# Multi-Directional Shape Change Analysis of Biotensegrity Model Mimicking Human Spine Curvature
## 2 Spine Biotensegrity Models  
Four-stage spine biotensegrity models, SB1, SB2, and SB3 proposed by [PERSON], [PERSON], [PERSON] and [PERSON] [5], were used in this study. This section briefly explains the anatomical parameters and topologies of the spine biotensegrity models. In this study, the biotensegrity model was modelled by mimicking the topology of the human spine where the width, height and the natural curvature of actual human spine vertebrae anatomical parameters are taken into consideration. The exact location of the muscles, fascial and ligaments as well as other parts of the body connected to the spine are not considered in the modelling. The Class-1 tensegrity mast is adopted to mimic the human spine represented by vertebrae (i.e., struts) which are gapped and cushioned by intervertebral discs and well surrounded by muscles (i.e., cables). Real human spine anatomical parameters studied by [PERSON], et al. [19] were adopted for the geometrical input data of spine biotensegrity models. These anatomical parameters are the vertebral end-plate width, vertebral central body height and intervertebral disc height (see Figure 1). The aggregation of both vertebral central body height and intervertebral disc height characterizes the height of the spine biotensegrity model, whereas the accumulation of intervertebral disc height is modelled using the saddle height which corresponds to the vertical distance between the upper triangular surface of the lower cell and the lower triangular surface of the upper cell in the biotensegrity model. The tapered section properties of a human spine are incorporated into the spine biotensegrity models. A total of twenty-two vertebrae, particularly vertebrae C3-C7, T1-T12 and L1-L5 within the cervical, thoracic and lumbar regions, respectively, were considered in the spine biotensegrity models. The spine biotensegrity models also incorporated the natural curvature of the human spine with cervical lordosis at an apice AP1 of 20.1\({}^{\circ}\), thoracic kyphosis at an apice AP2 of 34.5\({}^{\circ}\) and lumbar lordosis at an apice AP3 of 29.2\({}^{\circ}\). The SB1, SB2 and SB3 models were differentiated in terms of the different sizes of triangular cells, represented by half of their triangular vertex angles (\(\theta\)) with values of 20\({}^{\circ}\), 30\({}^{\circ}\) and 40\({}^{\circ}\), respectively (Figure 2). The triangular cell was chosen for the spine biotensegrity models due to its simple geometry. All of the SB models have the same magnitude of triangle altitude, \(d\), which was characterized by the vertebral end-plate width of the vertebrae. The spine biotensegrity model is made up of four triangular cells which are stacked on each other. Each triangular cell is bounded by triangular surfaces at the lower and upper end, respectively (Figure 2). In total, there are eight triangular surfaces in the spine tensegrity model: one at the base, two at each of the three apices and one at the top. Figure 2 shows the geometry of a triangular cell in the global Cartesian coordinate system \(O_{o}\)-\(X_{o}Y_{o}Z_{o}\). The position vector of nodal coordinates is denoted as \(\mathbf{X}\). The origins of the local Cartesian system \(O_{i-1}\)-\(X_{i-1}Y_{i-1}Z_{i-1}\) and \(O_{i}\)-\(X_{i}Y_{i}Z_{i}\) are positioned at the centroid of the lower (\(X_{6i\text{-}5}\,X_{6i\text{-}4}\,X_{6i\text{-}3}\)) and upper triangular surfaces (\(\mathbf{X}_{6i\text{-}2}\,X_{6i\text{-}1}\,X_{6i\text{-}1}\,X_{6i\text{-}1}\)) of \(i\)th (\(i\) = 1, 2, \(\dots\), 4) triangular cell, respectively. Each spine biotensegrity model consists of sixty-nine elements (i.e., twelve struts and fifty-seven cables). Details and the form-finding strategy of the self-equilibrated human spine biotensegrity models can be found in [PERSON], [PERSON], [PERSON] and [PERSON] [5]. Figure 3 shows the model of spine biotensegrity used for shape change analysis in this study. The topologies, material properties and configurations of these models are shown in Tables 1 and 2.