Kevin L. Troyer

Human cervical spine ligaments exhibit fully nonlinear viscoelastic behavior.

Spinal ligaments provide stability and contribute to spinal motion patterns. These hydrated tissues exhibit time-dependent behavior during both static and dynamic loading regimes. Therefore, accurate viscoelastic characterization of these ligaments is requisite for development of computational analogues that model and predict time-dependent spine behavior. The development of accurate viscoelastic models must be preceded by rigorous, empirical evidence of linear viscoelastic, quasi-linear viscoelastic (QLV) or fully nonlinear viscoelastic behavior. This study utilized multiple physiological loading rates (frequencies) and strain amplitudes via cyclic loading and stress relaxation experiments in order to determine the viscoelastic behavior of the human lower cervical spine anterior longitudinal ligament, the posterior longitudinal ligament and the ligamentum flavum. The results indicated that the cyclic material properties of these ligaments were dependent on both strain amplitude and frequency. This strain amplitude-dependent behavior cannot be described using a linear viscoelastic formulation. Stress relaxation experiments at multiple strain magnitudes indicated that the shape of the relaxation curve was strongly dependent on strain magnitude, suggesting that a QLV formulation cannot adequately describe the comprehensive viscoelastic response of these ligaments. Therefore, a fully nonlinear viscoelastic formulation is requisite to model these lower cervical spine ligaments during activities of daily living.

Viscoelastic effects during loading play an integral role in soft tissue mechanics

Viscoelastic relaxation during tensioning is an intrinsic protective mechanism of biological soft tissues. However, current viscoelastic characterization methodologies for these tissues either negate this important behavior or provide correction methods that are severely restricted to a specific viscoelastic formulation and/or assume an a priori (linear) strain ramp history. In order to address these shortcomings, we present a novel finite ramp time correction method for stress relaxation experiments (to incorporate relaxation manifested during loading) that is independent of a specific viscoelastic formulation and can accommodate an arbitrary strain ramp history. We demonstrate transferability of our correction method between viscoelastic formulations by applying it to quasi-linear viscoelastic (QLV) and fully nonlinear viscoelastic constitutive equations. The errors associated with currently accepted methodologies for QLV and fully nonlinear viscoelastic formulations are elucidated. Our correction method is validated by demonstrating the ability of its fitted parameters to predict an independent cyclic experiment across multiple strain amplitudes and frequencies. The results presented herein: (i) indicate that our correction method significantly reduces the errors associated with previous methodologies; and (ii) demonstrate the necessity for the use of a fully nonlinear viscoelastic formulation, which incorporates relaxation manifested during loading, to model the viscoelastic behavior of biological soft tissues.

Nonlinear viscoelasticty plays an essential role in the functional behavior of spinal ligaments

Despite the significant role ligament viscoelasticity plays in functional spinal biomechanics, relatively few studies have been performed to develop constitutive models that explicitly characterize this complex behavior. Unfortunately, the application and interpretation of these previous models are limited due to the use of simplified (quasi-linear) viscoelastic formulations or characterization techniques that have been shown to affect the predictive accuracy of the fitted coefficients. In order to surmount these previous limitations, the current study presents the application of a novel fitting technique (applied to stress relaxation experiments) and nonlinear viscoelastic constitutive formulation to human cervical spine anterior longitudinal ligament (ALL), posterior longitudinal ligament (PLL) and ligamentum flavum (LF). The fitted coefficients were validated by quantifying the ability of the constitutive equation to predict an independent cyclic data set across multiple physiologic strain amplitudes and frequencies. The resulting validated constitutive formulation indicated that the strain-dependent viscoelastic behavior of the longitudinal ligaments (ALL and PLL) was dominated by both the short-term (t=0.1s) and the steady-state (as t→∞) behavior. Conversely, the LF exhibited consistent relaxation behavior across the investigated temporal spectrum. From these data, it can be hypothesized that the unique strain-dependent temporal behavior of these spinal ligaments may be a functional adaptation that minimizes muscular expenditure during quasi-static postures while maximizing structural stability of the spine during transient loading events.

