A forward dynamics simulation of human lumbar spine flexion predicting the load sharing of intervertebral discs, ligaments, and muscles

Abstract

Determining the internal dynamics of the human spine’s biological structure is one essential step that allows enhanced understanding of spinal degeneration processes. The unavailability of internal load figures in other methods highlights the importance of the forward dynamics approach as the most powerful approach to examine the internal degeneration of spinal structures. Consequently, a forward dynamics full-body model of the human body with a detailed lumbar spine is introduced. The aim was to determine the internal dynamics and the contribution of different spinal structures to loading. The multi-body model consists of the lower extremities, two feet, shanks and thighs, the pelvis, five lumbar vertebrae, and a lumped upper body including the head and both arms. All segments are modelled as rigid bodies. 202 muscles (legs, back, abdomen) are included as Hill-type elements. 58 nonlinear force elements are included to represent all spinal ligaments. The lumbar intervertebral discs were modelled nonlinearly. As results, internal kinematics, muscle forces, and internal loads for each biological structure are presented. A comparison between the nonlinear (new, enhanced modelling approach) and linear (standard modelling approach, bushing) modelling approaches of the intervertebral disc is presented. The model is available to all researchers as ready-to-use C/C++ code within our in-house multi-body simulation code demoa with all relevant binaries included.

Appendices

Appendix 1: Generated anthropometric data set: vertebral position, orientation, and dimension

An average data set of vertebral positions was calculated from the literature data (Ashton-Miller and Schultz 1997; Belytschko 1978; El-Rich and Shirazi-Adl 2004; Kitazaki and Griffin 1997) to be used by calcman. They determined global orientation and position of vertebrae using skin markers. The vertebral position is defined as the position of the geometric centre of each vertebral body with respect to the global reference frame. S1 was positioned in the centre of the pelvis. Position data from each literature data set was normalised in length to one reference trunk height, i.e. each data set was normalised to the same caudocranial distance ($z$-value) of T1-L5, $h_{\text {T1-L5,Avg}}=458.1\hbox {mm} \approx h_{\text {T1-L5,Belytschko}}$.

Then, a polynomial fit was estimated for each data set (see Fig. 9, $f_{\text {Ashton}}$, $f_{\text {Belytschko}}$, $f_{\text {Rich}}$ and $f_{\text {Kitazaki}}$). These four fits for $x=f(z)$ were averaged by calculating the mean coefficients of each polynomial. The resulting function (sixth-order polynomial) writes

$$\begin{aligned} f_{\text {Avg}}(z)&= 0.015 \,\text {mm} \!+\!0.201 \!\times \! z\!-\!1.12 \times 10^{-3} \,\text {mm}^{-1} \times z^2\nonumber \\&-1.2 \!\times \! 10^{-5} \,\text {mm}^{-2} \!\times \! z^3 \!+\!7.0 \!\times \! 10^{-8} \,\text {mm}^{-3} \times z^4\nonumber \\&-1.5 \times 10^{-10} \,\text {mm}^{-4} \times z^5\nonumber \\&+1.0 \!\times \! 10^{-13} \,\text {mm}^{-5} \times z^6. \end{aligned}$$

(2)

The average position data on the longitudinal axis including standard deviation and the basis data sets are presented in Fig. 9. Calcman calculates the subject-derived height ($h_{\text {T1-L5}}$) and, with that, scales the z-position of each vertebra by the ratio $h_{\text {T1-L5}}/h_{\text {T1-L5,Avg}}$ while leaving the $x$-position unscaled (Table 1).

Fig. 9

Anthropometric data set functions fitted to the position data of vertebral bodies (centre) in healthy male humans. All polynomial fits $f_{\text {Ashton}}(z)$ (Ashton-Miller and Schultz 1997), $f_{\text {Belytschko}}(z)$ (Belytschko 1978), $f_{\text {Rich}}(z)$ (El-Rich and Shirazi-Adl 2004), and $f_{\text {Kitazaki}}(z)$ (Kitazaki and Griffin 1997) were scaled to the reference height of $h_{\text {T1-L5,Avg}}=458.1$ mm from T1 to L5. $f_{\text {Avg}}(z)$ depicts the mean data set used for scaling in calcman, the thin grey lines capture its standard deviation $f_{\text {Avg}}(z)+/-\sigma$. Plus sign indicates the position of each vertebral centre L5 to T1. The origin is equal to the position point of L5. The sagittal axis from dorsal to ventral is equal to the $x$-axis, while the longitudinal axis from caudal to cranial is equal to the $z$-axis

