Comparing Predictive Accuracy and Computational Costs for Viscoelastic Modeling of Spinal Cord Tissues

The constitutive equation used to characterize and model spinal tissues can significantly influence the conclusions from experimental and computational studies. Therefore, researchers must make critical judgements regarding the balance of computational efficiency and predictive accuracy necessary for their purposes. The objective of this study is to quantitatively compare the fitting and prediction accuracy of linear viscoelastic (LV), quasi-linear viscoelastic (QLV), and (fully) non-linear viscoelastic (NLV) modeling of spinal-cord-pia-arachnoid-construct (SCPC), isolated cord parenchyma, and isolated pia-arachnoid-complex (PAC) mechanics in order to better inform these judgements. Experimental data collected during dynamic cyclic testing of each tissue condition were used to fit each viscoelastic formulation. These fitted models were then used to predict independent experimental data from stress-relaxation testing. Relative fitting accuracy was found not to directly reflect relative predictive accuracy, emphasizing the need for material model validation through predictions of independent data. For the SCPC and isolated cord, the NLV formulation best predicted the mechanical response to arbitrary loading conditions, but required significantly greater computational run time. The mechanical response of the PAC under arbitrary loading conditions was best predicted by the QLV formulation.

Viscoelasticity of spinal cord and meningeal tissues

Compared to the outer dura mater, the mechanical behavior of spinal pia and arachnoid meningeal layers has received very little attention in the literature. This is despite experimental evidence of their importance with respect to the overall spinal cord stiffness and recovery following compression. Accordingly, inclusion of the mechanical contribution of the pia and arachnoid maters would improve the predictive accuracy of finite element models of the spine, especially in the distribution of stresses and strain through the cord's cross-section. However, to-date, only linearly elastic moduli for what has been previously identified as spinal pia mater is available in the literature. This study is the first to quantitatively compare the viscoelastic behavior of isolated spinal pia-arachnoid-complex, neural tissue of the spinal cord parenchyma, and intact construct of the two. The results show that while it only makes up 5.5% of the overall cross-sectional area, the thin membranes of the innermost meninges significantly affect both the elastic and viscous response of the intact construct. Without the contribution of the pia and arachnoid maters, the spinal cord has very little inherent stiffness and experiences significant relaxation when strained. The ability of the fitted non-linear viscoelastic material models of each condition to predict independent data within experimental variability supports their implementation into future finite element computational studies of the spine.

STATEMENT OF SIGNIFICANCE

The neural tissue of the spinal cord is surrounded by three fibrous layers called meninges which are important in the behavior of the overall spinal-cord-meningeal construct. While the mechanical properties of the outermost layer have been reported, the pia mater and arachnoid mater have received considerably less attention. This study is the first to directly compare the behavior of the isolated neural tissue of the cord, the isolated pia-arachnoid complex, and the construct of these individual components. The results show that, despite being very thin, the inner meninges significantly affect the elastic and time-dependent response of the spinal cord, which may have important implications for studies of spinal cord injury.

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.

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.