed patellar instability, knee injuries, surgical procedures or arthritic deformations. The Nicola’s Foundation Onlus Ethics Committee has given its approval for this study. The cadavers were dissected after they have been stored for 24 h at 4 °C and then were preserved in a sterile gauze, sealed in a polyethylene bag, labeled and stored at 18 °C. They were thawed at a temperature of 4 °C when required. Warren and Marshall’s three-layer classification of the medial side of the knee was used to dissect the specimens (Warren and Marshall, 1979). All dissections started with a midline incision of the skin detaching it from the subcutaneous fascia. The joint capsule was accessed extending through a lateral incision at the vastus lateralis muscle extending laterally at the parapatellar side and at the lateral compartment of the tibia which was cut proximally to distally. The femur was tipped up detaching the muscle bundles. The isolated single muscles of the quadriceps were left inserted in their distal insertions and used as landmarks. The patella was rotated and the posteromedial capsule was opened from the inside, detaching and isolating the synovial capsule in the 3rd layer from the 2nd one. The fibers of the MPFL were identified by palpation and direct vision. After, we proceeded from the external side of the dissection at layer 1, paying particular attention to not damage the ligament, with blunt and anatomical forceps, because of the extreme adhesion between these two layers.
G. Criscenti et al. / Journal of Biomechanics 48 (2015) 4297–4302 Then, the next step was the identification of the medial epicondyle, the adductors tubercle, the insertion of medial collateral ligament, the magnus adductor muscle and the femoral insertion of MPFL (Placella et al., 2014). Finally, The MPFL was isolated such that it was the only connection between the patella and the femur (Fig. 1a). The MPFL was found in all the examined knees. After dissection, the length, the thickness, the width and the cross-sectional areas (CSAs) of the MPFL at patellar insertion, mid-substance, and femoral insertion were measured using a digital caliper (accuracy of 0.02 mm and resolution of 0.01 mm) and a digital micrometer (accuracy of 0.01 mm and resolution of 0.01 mm). All the measurements were repeated five times and the average was used for our purpose. Before the tests, the specimens were left in 37° saline bath for 30 min.
2.4. Stress relaxation tests A total of 12 isolated ligaments were tested; specimens were rectangular shaped with surgical scalpel so that the length-to-width aspect ratio (4:1) provided uniform tensile stress in the region where the strain was measured (Woo et al., 1976). All the specimens had a constant CSA and we assumed they had a rectangular shape (Fig. 1b). The isolated ligament was fixed with cyanoacrylate and sandpaper in a standard clamps and it was aligned to a 5 kN load cell of an Instron 5965 materials-testing machine. A custom 2D optical system was used to evaluate the elongation of the isolated ligament. The system consisted of a camera and four markers (3 mm diameter) which have been positioned and vertically aligned on two different areas of the MPFL to evaluate the ligament elongation and eventual slippage and on two aligned points on the clamps as references. In order to reduce tissue hysteresis, the specimens were preloaded with a force of 1 N and preconditioned by a series of 10 cycles starting from the undeformed position (0%) up to the strain of 3% with a strain rate of 0.1% s 1 (Woo et al., 1986). Subsequently, a stress relaxation test was performed by elongating the MPFL to 6% strain with a strain rate of 0.3% s 1 and held for 60 min (Fig. 2). During the test, samples were gently wetted using a water spray in order to avoid dehydration and the consequent dimensional changes. The slow loading rate was selected to prevent the possible errors as inaccurate strain measurements related to fast strain rate (Funk et al., 2000). It implies a linear behavior of strain–time curve before t0 and a constant one for t4t0. The time until the peak load was t0 ¼19.9 s. The total percentage of stress relaxation was defined as (Eq (6)): %SR ¼
σ ðt 0 Þ σ ðt 1 Þ ∙100 σ ðt 0 Þ
ð6Þ
where σ ðt 0 Þ was the peak stress at t0 and σ ðt 1 Þ was the stress measured at the end of the test.
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3. Results During the preconditioning phase, the samples showed the typical hysteresis of a biological soft-tissue (Fig. 4). After 10 cycles, the loading and unloading curves became repeatable and the stress relaxation test was performed. The constants describing the instantaneous elastic response (A and B) and the reduced relaxation function (C, τ1 ; τ2 ) are listed in Table 1. The MPFL showed a linear increase of strain with time in the loading phase of the stress relaxation test and, at the end of this phase, the corresponding stress was 4.17 3.2 MPa and showed a non-linear trend. The most relaxation occurs within the first 20 min (Fig. 3) and, after 60 min, the total stress relaxation was 32.77 4.7%. The mQLV theory allowed a suitable fit of the experimental data for each sample; the R2 was always greater than 0.990 (Fig. 5). The late relaxation time constant τ2 is three orders of magnitude higher than τ1 suggesting that the assumption of a continuous relaxation spectrum is valid. The obtained constants A, B, C, τ1 , and τ2 validated the model predicting with high accuracy the response of the MPFL from the cyclic stress relaxation test. In particular, the initial peak stresses were predicted with higher accuracy than the final ones. In the final part of the cyclic test, the theoretical peaks reached the equilibrium faster than the experimental ones. Fig. 6 shows the best (a) and worst (b) predictions of the experimental peak stresses of a cyclic loading history based on the constant obtained from the stress relaxation experiment. The difference between theoretical and experimental peak stresses was between 0.5% and 3.1% for the best prediction and 9.7% and 16.4% for the worst one. In particular, the higher error was found in the prediction of the initial peak stress and it was 6.5 76.1%.