Each vertebral orientation was defined as the normal of the curve (Eq. 2) at its position aligning with the anterior–posterior direction of the vertebra. Each vertebra was modelled as a cylinder (see below), although real vertebrae are slightly wedge-shaped. The calculated orientations were compared with the experimental data from the literature (Ashton-Miller and Schultz 1997; Campbell-Kyureghyan et al. 2005), which are usually presented as upper endplate orientations in the sagittal plane. For this, we fitted graphic primitives from computed tomography (CT) data (Anatomium (TM), 21st Century Solutions Ltd/Gibraltar) that represent T1 to S1, to their respective cylinder primitives and estimated the upper endplate orientation (Table 1). All three data sets vary locally in the range of a few degrees, which is well represented by their standard deviations.

Vertebral dimensions, i.e. vertebral height, depth, and width, were determined for the entire spine, i.e. C3 to S1. The study by Gilad and Nissan (1986) was used as a basis. However, it only includes cervical and lumbar dimensions. Missing data were determined by parameter scaling of data sets from Panjabi et al. (1991, 1992), White and Panjabi (1990). For example, depths for the whole spine included in Panjabi et al. (1992, 1991), White and Panjabi (1990) were scaled to the basis depths in Gilad and Nissan (1986) in order to determine missing vertebral depths in the thoracic spine.

The cylinder’s height was set to the maximum of dorsal and ventral height (Eq. 3 $_{5}$), its radius to the maximum of depth and width (Eq. 3 $_6$). This allows an easy inclusion of the mass of the spinous and transverse processes. It is assumed that their mass approximately balances the overestimation by using the maximum height and ignoring the wedge shape of each vertebral body. The vertebral mass (Eq. 3 $_3$) was calculated on the basis of each cylinder’s volume (Eq. 3 $_4$) and the lumbar vertebrae bone density $\rho _{\text {lumbar}} = 600{-}1{,}000\,\text {kg/m}3$ (Nigg and Herzog 2007). The moment of inertia tensor follows from the cylinder’s parameters (see Eqs. (3 $)_{1,2}$):

$$\begin{aligned} I_{\text {cyl,xx}} = I_{\text {cyl,yy}}&= 1/12 \times m_{\text {vert,i}} \times h_{\text {vert,i}}^2+ 1/4\nonumber \\&\times \,\, m_{\text {vert,i}} \times r_{\text {vert,i}}^2, \nonumber \\ I_{\text {cyl,zz}}&= 1/2 \times m_{\text {vert,i}} \times r_{\text {vert,i}}^2 \nonumber \\ \text{ with }\qquad m_{\text {vert,i}}&= V_{\text {vert,i}} \times \rho _{\text {lumbar}} \times 3/2, \nonumber \\ V_{\text {vert,i}}&= \pi \times r_{\text {vert,i,width}}^{2} \times h_{\text {vert,i}}, \nonumber \\ h_{\text {vert,i}}&= \max \{h_{\text {vert,i,dorsal}},h_{\text {vert,i,ventral}}\}, \nonumber \\ r_{\text {vert,i}}&= \max \{r_{\text {vert,i,depth}},r_{\text {vert,i,width}}\}. \end{aligned}$$

(3)

Appendix 2: Nonlinear ligament model

Various studies on ligaments confirm a nonlinear stress–strain characteristic for ligaments (Chazal et al. 1985; Pintar et al. 1992). For our lumbar spine model, the spinal ligaments were modelled as nonlinear straight-line elements based on the experimental results by Chazal et al. (1985), Panjabi et al. (1982). The function describing the spinal ligament’s stress–strain characteristics was assumed to be continuously differentiable ($C^1$ continuity) between an initial nonlinear region (up to A) and a constant stiffness at higher loads (A-B, see Fig. 10). The concave region at even higher forces is not shown here. Hence, the force–length characteristic of a spinal ligament is modelled as

