A Hyper-Viscoelastic Continuum-Level Finite Element Model of the Spinal Cord Assessed for Transverse Indentation and Impact Loading

Abstract

Finite Element (FE) modelling of spinal cord response to impact can provide unique insights into the neural tissue response and injury risk potential. Yet, contemporary human body models (HBMs) used to examine injury risk and prevention across a wide range of impact scenarios often lack detailed integration of the spinal cord and surrounding tissues. The integration of a spinal cord in contemporary HBMs has been limited by the need for a continuum-level model owing to the relatively large element size required to be compatible with HBM, and the requirement for model development based on published material properties and validation using relevant non-linear material data. The goals of this study were to develop and assess non-linear material model parameters for the spinal cord parenchyma and pia mater, and incorporate these models into a continuum-level model of the spinal cord with a mesh size conducive to integration in HBM. First, hyper-viscoelastic material properties based on tissue-level mechanical test data for the spinal cord and hyperelastic material properties for the pia mater were determined. Secondly, the constitutive models were integrated in a spinal cord segment FE model validated against independent experimental data representing transverse compression of the spinal cord-pia mater complex (SCP) under quasi-static indentation and dynamic impact loading. The constitutive model parameters were fit to a quasi-linear viscoelastic model with an Ogden hyperelastic function, and then verified using single element test cases corresponding to the experimental strain rates for the spinal cord (0.32–77.22 s−1) and pia mater (0.05 s−1). Validation of the spinal cord model was then performed by re-creating, in an explicit FE code, two independent ex-vivo experimental setups: 1) transverse indentation of a porcine spinal cord-pia mater complex and 2) dynamic transverse impact of a bovine SCP. The indentation model accurately matched the experimental results up to 60% compression of the SCP, while the impact model predicted the loading phase and the maximum deformation (within 7%) of the SCP experimental data. This study quantified the important biomechanical contribution of the pia mater tissue during spinal cord deformation. The validated material models established in this study can be implemented in computational HBM.

Symmetric mixed discontinuous Galerkin methods for linear viscoelasticity

“…To conclude this section we consider a simple viscoelastic model for a segment of cervical spinal cord (consisting of white and grey matter), surrounded by the pia mater (represented as a thin layer of elastic material). The problem setup mimics indentation tests as in, e.g., [30,37], which in turn replicate problems arising due to degenerative factors. We only take a transversal cross-section of approximately 13 mm in maximal diameter, and in this case the indentation region is simply a curved subset of the anterior part of the pia mater, having length 4 mm.…”

“…An advantage of the DG-based formulation advanced herein is that it permits us to readily consider [21,33,37], see also [20,30]), Note that in [37] the pia matter is considered elastic and the white and grey matter subdomains are considered hyperelastic, in [30] there is only pia mater and homogeneous spinal cord (all visco-hyperelastic), in [21] the spinal cord is homogeneous and linear viscoelastic, whereas in [33] a poroelastic model has been used for all layers. Here the elastic behaviour of the pia mater is modelled with a much smaller value than in the rest of the domain, ω pia = 1/1000 s < ω white,grey = 1/6.7 s).…”

Quantifying tethered spinal cord tension: Ultrasound elastography results from computational, cadaveric, and intraoperative first-in-human studies

“…In line with previous studies, the spinal cord was assumed to be homogeneous, linear elastic, and orthotropic [i.e., mechanical properties in its lateral (lengthy) direction differed from the two orthogonal, radial directions] 56,57 . The transducer was acoustically coupled to the cord by an isotropic water medium.…”

Comparison of numerical methods for cerebrospinal fluid representation and fluid–structure interaction during transverse impact of a finite element spinal cord model

“…The dura was meshed with an average shell element length of 0.8 mm (11,200 shell elements), the spinal cord was meshed with an average element size of 0.8 mm (37,100 hexahedral elements) and the pia mater was meshed with 9100 hexahedral elements, using coincident nodes to the outer surface of the spinal cord. Convergence of the spinal cord‐pia mater complex mesh was conducted previously, 20 and was used in the current study.…”

“…13,[17][18][19] Previous research has established that the pia mater contributes to the non-linear mechanical response of the spinal cord complex during impact. 20,21 To model this behavior, the spinal cord and pia mater mechanical properties from experimental data 22,23 have been represented using an Ogden hyperelastic constitutive model, as implemented in a commercial FE code with a modified strain density energy function (Equation 1). 24 In addition, experimental data of the spinal cord tissue indicates strong viscoelastic effects.…”