4. Discussion In this study, the time- and history-dependent viscoelastic behavior of the human medial patello-femoral ligament was
2.5. Validation of the constants To validate the obtained constants for the MPFL, a cyclic stress relaxation test was performed on 3 samples. In particular, the samples were subjected to 10 cycles of strain between 4% and 6% with a strain rate of 0.3% s 1 and the corresponding peak stresses were recorded (Fig. 3) (Woo et al., 1981). The difference between theoretical and experimental peak stresses (%diff) was defined as (Eq. (7)) (Abramowitch and Woo, 2004): %diff ¼
σ Theoretical σ Experimental ∙100 σ Experimental
ð7Þ
The constants were considered acceptable if the difference was lower than 15%.
Fig. 1. MPFL dissected (a) and isolated (b) specimen.
Fig. 2. Strain and stress data versus time depicting a typical static stress relaxation test.
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Fig. 4. Preconditioning phase of the isolated MPFL.
Fig. 3. Strain and stress data versus time depicting a typical cyclic stress relaxation test.
described by a modified Fung’s quasi-linear viscoelastic theory based on Abramowitch approach (Abramowitch and Woo, 2004). In particular, the five constants of QLV theory (A, B, C, τ1 , τ2 ) were determined through stress relaxation experiments and utilizing slow strain rate in order to reduce errors related to fast ramp times. Finally, a cyclic stress relaxation test was performed to validate the constants comparing theoretical and experimental peak stresses. This result demonstrated the validity of the QLV constants obtained for the MPFL for all specimens at the strain level used in this study. However, until the 90’s, limited information was known about the function of MPFL, and many authors discussed about its existence or considered it as an inconsistent structure of the knee (Reider et al., 1981). No studies related to the viscoelastic properties of the medial patello-femoral ligament are present in literature whereas different studies were performed to evaluate the viscoelastic behavior of other tissues using the quasi-linear viscoelastic theory. Elliott et al. (2003) studied the quasi-linear viscoelastic properties in transgenic mouse tail tendons providing quantitative evidence for structure–function relationship and the role of proteoglycan in viscoelasticity. The viscoelastic properties of human patellar tendon in young and old specimens were evaluated by Johnson et al. (1994) and by Lyon et al. (1988) while Sauren et al. (1983) studied the viscoelastic behavior of porcine aortic valve tissues. BonifasiLista et al. (2005) studied the viscoelasticity of the human medial collateral ligament under longitudinal, transverse and shear loading while Abramowitch et al. (2004) performed the quasilinear viscoelastic characterization of MCL in a goat model. Comparing the five constants obtained in the previous studies related to ligaments and tendons, our estimations were similar to the ones obtained in Abramowitch and Woo (2004) and Abramowitch et al. (2004). In particular, the late relaxation constant τ2 was highly sensitive to the strain rate (Abramowitch and Woo, 2004) and it had the same order of magnitude of the one estimated in the previous works.
Considering the validation phase, the obtained results were similar to those reported by Abramowitch et al. (2004) and Woo et al. (1981). These results demonstrated the accuracy of constants estimation and allowed to compare the properties of the MPFL with other ligamentous tissues, confirming the hypothesis to consider it as a ligament. In addition, the quasi-linear viscoelastic properties of MPFL obtained in this study can help in the selection of the best repair, not only evaluating the quasi static parameters (such us the ultimate load) but comparing the graft behavior during the entire test. Furthermore, recent developments in TE demonstrated that the appropriate tensile and viscoelastic properties of 3D scaffolds are fundamental for cell proliferation and differentiation (Engler et al., 2006; Tse and Engler, 2011). For this reason, the complete mechanical characterization of the tissue is a necessary starting point for the design of a scaffold for the regeneration of the MPFL. Since natural ligament structure is characterized by bands of dense collagen fibers, textile grafts, with several geometries (woven, knitted and braided) are currently under investigation: in general, their properties are closely related to the characteristics of the native fibers and their macroscopic organization (Ge et al., 2005). Another technique that is currently becoming very popular in ligament TE is electrospinning (ESP), a process consists in the use of an high voltage electrostatic field to produce an electrically charged jet from a polymer solution, which by drying during evaporation, lead to the formation of nanofibers (Hammoudi and Temenoff, 2011). One of the more remarkable advantages of this technique is the possibility to produce nanofibrous matrices in a continuous and scalable way. ESP allows the fabrication of fibers with a diameter within the range of both nanometers and micrometers that could be deposited in a random or aligned fashion. Furthermore, it is a cost effective procedure, compared to the other methods (Lee et al., 2005; Hwang et al., 2009; Lanfer et al., 2009). After the seeding on fibrous scaffolds, cells are proved to spontaneously align with the direction of the fibers, depositing abundant ECM rich in collagen type I and III (Vaquette et al., 2010). By varying the ESP working parameters (e.g. the electric voltage, the flow rate), as well as the geometry of the collector (and thus the fiber alignment), and raw materials it is possible to modulate the mechanical properties of the scaffold in a wide range (Huang et al., 2003). Hence, the presented characterization of the MPFL can be used to create a successful scaffold, with similar mQLV constants, by controlling the fabrication parameters.