$$\begin{aligned}&F_{\text {el,LIG}}(l_{\text {LIG}})\nonumber \\&\quad = \left\{ \begin{array}{l@{\quad }l} 0 &{} \, l_{\text {LIG}} < l_{\text {LIG,0}} \\ K_{\text {LIG,nl}} \, (l_{\text {LIG}}-l_{\text {LIG,0}})^{\nu _{\text {LIG,nll}}} &{}\, l_{\text {LIG}} < l_{\text {LIG,nll}} \\ \Delta F_{\text {LIG,0}} + K_{\text {LIG,l}} \,(l_{\text {LIG}} - l_{\text {LIG,nll}}) &{} \, l_{\text {LIG}} \ge l_{\text {LIG,nll}} \end{array} \right. \!, \nonumber \\ \end{aligned}$$

(4)

where $l_{\text {LIG}}$ is its length. Its parameters are

$$\begin{aligned} l_{\text {LIG,nll}}&= (1+\Delta U_{\text {LIG,nll}}) \, l_{\text {LIG,0}}, \nonumber \\ \nu _{\text {LIG,nll}}&= \Delta U_{\text {LIG,nll}} / \Delta U_{\text {LIG,l}}, \nonumber \\ K_{\text {LIG,nl}}&= \Delta F_{\text {LIG,0}}/(\Delta U_{\text {LIG,nll}}\,l_{\text {LIG,0}})^{\nu _{\text {LIG,nll}}}, \nonumber \\ K_{\text {LIG,l}}&= \Delta F_{\text {LIG,0}}/(\Delta U_{\text {LIG,l}}\,l_{\text {LIG,0}}) . \end{aligned}$$

(5)

Fig. 10

Force–length relation of the spinal ligament. Four independent parameters were used such as $l_{\text {LIG,0}}$ (rest length), $\Delta U_{\text {LIG,nll}}$ (relative stretch at nonlinear–linear transition), $\Delta F_{\text {LIG,0}}$ (both force at the transition and force increase in the linear part), and $\Delta U_{\text {LIG,l}}$ (relative additional stretch in the linear part providing a force increase in $\Delta F_{\text {LIG,0}}$). The transition between the initial nonlinear and the linear region is located at $l_{\text {LIG,nll}} = (1+\Delta U_{\text {LIG,nll}})$, the exponent for the initial nonlinear relation is $\nu _{\text {LIG,nll}} = \Delta U_{\text {LIG,nll}} / \Delta U_{\text {LIG,l}}$. The parameters of relevant and included spinal ligaments are given in Table 5. They were derived from experimental data (Chazal et al. 1985)

They were derived from four independent parameters such as $l_{\text {LIG,0}}$ (rest length of ligament), $\Delta U_{\text {LIG,nll}}$ (relative stretch at nonlinear–linear transition), $\Delta F_{\text {LIG,0}}$ (both force at transition and force increase in linear part), and $\Delta U_{\text {LIG,l}}$ (relative additional stretch in linear part providing the force increase $\Delta F_{\text {LIG,0}}$), cf.  Fig. 10. These parameters were partly extracted from tensile experiments on 43 human spinal ligaments performed by Chazal et al. (1985). For five spinal ligaments, i.e. anterior longitudinal (ALL) and posterior longitudinal ligament (PLL), ligamentum flavum (LF), supra-spinal (SSL), and interspinal (ISL) ligament, Chazal et al. (1985) estimated tension and elongation at transition (point A), at the end of the linear phase (point B) and at the apex of the tension elongation curve. The arithmetic means of the extracted force and elongation data at A and B ($F_{\text {A}}$, $l_{\text {A}}$, $F_{\text {B}}$, $l_{\text {B}}$) were used to calculate three input parameters of the ligament characteristic ($\Delta U_{\text {LIG,nll}}$, $\Delta U_{\text {LIG,l}}$, $\Delta F_{\text {LIG,0}}$). These input parameters were determined separately for each of the five ligaments.