Experimental Characterization and Finite Element Implementation of Soft Tissue Nonlinear Viscoelasticity

Finite element (FE) models of articular joint structures do not typically implement the fully nonlinear viscoelastic behavior of the soft connective tissue components. Instead, contemporary whole joint FE models usually represent the transient soft tissue behavior with significantly simplified formulations that are computationally tractable. The resultant fidelity of these models is greatly compromised with respect to predictions under temporally varying static and dynamic loading regimes. In addition, models based upon experimentally derived nonlinear viscoelastic coefficients that do not account for the transient behavior during the loading event(s) may further reduce the models predictive accuracy. The current study provides the derivation and validation of a novel, phenomenological nonlinear viscoelastic formulation (based on the single integral nonlinear superposition formulation) that can be directly inputted into FE algorithms. This formulation and an accompanying experimental characterization technique, which incorporates relaxation manifested during the loading period of stress relaxation experiments, is compared to a previously published characterization method and validated against an independent analytical model. The results demonstrated that the static and dynamic FE approximations are in good agreement with the analytical solution. Additionally, the predictive accuracy of these approximations was observed to be highly dependent upon the experimental characterization technique. It is expected that implementation of the novel, computationally tractable nonlinear viscoelastic formulation and associated experimental characterization technique presented in the current study will greatly improve the predictive accuracy of the individual connective tissue components for whole joint FE simulations subjected to static and dynamic loading regimes.

Mechanical Comparison of Two Suture Constructs For Extra-Capsular Stifle Stabilization

OBJECTIVE Mechanical evaluation of 2 suture constructs for extracapsular stifle stabilization. STUDY DESIGN In vitro study. SAMPLE POPULATION Crimped interlocking loop constructs (ILC) of 45 kg nylon leader line (NLL) and Orthofiber® (OF). METHODS ILC were tightened to 100 N, then crimp secured. Ramp to failure (n=10/group)-Data were derived from force/displacement plots. Stress-relaxation testing (n=10/group)-ILCs were nondestructively loaded and held at resultant displacement as force data were recorded. Incremental, cyclic loading (n=10/group)-ILCs were loaded (5 cycles/set) starting at 100 N and incrementally increased by 50 N (1 and 3 Hz protocols). Loop tension and elongation were recorded after each set. RESULTS Ramp to failure-initial loop tension was similar (NLL 75.5 ± 9.5 N; OF 68.7 ± 10.4 N, P=.140). Tested OF constructs were stiffer (NLL 125.7 ± 4.0; OF 234.6 ± 25.0 N/mm, P≤.001), had lower yield load (NLL 193.6 ± 13.8; OF 137.3 ± 94.3 N, P≤.001), lower peak load (NLL 873.7 ± 68.6; OF 653.6 ± 80.2 N, P≤.001), and lower elongation at failure (NLL 19.1 ± 1.4; OF 5.2 ± 1.0 mm, P≤.001) and at yield (NLL 1.52 ± 0.2; OF 0.3 ± 0.6 mm, P=.003) than NLL constructs. Yield in NLL ILCs was variable knot tightening/crimp slippage, but only crimp-suture slippage in OF. Stress-relaxation testing-OF demonstrated greater relaxation. Incremental, cyclic loading-induced ILC elongation and tension loss in both groups, independent of loading frequency. NLL lost tension at lower rate, but elongated more than OF. CONCLUSIONS NLL construct is mechanically superior to OF construct.

Nonlinear Viscoelastic Characterization of the Porcine Spinal Cord

Although quasi-static and quasi-linear viscoelastic properties of the spinal cord have been reported previously, there are no published studies that have investigated the fully (strain-dependent) nonlinear viscoelastic properties of the spinal cord. In this study, stress relaxation experiments and dynamic cycling were performed on six fresh porcine lumbar cord specimens to examine their viscoelastic mechanical properties. The stress relaxation data were fitted to a modified superposition formulation and a novel finite ramp time correction technique was applied. The parameters obtained from this fitting methodology were used to predict the average dynamic cyclic viscoelastic behavior of the porcine cord. The data indicate that the porcine spinal cord exhibited fully nonlinear viscoelastic behavior. The average weighted root mean squared error for a Heaviside ramp fit was 2.8 kPa, which was significantly greater (p<0.001) than that of the nonlinear (comprehensive viscoelastic characterization method) fit (0.365 kPa). Further, the nonlinear mechanical parameters obtained were able to accurately predict the dynamic behavior, thus exemplifying the reliability of the obtained nonlinear parameters. These parameters will be important for future studies investigating various damage mechanisms of the spinal cord and studies developing high-resolution finite elements models of the spine.

Implantable microelectromechanical sensors for diagnostic monitoring and post-surgical prediction of bone fracture healing.