G. Criscenti et al. / Journal of Biomechanics 48 (2015) 4297–4302
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Table1 Constants estimation of human MPFL, and comparison with goat MCL, used in by Abramowitch for model validation (Abramowitch et al., 2004). Values are indicated as mean 7 standard deviation.
Human MPFL Goat MCL
A (MPa)
B
C
τ 1 (s)
τ 2 (s)
1.217 0.96 0.7 70.6
26.03 7 4.16 54.1 718.9
0.117 0.02 0.201 70.106
6.32 7 1.76 0.80 7 0.43
903.477 504.73 1269 7 291
Fig. 5. Typical curve fitting of the experimental data with the mQLV theory.
Fig. 6. Best (a) and worst (b) predictions of the experimental peak stresses of a cyclic loading history based on the constant obtained from the stress relaxation experiment.
Conflict of interest statement All authors confirm they have no financial or other conflict of interest relevant to this study.
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Nomura, E., 1999. Classification of lesions of the medial patello-femoral ligament in patellar dislocation. Int. Orthop. 23 (5), 260–263. Ostermeier, S., Stukenborg-Colsman, C., Hurschler, C., Wirth, C.-J., 2006. In vitro investigation of the effect of medial patellofemoral ligament reconstruction and medial tibial tuberosity transfer on lateral patellar stability. Arthroscopy 22 (3), 308–319. http://dx.doi.org/10.1016/j.arthro.2005.09.024. Panagiotopoulos, E., Strzelczyk, P., Herrmann, M., Scuderi, G., 2006. Cadaveric study on static medial patellar stabilizers: the dynamizing role of the vastus medialis obliquus on medial patellofemoral ligament. Knee Surg. Sports Traumatol. Arthrosc. 14 (1), 7–12. http://dx.doi.org/10.1007/s00167-005-0631-z. Placella, G., Tei, M.M., Sebastiani, E., Criscenti, G., Speziali, A., Mazzola, C., Georgoulis, A., Cerulli, G., 2014. Shape and size of the medial patellofemoral ligament for the best surgical reconstruction: a human cadaveric study. Knee Surg. Sports Traumatol. Arthrosc. 22, 2327–2333. Provenzano, P., Lakes, R., Keenan, T., Vanderby Jr., R., 2001. Nonlinear ligament viscoelasticity. Ann. Biomed. Eng. 29, 908–914. Reider, B., Marshall, J.L., Koslin, B., Ring, B., Girgis, F.G., 1981. The anterior aspect of the knee joint. J. Bone Joint Surg. Am. 63, 351–356. Rodrigues, M.T., Reis, R.L., Gomes, M.E., 2013. Engineering tendon and ligament tissues: present developments towards successful clinical products. J. Tissue Eng. Regen. Med. 7, 673–686. Rousseau, E.P.M., Sauren, A.A.H.J., Van Hout, M.C., van Steenhoven, A.A., 1983. Elastic and viscoelastic material behavior of fresh and glutaraldehyde fixed porcine aortic valve tissue. J. Biomech. 16, 339–348. Sachlos, E., Czernuszka, J.T., 2003. Making tissue engineering scaffolds work. Review: the application of solid freeform fabrication technology to the production of tissue engineering scaffolds. Eur. Cells Mater. 5, 29–39, discussion 39–40. Sauren, A.A., Rousseau, E.P., 1983. A concise sensitivity analysis of the quasi-linear viscoelastic model proposed by Fung. J. Biomech. Eng. 105, 92–95.