The ligament’s rest length ($l_{\text {LIG,0}}$) was extracted from the model’s geometry. It is accepted that spinal ligaments are not in a strain-free state when the spine is in its neutral anatomical position during upright stance. Pre-strain data for the spinal ligaments were taken from the literature (Aspden 1992; Nachemson and Evans 1968; Robertson et al. 2013) and used to determine the ligament’s rest length in the neutral position. The neutral position was approximated by simulating an initial settling of the lumbar spine model (see 2.6, neutral position). In that state, each ligament’s rest length was calculated by the distance between two ligament’s attachment points ($l_{\text {LIG,SET}}$ [m]) and the respective pre-strain ($\epsilon _{\text {LIG}}$ [ ]): $l_{\text {LIG,0}}=l_{\text {LIG,SET}}/(1+\epsilon _{\text {LIG}})$.

As proposed by Panjabi et al. (1982), ligaments with planar attachment areas, like the ALL, LF, and ISL, can be approximated by more than one pair of attachment points, i.e. multiple lines for one ligament. Their characteristic (Eq. 4) is then equally distributed by equally distributing $\Delta F_{\text {LIG,0}}$ between multiple lines.

A velocity-dependent damper was also included in order to account for the ligament’s nonlinear viscoelastic behaviour, i.e.

$$\begin{aligned}&F_{\text {damp,LIG}}= F_{\text {el,LIG}} \times d_{\text {LIG}} \times v_{\text {LIG}} \;\; \text {with} \nonumber \\&\text {with} \;\; \;\; v_{\text {LIG}}=\frac{{\mathrm {dl}}_{\text {LIG}(t)}}{{\mathrm {dt}}}, \;\; d_{\text {LIG}} = 1.0 \,\text {s}\,\text {m}^{-1} \nonumber \\&\text {and consequently} \;\; \;\; F_{\text {tot,LIG}} = F_{\text {el,LIG}} \times (1 + d_{\text {LIG}} \times v_{\text {LIG}})\mathrm {.}\nonumber \\ \end{aligned}$$

(6)

According to Gerritsen et al. (1995), the damping force depends on the elastic force $F_{\text {el,LIG}}$ and is scaled by a damping coefficient $d_{\text {LIG}}$. This avoids discontinuity and numerical stabilities.

Appendix 3: Ligament parameters

See Table 5.

Table 5 Ligament modelling parameters for the nonlinear MB model version, i.e. $l_{\text {LIG,0}}$ (rest length of ligament), $\Delta U_{\text {LIG,nll}}$ (relative stretch at nonlinear–linear transition), $\Delta F_{\text {LIG,0}}$ (both force at transition and force increase in linear part), $\Delta U_{\text {LIG,l}}$ (relative additional stretch in linear part providing the force increase $\Delta F_{\text {LIG,0}}$) and $\epsilon$ (pre-strain of ligament), for all included spinal ligaments, i.e. anterior longitudinal (ALL) and posterior longitudinal ligament (PLL), ligamentum flavum (LF), supraspinal (SSL), and interspinal (ISL) ligament

Appendix 4: Development of specific IVD model

In MB models, IVDs are either modelled as one force element or one force element is used to capture the mechanical response of the IVD as well as its surrounding ligaments and muscles, i.e. lumped modelling approach for one functional spinal unit. The mechanical response of an IVD (Huynh et al. 2010) or a functional spinal unit (Christophy et al. 2011; Monteiro 2009) is usually modelled using a linear and decoupled bushing element (6-DOF spring-dashpot element) as a connecting force element. Stiffness and damping parameters are taken from experimental measurements (Ashton-Miller and Schultz 1997; Berkson et al. 1979; Eberlein et al. 2004; Gardner-Morse and Stokes 2004; Hirsch and Nachemson 1954; Schultz et al. 1979; Virgin 1951). Experimental data clearly demonstrate the nonlinear nature of these biological tissues. Hence, it is linearised to be used as model parameters for the linear elements in other MB models (Huynh et al. 2010; Monteiro 2009). So far, model parameters differ, as experimental results cover a wider range.