The relationship between modern clinical diagnostic data, such as from radiographs or computed tomography, and the temporal biomechanical integrity of bone fracture healing has not been well‐established. A diagnostic tool that could quantitatively describe the biomechanical stability of the fracture site in order to predict the course of healing would represent a paradigm shift in the way fracture healing is evaluated. This paper describes the development and evaluation of a wireless, biocompatible, implantable, microelectromechanical system (bioMEMS) sensor, and its implementation in a large animal (ovine) model, that utilized both normal and delayed healing variants. The in vivo data indicated that the bioMEMS sensor was capable of detecting statistically significant differences (p‐value <0.04) between the two fracture healing groups as early as 21 days post‐fracture. In addition, post‐sacrifice micro‐computed tomography, and histology data demonstrated that the two model variants represented significantly different fracture healing outcomes, providing explicit supporting evidence that the sensor has the ability to predict differential healing cascades. These data verify that the bioMEMS sensor can be used as a diagnostic tool for detecting the in vivo course of fracture healing in the acute post‐treatment period.

The development and validation of a numerical integration method for non-linear viscoelastic modeling

Compelling evidence that many biological soft tissues display both strain- and time-dependent behavior has led to the development of fully non-linear viscoelastic modeling techniques to represent the tissue’s mechanical response under dynamic conditions. Since the current stress state of a viscoelastic material is dependent on all previous loading events, numerical analyses are complicated by the requirement of computing and storing the stress at each step throughout the load history. This requirement quickly becomes computationally expensive, and in some cases intractable, for finite element models. Therefore, we have developed a strain-dependent numerical integration approach for capturing non-linear viscoelasticity that enables calculation of the current stress from a strain-dependent history state variable stored from the preceding time step only, which improves both fitting efficiency and computational tractability. This methodology was validated based on its ability to recover non-linear viscoelastic coefficients from simulated stress-relaxation (six strain levels) and dynamic cyclic (three frequencies) experimental stress-strain data. The model successfully fit each data set with average errors in recovered coefficients of 0.3% for stress-relaxation fits and 0.1% for cyclic. The results support the use of the presented methodology to develop linear or non-linear viscoelastic models from stress-relaxation or cyclic experimental data of biological soft tissues.

Comparison of in vivo and ex vivo viscoelastic behavior of the spinal cord

Despite efforts to simulate the in vivo environment, post-mortem degradation and lack of blood perfusion complicate the use of ex vivo derived material models in computational studies of spinal cord injury. In order to quantify the mechanical changes that manifest ex vivo, the viscoelastic behavior of in vivo and ex vivo porcine spinal cord samples were compared. Stress-relaxation data from each condition were fit to a non-linear viscoelastic model using a novel characterization technique called the direct fit method. To validate the presented material models, the parameters obtained for each condition were used to predict the respective dynamic cyclic response. Both ex vivo and in vivo samples displayed non-linear viscoelastic behavior with a significant increase in relaxation with applied strain. However, at all three strain magnitudes compared, ex vivo samples experienced a higher stress and greater relaxation than in vivo samples. Significant differences between model parameters also showed distinct relaxation behaviors, especially in non-linear relaxation modulus components associated with the short-term response (0.1-1 s). The results of this study underscore the necessity of utilizing material models developed from in vivo experimental data for studies of spinal cord injury, where the time-dependent properties are critical. The ability of each material model to accurately predict the dynamic cyclic response validates the presented methodology and supports the use of the in vivo model in future high-resolution finite element modeling efforts. STATEMENT OF SIGNIFICANCE Neural tissues (such as the brain and spinal cord) display time-dependent, or viscoelastic, mechanical behavior making it difficult to model how they respond to various loading conditions, including injury. Methods that aim to characterize the behavior of the spinal cord almost exclusively use ex vivo cadaveric or animal samples, despite evidence that time after death affects the behavior compared to that in a living animal (in vivo response). Therefore, this study directly compared the mechanical response of ex vivo and in vivo samples to quantify these differences for the first time. This will allow researchers to draw more accurate conclusions about spinal cord injuries based on ex vivo data (which are easier to obtain) and emphasizes the importance of future in vivo experimental animal work.

Quasi-Linear Viscoelastic Theory is Insufficient to Comprehensively Describe the Time-Dependent Behavior of Human Cervical Spine Ligaments

Stress relaxation experiments were conducted on cervical spine ligaments at multiple strain magnitudes to determine the validity and applicability of the quasi-linear viscoelastic (QLV) theory to model their dynamic behavior. The results indicate that the shape of the stress relaxation curve is dependent upon the magnitude of the applied strain. Thus, a more general, nonlinear formulation is required to model these ligaments within the physiological strain range.Copyright