Sauren, A.A.H.J., van Hout, M.C., van Steenhoven, A.A., Veldpaus, F.E., Janssen, J.D., 1983. The mechanical properties of porcine aortic valve tissues. J. Biomech. 16, 327–337. Thomopoulos, S., Williams, G.R., Soslowsky, L.J., 2003. Tendon to bone healing: differences in biomechanical, structural, and compositional properties due to a range of activity levels. J. Biomech. Eng., 125; , pp. 106–113 2003. Tse, J.R., Engler, A.J., 2011. Stiffness gradients mimicking in vivo tissue variation regulate mesenchymal stem cell fate. PloS One 6 (1), e15978. Vaquette, C., Kahn, C., Frochot, C., Nouvel, C., Six, J.-L., De Isla, N., Luo, L.-H., CooperWhite, J., Rahouadj, R., Wang, X., 2010. Aligned poly(L-lactic- co-e-caprolactone) electrospun microfibers and knitted structure: a novel composite scaffold for ligament tissue engineering. J. Biomed. Mater. Res. A 94, 1270–1282. Warren, L.F., Marshall, J.L., 1979. The supporting structures and layers on the medial side of the knee: an anatomical analysis. J. Bone Joint Surg. Am. 61, 56–62. Woo, S.L., Akeson, W.H., Jemmott, G.F., 1976. Measurements of nonhomogeneous, directional mechanical properties of articular cartilage in tension. J. Biomech. 9, 785–791. Woo, S.L., Debski, R.E., Withrow, J.D., Janaushek, M.A., 1999. Current concepts biomechanics of knee ligaments. Am. J. Sports Med. 27, 533–543. Woo, S.L., Orlando, C.A., Gomez, M.A., Frank, C.B., Akeson, W.H., 1986. Tensile properties of the medial collateral ligament as a function of age. J. Orthop. Res. 4, 133–141. Woo, S.L.-Y., Gomez, M.A., Akeson, W.H., 1981. The time and history-dependent viscoelastic properties of the canine medial collateral ligament. J. Biomech. Eng. 103, 293–298. Woo, S.L.-Y., Simon, B.R., Kuei, S.C., Akeson, W.H., 1980. Quasi-linear viscoelastic properties of normal articular cartilage. J. Biomech. Eng. 102, 85–90.
Contents lists available at ScienceDirect
Journal of Biomechanics journal homepage: www.elsevier.com/locate/jbiomech www.JBiomech.com
Quasi-linear viscoelastic properties of the human medial patello-femoral ligament G. Criscenti a,b, C. De Maria b,n, E. Sebastiani a,d, M. Tei a,d, G. Placella a,d, A. Speziali a,d, G. Vozzi b,c, G. Cerulli a,d a
Istituto di Ricerca Traslazionale per l’Apparato Locomotore – Nicola Cerulli – LPMRI, via A. Einstein 12, 52100 Arezzo, Italy Research Center “E. Piaggio”, University of Pisa, Largo Lucio Lazzarino 1, 56126 Pisa, Italy c Department of Ingegneria dell’Informazione, University of Pisa, via G. Caruso 16, 56126 Pisa, Italy d Department of Orthopedic surgery, University of Perugia, via S. Andrea delle Fratte 1, 06134 Perugia, Italy b
art ic l e i nf o
a b s t r a c t
Article history: Accepted 26 October 2015
The evaluation of viscoelastic properties of human medial patello-femoral ligament is fundamental to understand its physiological function and contribution as stabilizer for the selection of the methods of repair and reconstruction and for the development of scaffolds with adequate mechanical properties. In this work, 12 human specimens were tested to evaluate the time- and history-dependent non linear viscoelastic properties of human medial patello-femoral ligament using the quasi-linear viscoelastic (QLV) theory formulated by Fung et al. (1972) and modified by Abramowitch and Woo (2004). The five constant of the QLV theory, used to describe the instantaneous elastic response and the reduced relaxation function on stress relaxation experiments, were successfully evaluated. It was found that the constant A was 1.21 70.96 MPa and the dimensionless constant B was 26.03 74.16. The magnitude of viscous response, the constant C, was 0.11 70.02 and the initial and late relaxation time constants τ1 and τ2 were 6.32 71.76 s and 903.47 7504.73 s respectively. The total stress relaxation was 32.7 74.7%. To validate our results, the obtained constants were used to evaluate peak stresses from a cyclic stress relaxation test on three different specimens. The theoretically predicted values fit the experimental ones demonstrating that the QLV theory could be used to evaluate the viscoelastic properties of the human medial patello-femoral ligament. & 2015 Elsevier Ltd. All rights reserved.
Keywords: Medial patello-femoral ligament (MPFL) Ligament biomechanics Ligament mechanical properties Quasi-linear viscoelastic theory QLV theory Tissue Engineering
1. Introduction The medial patello-femoral ligament (MPFL) is the primary static stabilizer of the patello-femoral joint and is commonly injured during a first time lateral patellar dislocation (Ahmad et al, 2000; Desio et al., 1998; Greiwe et al., 2010; Nomura 1999; Ostermeier et al., 2006; Panagiotopoulos et al., 2006). The ligament natural healing process is often inadequate to restore its structural, mechanical and biological properties and, in most cases, it results in the formation of scar tissue with inferior mechanical properties (Hammoudi and Temenoff, 2011). Two different strategies can be followed to reach a complete and long-term functional repair of the damaged tissue: reconstructive and regenerative approaches. Different points of view regarding reconstructive methods have been presented, focusing on the graft choice and tension, knee flexion angle and fixation n
Corresponding author: Tel.: þ 39 50 2217056; fax: þ 39 50 2217051. E-mail address: [email protected] (C. De Maria).