Instead, a recently developed IVD model was used in our MB model and implemented in demoa. It captures the nonlinear and coupled mechanical response of IVDs in the lumbar region from pelvis to vertebra L1 (Karajan et al. 2013). In particular, a FE model of the IVD (Karajan 2012) was used to pre-compute the mechanical behaviour of the IVD in terms of stresses and strains. The FE model of the IVD is based on the theory of porous media. It is capable of capturing the porous microstructure of the IVD. The mechanical response of the IVD due to several applied deformation states was captured at its centre of gravity by homogenisation via an integration of the resulting stresses at the connecting surface of the adjacent vertebra. Thus, each of the homogenised results represents a reaction force and moment due to an applied deformation state (combination of displacement and rotation). A parameterisation using a cubic polynomial turned out to deliver the best fit of the resulting forces and moments gathered at the varying deformation states, i. e. sampling points, which can be related to the displacements and rotations of the MB system defined in demoa. For more detailed information on the representation, the reader is referred to Karajan et al. (2013).

Appendix 5: Hill-type muscles in MB models

In forward dynamics MB models, muscles are often modelled as Hill-type MTCs (Haeufle et al. 2014). This labelling refers to A.V. Hills’s experimental finding (Hill 1938) that the force–velocity relation of a muscles fibre is a hyperbola. Thus, Hill-type models are phenomenologically based macroscopic lumped parameter models. Such a model usually consists of a contractile element (CE), a serial, and a parallel elastic element. The corresponding contraction (Eq. 7 $_1$, CE length $l_{\text {CE}}),$ and activation dynamics (Eq. 7 $_2$, activity $q$ with $0\le q\le 1$) of a muscle are described by first-order differential equations:

$$\begin{aligned}&\dot{l}_{\text {CE}}=f_{l}(l_{\text {CE}},l_{\text {MTC}},q),\nonumber \\&\dot{q}=f_{\text {q}}(q,\hbox {STIM}) \end{aligned}$$

(7)

with $l_{\text {MTC}}$: length of MTC, $\hbox {STIM}$ stimulation input. $\hbox {STIM}$ is calculated by our simple lambda-control model, a bio-inspired motor control model. It represents the electrical stimulation signal (summed transmembrane potential) on the muscles’ surface and can be compared with surface electromyography (sEMG). Günther et al. (2007) included an additional damping component parallel to the serial elastic element. Hence, the right-hand side of Eq. (7)$_1$ also depends on $\dot{l}_{\text {MTC}}$. By using this modified Hill-type muscle model, movement control is expected to be more feasible, because eigenoscillations are damped due to the interaction of muscle and segment inertia (Günther et al. 2007). We refer to Günther et al. (2007), Haeufle et al. (2014), Mörl et al. (2012) for a detailed description of the muscle model. It was implemented in demoa and used for all MTCs in the presented full human MB model.

Appendix 6: Muscle parameters

See Table  6.

Table 6 Muscle modelling parameters: maximum isometric force of the contractile element $F_{\text {max}}$, optimal fibre length $l_{\text {CE,opt}}$, and serial elastic element $l_{\text {SEE,0}}$

Appendix 7: Comparison with the literature data

Kinematic data from the literature cannot be compared directly with results from experimental studies because investigated overall movements differ. Some studies present flexion–extension results in a combined way, others present data, measured with less or more flexion amplitudes. Therefore, relative changes and the order of magnitude were used for comparison. In the following, vertical translational and angular displacements, and IVD forces of the presented lumbar spine model are compared with the literature data. Additionally, the resultant IVD forces of the nonlinear intervertebral disc model included into the spine model are compared with results of three lumbar spine model variations including the classical, linear intervertebral disc model for three different parameter sets. The resultant IVD forces again are compared with the literature data (Fig. 11).

Fig. 11

A Vertical displacements in the local sagittal plane (along the local longitudinal axis ($z$: caudocranial direction)) of the subjacent body and B angular displacements of intervertebral discs $\varphi _{\text {y}}$ in the local sagittal plane of the subjacent body (its local $x$–$z$-plane) as a function of the negative lumbar lordosis angle. Vertical and rotational displacements are depicted relative to displacement at $t_{\text {SET}}$, i.e. $r_{\text {z},t_{\text {SET}}} = 0$ m and $\Delta \varphi _{\text {y},t_{\text {SET}}} = 0 ^\circ$. Bold lines are used to emphasise translational and rotational displacements in L4/5