http://dx.doi.org/10.1016/j.jbiomech.2015.10.042 0021-9290/& 2015 Elsevier Ltd. All rights reserved.
methods. Many of these methods have been compared, studying the stiffness of the reconstructed ligament and the elongation of the graft after cyclic tests, but results are controversial (Mountney et al., 2005; Lenschow et al., 2013). The regenerative approach represents a new strategy based on Tissue Engineering (TE) that aims at fabricating an immunological tolerant tissue substitute to permanently restore the functionality of the damaged one, without the need for supplementary therapies (Sachlos and Czernuszka, 2003). Nevertheless, important challenges must be solved to obtain complete ligament repair that will lead to a clinically effective and commercially successful application (Rodrigues et al., 2013): it is well known that the fate of cells seeded onto a scaffold is extremely dependent on the mechanical properties of this substrate (Engler et al., 2006; Tse and Engler, 2011). Thus, determining the viscoelastic properties of the MPFL is a critical point to understand its physiological function and its contribution as patellar stabilizer in order to select the optimal replacement graft and to define the correct mechanical parameters for the design and fabrication of scaffolds.
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The viscoelastic behavior of biological tissues depends on the complex interaction among collagen, elastin, proteoglycans and water (Woo et al., 1999) and it was studied for different tissues as aortic valve tissues (Rousseau et al., 1983; Sauren et al., 1983), cartilage (Woo et al., 1980), skeletal muscle (Best et al., 1994), bone (Lakes et al., 1979; Lakes and Katz, 1979), ligament (Abramowitch and Woo, 2004; Haut and Little 1969; Provenzano et al., 2001; Woo et al., 1981) and tendon (Atkinson et al., 1999; Haut and Little 1972). Ligaments are viscoelastic and display time-dependent and load-history-dependent mechanical behavior (Fung et al., 1972) described by hysteresis, creep and stress relaxation. Hysteresis is the energy dissipation with continuous phases of loading and unloading; creep is the increase of length over time under a constant load while stress relaxation is a decrease in the load when the ligament is held at a fixed elongation (Woo et al., 1999). The quasi-linear viscoelastic (QLV) theory formulated by Fung and modified by Abramowitch and Woo (2004) is the most used approach to describe the viscoelastic behavior of biological tissues (Carew et al., 1999; Elliott et al., 2003; Fung et al., 1972; Fung 1993; Funk et al., 2000; Kwan et al., 1993; Sauren et al., 1983; Thomopoulos et al., 2003; Woo et al., 1981), especially ligaments and tendons as the porcine anterior cruciate ligament (Kwan et al., 1993), human patellar tendon (Johnson et al., 1994) and the canine medial collateral ligament (MCL) (Abramowitch et al., 2004). The theory is based on the assumption of an instantaneous change of strain in the load application phase, although this is experimentally impossible to be performed. For this reason, many researchers tried to use high strain rates assuming that no relaxation occurred during the loading phase. With this approach, the parameters estimation was affected by high variability because experimental errors related to high strain rate (vibration of the testing machine, errors in reaching the final position, such as the overshoot) (Funk et al., 2000), obtaining estimated constants that were not comparable with other ones obtained from other biological tissues and in other laboratories. The modified quasi-linear viscoelastic theory (mQLV) allows to evaluate the reduced relaxation function and elastic response performing experiments with slow strain rate (Abramowitch and Woo, 2004). In the present work, we used this approach, avoiding errors related to the fast movements of the testing machines. Thus, the objective of this study was to determine the five constant of the modified QLV theory related to the MPFL, used to describe the instantaneous elastic response and the reduced relaxation function in stress relaxation experiments, and thus provide a complete characterization of its viscoelastic behavior. These values can be the target point for the selection of the optimal reconstructive and regenerative procedures to restore MPFL function. 2. Materials and methods 2.1. Modified quasi-linear viscoelastic (mQLV) theory The QLV theory formulated by Fung et al. (1972) and modified by Abramowitch and Woo (2004) was used to describe the time- and history-dependent viscoelastic and nonlinear mechanical properties of the MPFL. The Fung’s theory assumes that the general stress relaxation behavior of soft-tissue is expressed by the convolution integral of the reduced relaxation function G(t) and the elastic response σe(ε) (Eq. (1)): Z t ∂σ e ðεÞ ∂ε σ ðt Þ ¼ G ðt τ Þ ∂τ ð1Þ ∂ε ∂ τ 1 The instantaneous elastic response represents the maximum stress in response to an instantaneous step input of strain (Abramowitch and Woo, 2004) and is described by the exponential function (Eq. (2)): σ e ðϵÞ ¼ A eBϵ 1 ð2Þ where A and B are the material linear and nonlinear constants respectively. In
particular, the constant B represents the rate of change of the slope of the stress– strain curve and the product AB is the initial slope of the curve (Woo et al., 1981). The reduced relaxation function represents the time dependent stress response of the tissue normalized by the stress present at the time of the strain step input (Fung et al., 1972) and it was proposed by Fung to describe the sensitivity of soft tissues to strain rate. This equation is derived from the standard linear solid model that was chosen because it presents a finite plateau for a long relaxation time. In order to describe a constant spectrum of relaxation, the form of the reduced relaxation function is (Abramowitch and Woo, 2004) (Eq. (3)):
Gðt Þ ¼
h i 1þ C E1 τt2 E1 τt1
ð3Þ
1 þ Clnðττ21 Þ
R 1 z where E1 ðyÞ ¼ y e z dz is the exponential integral, and C, τ1 and τ2 are material constants with τ1 { τ2 . The dimensionless constant C represents the magnitude of viscous effects and is related to the percentage of relaxation (Abramowitch et al., 2004) while the time constants τ1 and τ2 describe the initial and the late relaxation of the slope of the stress-relaxation curve at early and late time periods, respectively (Sauren and Rousseau, 1983). According to the methodology proposed by Abramowitch and Woo (2004) slow strain rates during the loading phase of a stress relaxation test can be used. In particular, the stress resulting from the loading phase of the stress relaxation test with a constant strain rate γ over the time 0o t o t 0 is (Eq. (4)):
σ ðt Þ ¼
ABγ 1þ C ln τ2 =τ1
Z
t
tτ t τ 1 þ C E1 E1 eBγτ ∂τ
τ2
0
τ1
ð4Þ
The stress resulting from the relaxation phase (from t 0 to t ¼ 1) of the stress relaxation test is (Eq. (5)):
σ ðt Þ ¼
ABγ 1þ C ln τ2 =τ1
Z
1
t0
t τ t τ 1 þ C E1 E1 eBγτ ∂τ
τ2
τ1
ð5Þ
where A, B, C, τ1 , τ2 are the five constants of the QLV theory that need to be determined. This method showed a unique solution that was minimally sensitive to systematic and random experimental noise (Abramowitch et al., 2004). 2.2. Constant estimation and validation The five constants of the QLV theory were determined from the experimental data. Stress–strain data obtained from the loading portion of the stress relaxation experiments were used to fit Eq. (2) in order to determine constant A (Abramowitch et al., 2004). Constants B, C, τ1 and τ2 were obtained substituting constant A in Eqs. (4) and (5) and simultaneously fitting the stress–time data related to the loading and relaxation portions of the stress relaxation experiment (Abramowitch and Woo, 2004; Abramowitch et al., 2004). A modified Levenberg–Marquardt algorithm was used for non-linear optimization process (Abramowitch et al., 2004). To validate the results, the obtained constants were used to evaluate peak stresses from separate independent cyclic stress relaxation tests. In particular, the constants obtained from each specimen with the stress relaxation experiments were separately used in Eq. (1) to define the stress response of each sample and theoretical peak stresses at each cycle were determined (Woo et al., 1981). Subsequently, the theoretical peak stresses and experimental ones obtained during cyclic stress-relaxation test for each specimen were compared. 2.3. Preparation of the specimens A total of 15 human cadaveric knees from 5 women and 9 men with a mean age of 7678 years (range 68–84 years) were used in this experiment. None of these showed patellar instability, knee injuries, surgical procedures or arthritic deformations. The Nicola’s Foundation Onlus Ethics Committee has given its approval for this study. The cadavers were dissected after they have been stored for 24 h at 4 °C and then were preserved in a sterile gauze, sealed in a polyethylene bag, labeled and stored at 18 °C. They were thawed at a temperature of 4 °C when required. Warren and Marshall’s three-layer classification of the medial side of the knee was used to dissect the specimens (Warren and Marshall, 1979). All dissections started with a midline incision of the skin detaching it from the subcutaneous fascia. The joint capsule was accessed extending through a lateral incision at the vastus lateralis muscle extending laterally at the parapatellar side and at the lateral compartment of the tibia which was cut proximally to distally. The femur was tipped up detaching the muscle bundles. The isolated single muscles of the quadriceps were left inserted in their distal insertions and used as landmarks. The patella was rotated and the posteromedial capsule was opened from the inside, detaching and isolating the synovial capsule in the 3rd layer from the 2nd one. The fibers of the MPFL were identified by palpation and direct vision. After, we proceeded from the external side of the dissection at layer 1, paying particular attention to not damage the ligament, with blunt and anatomical forceps, because of the extreme adhesion between these two layers.
G. Criscenti et al. / Journal of Biomechanics 48 (2015) 4297–4302 Then, the next step was the identification of the medial epicondyle, the adductors tubercle, the insertion of medial collateral ligament, the magnus adductor muscle and the femoral insertion of MPFL (Placella et al., 2014). Finally, The MPFL was isolated such that it was the only connection between the patella and the femur (Fig. 1a). The MPFL was found in all the examined knees. After dissection, the length, the thickness, the width and the cross-sectional areas (CSAs) of the MPFL at patellar insertion, mid-substance, and femoral insertion were measured using a digital caliper (accuracy of 0.02 mm and resolution of 0.01 mm) and a digital micrometer (accuracy of 0.01 mm and resolution of 0.01 mm). All the measurements were repeated five times and the average was used for our purpose. Before the tests, the specimens were left in 37° saline bath for 30 min.