Vertical displacements in the IVDs increase from cranial to caudal (top to bottom, see Fig. 12). For a simple flexion movement simulated with our full human body model, the displacement amplitudes increase from 0.34 mm (L1/2), to 0.45 mm (L2/3), to 0.58 mm (L3/4), to 0.62 mm (L4/5). In vitro studies by Hayes et al. (1989) report displacements for a complete flexion–extension movement in the same order: L1/2 (1.9 mm) $\rightarrow$ L2/3 (2.4 mm) $\rightarrow$ L3/4 (2.5 mm) $\rightarrow$ L4/5 (3.0 mm). The same way, Li et al. (2009) report displacements for a flexion–extension movement in the order: L2/3 ($0.2\pm 0.2$ mm) $\rightarrow$ L3/4 ($0.6\pm 0.4$ mm) $\rightarrow$ L4/5 ($0.7\pm 0.6$ mm). Additionally, the order of magnitude of all three studies lies in the same range.

Fig. 12

Vertical displacements in the lumbar joints, w.r.t local joint coordinate system, increase from cranial to caudal. In vitro studies (Hayes et al. 1989; Li et al. 2009) report displacements in the same order; however, they measured flexion–extension movements of higher amplitudes. Additional lines depict trend of values

More experimental data exist for the comparison of angular displacements in the IVDs. Ibarz et al. (2013), White and Panjabi (1990) present various in vivo measurements of angular joint motion during flexion–extension movement. Interestingly, data presented in the literature do not match, i.e. some present increasing angular motion from cranial to caudal (Hayes et al. 1989; Ibarz et al. 2013; Pearcy and Portek 1984; White and Panjabi 1990) while others present the opposite (Lee et al. 2002; Li et al. 2009; Wong et al. 2004, 2006). The simulation results that were presented in this study showed an increase in angular displacement during flexion in the order L1/2 ($2.6^{\circ }$) $\rightarrow$ L2/3 ($3.6^{\circ }$) $\rightarrow$ L3/4 ($4.4^{\circ }$) $\rightarrow$ L4/5 ($4.6^{\circ }$). This caudocranial increase in angular IVD joint motion is in accordance with flexion–extension studies by Hayes et al. (1989), Ibarz et al. (2013), Pearcy and Portek (1984), White and Panjabi (1990): L1/2 ($5-16^{\circ }$) $\rightarrow$ L2/3 ($7.8-18^{\circ }$) $\rightarrow$ L3/4 ($6-17^{\circ }$) $\rightarrow$ L4/5 ($9-21^{\circ }$). Again, only relative changes could be presented (see Fig. 13).

Fig. 13

Angular displacements in the lumbar joints, w.r.t local joint coordinate system, increase from cranial to caudal. In vitro studies (Hayes et al. 1989; Ibarz et al. 2013; Pearcy and Portek 1984; White and Panjabi 1990) report displacements in the same order; however, they measured flexion–extension movements of higher amplitudes. Note, we did not include studies (Lee et al. 2002; Li et al. 2009; Wong et al. 2004, 2006) with the opposite trend in this figure. Additional lines depict trend of values

Additionally, we compared our angular displacements with the intervertebral flexion–extension (IVFE) presented by Lee et al. (2002), Wong et al. (2004, 2006). It indicates the relationship between net lumbar motion and motion of the individual vertebrae. The IVFE is often used as input data for modelling of the IVD element in inverse dynamic models, cf. slope $k$ in Christophy et al. (2011). Our presented joint kinematics plotted against the lumbar lordosis angle are equivalent to the IVFE. Our simulation results for the IVFE also show a linear-like pattern, like experiments determining the IVFE characteristics. However, simulated IVFE in all joints of our model increase caudocranial, whereas in experiments IVFE slopes increase with investigated joints moving cranial. This is due to the fact that so far the new IVD model (Karajan et al. 2013) was used for all lumbar IVDs with the same parameters. The new, nonlinear IVD model was developed for the L4/5 intervertebral disc and then applied to all other IVDs. We argue that by applying different scaled material properties and an enhanced IVD element’s geometry, a more realistic IVFE pattern would occur. Additionally, it will furthermore result in a more physiological angular displacement behaviour. An enhanced nonlinear IVD model accounting for the mentioned shortcomings is already available, but was not ready to be included in this study yet (Karajan et al. 2014).