2.4. Stress relaxation tests A total of 12 isolated ligaments were tested; specimens were rectangular shaped with surgical scalpel so that the length-to-width aspect ratio (4:1) provided uniform tensile stress in the region where the strain was measured (Woo et al., 1976). All the specimens had a constant CSA and we assumed they had a rectangular shape (Fig. 1b). The isolated ligament was fixed with cyanoacrylate and sandpaper in a standard clamps and it was aligned to a 5 kN load cell of an Instron 5965 materials-testing machine. A custom 2D optical system was used to evaluate the elongation of the isolated ligament. The system consisted of a camera and four markers (3 mm diameter) which have been positioned and vertically aligned on two different areas of the MPFL to evaluate the ligament elongation and eventual slippage and on two aligned points on the clamps as references. In order to reduce tissue hysteresis, the specimens were preloaded with a force of 1 N and preconditioned by a series of 10 cycles starting from the undeformed position (0%) up to the strain of 3% with a strain rate of 0.1% s 1 (Woo et al., 1986). Subsequently, a stress relaxation test was performed by elongating the MPFL to 6% strain with a strain rate of 0.3% s 1 and held for 60 min (Fig. 2). During the test, samples were gently wetted using a water spray in order to avoid dehydration and the consequent dimensional changes. The slow loading rate was selected to prevent the possible errors as inaccurate strain measurements related to fast strain rate (Funk et al., 2000). It implies a linear behavior of strain–time curve before t0 and a constant one for t4t0. The time until the peak load was t0 ¼19.9 s. The total percentage of stress relaxation was defined as (Eq (6)): %SR ¼
σ ðt 0 Þ σ ðt 1 Þ ∙100 σ ðt 0 Þ
ð6Þ
where σ ðt 0 Þ was the peak stress at t0 and σ ðt 1 Þ was the stress measured at the end of the test.
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3. Results During the preconditioning phase, the samples showed the typical hysteresis of a biological soft-tissue (Fig. 4). After 10 cycles, the loading and unloading curves became repeatable and the stress relaxation test was performed. The constants describing the instantaneous elastic response (A and B) and the reduced relaxation function (C, τ1 ; τ2 ) are listed in Table 1. The MPFL showed a linear increase of strain with time in the loading phase of the stress relaxation test and, at the end of this phase, the corresponding stress was 4.17 3.2 MPa and showed a non-linear trend. The most relaxation occurs within the first 20 min (Fig. 3) and, after 60 min, the total stress relaxation was 32.77 4.7%. The mQLV theory allowed a suitable fit of the experimental data for each sample; the R2 was always greater than 0.990 (Fig. 5). The late relaxation time constant τ2 is three orders of magnitude higher than τ1 suggesting that the assumption of a continuous relaxation spectrum is valid. The obtained constants A, B, C, τ1 , and τ2 validated the model predicting with high accuracy the response of the MPFL from the cyclic stress relaxation test. In particular, the initial peak stresses were predicted with higher accuracy than the final ones. In the final part of the cyclic test, the theoretical peaks reached the equilibrium faster than the experimental ones. Fig. 6 shows the best (a) and worst (b) predictions of the experimental peak stresses of a cyclic loading history based on the constant obtained from the stress relaxation experiment. The difference between theoretical and experimental peak stresses was between 0.5% and 3.1% for the best prediction and 9.7% and 16.4% for the worst one. In particular, the higher error was found in the prediction of the initial peak stress and it was 6.5 76.1%.
4. Discussion In this study, the time- and history-dependent viscoelastic behavior of the human medial patello-femoral ligament was
2.5. Validation of the constants To validate the obtained constants for the MPFL, a cyclic stress relaxation test was performed on 3 samples. In particular, the samples were subjected to 10 cycles of strain between 4% and 6% with a strain rate of 0.3% s 1 and the corresponding peak stresses were recorded (Fig. 3) (Woo et al., 1981). The difference between theoretical and experimental peak stresses (%diff) was defined as (Eq. (7)) (Abramowitch and Woo, 2004): %diff ¼
σ Theoretical σ Experimental ∙100 σ Experimental
ð7Þ
The constants were considered acceptable if the difference was lower than 15%.
Fig. 1. MPFL dissected (a) and isolated (b) specimen.
Fig. 2. Strain and stress data versus time depicting a typical static stress relaxation test.
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Fig. 4. Preconditioning phase of the isolated MPFL.