In vivo measurements of lumbar joint dynamics are rare. For a first validation, we used intradiscal pressure measurement data of the IVD L4/5. Measurements were taken with one healthy male test subject in different postures by Wilke et al. (1999, 2001). They measured a disc pressure of 0.43–0.5 MPa during relaxed standing of the test subject (body mass 70 kg, body height 1.74 m). In a $36^\circ$ flexed standing posture, the disc pressure increased almost linearly to 1.08 MPa. The flexion angle was defined differently, i.e. between the thoracolumbar junction and the sacrum ($\alpha _{\text {fl,S1--T12}}$). The disc area, measured using magnetic resonance imaging, was 18 cm$^2$. Thus, the respective axial joint force was 774–900 N in relaxed standing and 1,944 N in a flexed position. In our model, axial forces increased from $F_{\text {z,upright}}=942$ N during upright stance to $F_{\text {z,flex}}=1{,}681$ N at $t_{\text {sim,end-nl}}$ with $\alpha _{\text {fl,S1--T12}}=3^\circ$ and $\alpha _{\text {fl,S1--T12}}=33^\circ$, respectively. Taking into account our slightly different flexion angle (higher at the beginning, smaller at the end), simulated longitudinal forces using the nonlinear IVD model in L4/5 show good agreement with experimental data.

In comparison with in vivo measurements of joint dynamics (Wilke et al. 1999, 2001), our new, nonlinear IVD model included in the spine model shows the best agreement (see 3.1). The classical, linear IVD model variations using either Monteiro parameters or Gardner–Morse parametres result in higher forces while standing and lower forces when reaching a flexed position. Restoring forces during stance in the Huynh model show good agreement with our model results, while forces in flexion are underestimated.

Appendix 8: A simulation of heel impact with the ground

We simulated a double heel contact of the model with the ground after a free fall of about 8 mm, i.e. after about 0.04 s, starting from a completely symmetrical initial position. To insure ball contacts would not interfere with the impact, the ankle joints were fixed in a slightly dorsally flexed orientation as bracket joints, the latter being a method to set the number of degrees of freedom in a joint to zero. Just alike, we fixed the upper body to L1, resulting in a lumped L1-plus upper body. Due to the stiffness $4\times 10^{5}$ N/m of the linear elastic spring modelling the normal ground reaction force component of each heel contact element, both heel elements were deformed about 2 mm in vertical direction during the impact. The elements’ vertical damping force was modelled in proportion to the product of vertical deformation and velocity, with the corresponding coefficient of proportionality set to $4\times 10^{5}$ Ns/m$^2$. Correspondingly, the net vertical ground reaction force reached a peak of about 65 m/s$^2$ (Fig. 14).

The mass distribution in the model was chosen such that all segment masses represented just the bony parts. This is because it is only the skeleton that is decelerated during this impact period of about 20.0 ms (Denoth et al. 1985; Nigg and Denoth 1980), whereas all soft tissue material like the muscles is coupled viscoelastically to the bones (Denoth 1985; Gruber et al. 1998; Günther et al. 2003; Schmitt and Günther 2011) and decelerated with a time delay. In modelling its mechanical response, the soft tissue material is denoted ‘wobbling mass’ (Denoth 1985). The ratio of wobbling to segment mass was set to zero in the feet, to $1/2$ in the shanks, and to $2/3$ in the thighs (Günther et al. 2003; Schmitt and Günther 2011). For the pelvis’ bony part, we assumed 3.0 kg, neglecting an assumed wobbling mass portion of 5.3 kg. The bony masses of the 24 vertebrae were assumed to decrease from about 50 g in L5 to 10 g in C1. In our model, the inertia of all vertebrae from T12 up to C1 is included in the upper body. For the upper body, we applied the ‘effective mass’ concept introduced by Denoth (1986), Denoth et al. (1985): during the 20.0 ms impact period, the effectively decelerated mass may depend on both net bony mass of the body and the initial joint angular configuration. We estimated that one-third of the 36.8 kg act effectively during impact. Altogether, the decelerated body masses summed to 25.0 kg in our impact model.

The resulting distribution of vertical acceleration components measured in the global reference frame is plotted in Fig. 14. There is a shock wave propagating from the feet to the head. The acceleration amplitudes decrease from distal to proximal in the legs, with a slight deviation from this rule: the foot acceleration is a little lower than the shank acceleration. This fact is due to the foot centre of mass being located about 10 cm anterior to the ankle and its upward movement determined by a superposition of heel translation and foot–shank rotation. Above the pelvis, the amplitudes decrease along the lumbar spine from caudal to cranial. Quantitatively, the acceleration maxima are time-shifted according to the following segmental sequence: shank(s) (0.06 ms), thigh(s) (0.02 ms), pelvis ($-$0.01 ms) L5 (0.08 ms) L4 (0.26 ms) L3 (0.18 ms), whole body (0.22 ms) L2 (8.73 ms) L1-plus upper body. The L1-plus upper body mass is the highest of all segments and located most distantly from the ground. Its maximum acceleration occurs clearly later than in any other segment including the whole body.

Fig. 14

Vertical components of the model’s net ground reaction force and each segment’s centre of mass acceleration during an impact of the upright model on the heels from 8 mm falling height

These numbers can be compared to known experimental data of shock wave propagation from the ankle to the head (Lafortune et al. 1996). For this distance, these authors found propagation times between 5.9 and 4.7 ms, for knee angles between full extension and $40^{\circ }$ flexion, respectively. Our simulation results demonstrate that the propagation along the chain of the four distal bodies connected by hard joint constraints (foot, shank, thigh, pelvis) is faster than the propagation via viscoelastically deforming soft constraints (like IVDs in our case). Therefore, we would expect that summing up the time shifts of the $24$ IVDs will dominate the shock wave propagation time from heel to head, because physiologically there are only four additional distal cartilage connections from heel to pelvis: lower and upper ankle joints, knee, and hip. To estimate the propagation velocity depending on IVD properties, we first calculate the mean time shift per IVD for the four IVDs located between pelvis and L2 as $\frac{-0.01+0.08+0.26+0.40\,\hbox {ms}}{4}=\frac{0.73\,\hbox {ms}}{4}$. For the propagation time along the vertebral column comprising 24 IVDs, we found $24\times \frac{0.73\,\hbox {ms}}{4}\approx 4.4\,\hbox {ms}$. The deviation from the data by Lafortune et al. (1996) can be easily explained by two delaying contributions missing in our model so far: (1) those by the cartilage in the leg joints and (2) those by acoustic wave propagation in the bones. However, the latter effects should be negligible because the elastic modulus of bones is about 4 orders of magnitude higher than that of IVDs, with approximately the same mass density.

The distance between L5 and C1 is roughly 60 cm. Thus, we can estimate the propagation velocity from our computer experiment to be $c_{\mathrm{long}}=60\,\hbox {cm}/4.4\,\hbox {ms}=136\,\hbox {m}/\hbox {s}$. Based on theory, we may predict this number based on elastic modulus $E=\frac{\sigma }{\varepsilon }=\frac{\Delta \,F}{\Delta \,l}\times \frac{A_{0}}{l_{0}}$ with the stress $\sigma =\frac{\Delta \,F}{A_{0}}$, the strain $\varepsilon =\frac{\Delta \,l}{l_{0}}$, and the stiffness $K=\frac{\Delta \,F}{\Delta \,l}$. The area of a vertebra’s upper endplate is roughly 1 cm$^{2}$ and the range of IVD thicknesses is about 1 cm$\ldots$0.5 cm from L5 to C1. In general, wave propagation velocity $c_{\mathrm{long}}$ in solids depends also on the Poisson ratio $\mu =-\frac{\Delta \,A}{A_{0}}/\frac{\Delta \,l}{l_{0}}$. For longitudinal waves, theory predicts $c_{\mathrm{long}}=\sqrt{\frac{E\times (1-\mu )}{\rho \times (1-\mu -\mu ^{2})}}$, where $\rho$ symbolises the mass density. With a lumbar IVD stiffness estimation of $5\times 10^{5}$ N/m (compare Table 2), lumbar thickness of 1 cm, and a small Poisson ratio $\mu \ll 0.5$, a value of $c_{\mathrm{long}}=73$ m/s can be predicted from theory. This is nearby the theoretical minimum value for $\mu =0$. This theoretical $c_{\mathrm{long}}$ increases for $\mu$ approaching the pole $0.5$. For $\mu =0.45$, $c_{\mathrm{long}}$ would be about twice the minimum value.