Fig. 3. Strain and stress data versus time depicting a typical cyclic stress relaxation test.
described by a modified Fung’s quasi-linear viscoelastic theory based on Abramowitch approach (Abramowitch and Woo, 2004). In particular, the five constants of QLV theory (A, B, C, τ1 , τ2 ) were determined through stress relaxation experiments and utilizing slow strain rate in order to reduce errors related to fast ramp times. Finally, a cyclic stress relaxation test was performed to validate the constants comparing theoretical and experimental peak stresses. This result demonstrated the validity of the QLV constants obtained for the MPFL for all specimens at the strain level used in this study. However, until the 90’s, limited information was known about the function of MPFL, and many authors discussed about its existence or considered it as an inconsistent structure of the knee (Reider et al., 1981). No studies related to the viscoelastic properties of the medial patello-femoral ligament are present in literature whereas different studies were performed to evaluate the viscoelastic behavior of other tissues using the quasi-linear viscoelastic theory. Elliott et al. (2003) studied the quasi-linear viscoelastic properties in transgenic mouse tail tendons providing quantitative evidence for structure–function relationship and the role of proteoglycan in viscoelasticity. The viscoelastic properties of human patellar tendon in young and old specimens were evaluated by Johnson et al. (1994) and by Lyon et al. (1988) while Sauren et al. (1983) studied the viscoelastic behavior of porcine aortic valve tissues. BonifasiLista et al. (2005) studied the viscoelasticity of the human medial collateral ligament under longitudinal, transverse and shear loading while Abramowitch et al. (2004) performed the quasilinear viscoelastic characterization of MCL in a goat model. Comparing the five constants obtained in the previous studies related to ligaments and tendons, our estimations were similar to the ones obtained in Abramowitch and Woo (2004) and Abramowitch et al. (2004). In particular, the late relaxation constant τ2 was highly sensitive to the strain rate (Abramowitch and Woo, 2004) and it had the same order of magnitude of the one estimated in the previous works.
Considering the validation phase, the obtained results were similar to those reported by Abramowitch et al. (2004) and Woo et al. (1981). These results demonstrated the accuracy of constants estimation and allowed to compare the properties of the MPFL with other ligamentous tissues, confirming the hypothesis to consider it as a ligament. In addition, the quasi-linear viscoelastic properties of MPFL obtained in this study can help in the selection of the best repair, not only evaluating the quasi static parameters (such us the ultimate load) but comparing the graft behavior during the entire test. Furthermore, recent developments in TE demonstrated that the appropriate tensile and viscoelastic properties of 3D scaffolds are fundamental for cell proliferation and differentiation (Engler et al., 2006; Tse and Engler, 2011). For this reason, the complete mechanical characterization of the tissue is a necessary starting point for the design of a scaffold for the regeneration of the MPFL. Since natural ligament structure is characterized by bands of dense collagen fibers, textile grafts, with several geometries (woven, knitted and braided) are currently under investigation: in general, their properties are closely related to the characteristics of the native fibers and their macroscopic organization (Ge et al., 2005). Another technique that is currently becoming very popular in ligament TE is electrospinning (ESP), a process consists in the use of an high voltage electrostatic field to produce an electrically charged jet from a polymer solution, which by drying during evaporation, lead to the formation of nanofibers (Hammoudi and Temenoff, 2011). One of the more remarkable advantages of this technique is the possibility to produce nanofibrous matrices in a continuous and scalable way. ESP allows the fabrication of fibers with a diameter within the range of both nanometers and micrometers that could be deposited in a random or aligned fashion. Furthermore, it is a cost effective procedure, compared to the other methods (Lee et al., 2005; Hwang et al., 2009; Lanfer et al., 2009). After the seeding on fibrous scaffolds, cells are proved to spontaneously align with the direction of the fibers, depositing abundant ECM rich in collagen type I and III (Vaquette et al., 2010). By varying the ESP working parameters (e.g. the electric voltage, the flow rate), as well as the geometry of the collector (and thus the fiber alignment), and raw materials it is possible to modulate the mechanical properties of the scaffold in a wide range (Huang et al., 2003). Hence, the presented characterization of the MPFL can be used to create a successful scaffold, with similar mQLV constants, by controlling the fabrication parameters.
G. Criscenti et al. / Journal of Biomechanics 48 (2015) 4297–4302
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Table1 Constants estimation of human MPFL, and comparison with goat MCL, used in by Abramowitch for model validation (Abramowitch et al., 2004). Values are indicated as mean 7 standard deviation.
Human MPFL Goat MCL
A (MPa)
B
C
τ 1 (s)
τ 2 (s)
1.217 0.96 0.7 70.6
26.03 7 4.16 54.1 718.9
0.117 0.02 0.201 70.106
6.32 7 1.76 0.80 7 0.43
903.477 504.73 1269 7 291
Fig. 5. Typical curve fitting of the experimental data with the mQLV theory.
Fig. 6. Best (a) and worst (b) predictions of the experimental peak stresses of a cyclic loading history based on the constant obtained from the stress relaxation experiment.
Conflict of interest statement All authors confirm they have no financial or other conflict of interest relevant to this study.
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