Embodiment of intra-abdominal pressure in a flexible multibody model of the trunk and the spinal unloading effects during static lifting tasks

Abstract

The role of intra-abdominal pressure (IAP) in spinal load reduction has remained controversial, partly because previous musculoskeletal models did not introduce the pressure generating mechanism. In this study, an integrated computational methodology is proposed to combine the IAP change with core muscle activations. An ideal gas relationship was introduced to calculate pressure distribution within the abdominal cavity. Additionally, based on flexible multibody dynamics, a muscle membrane element was developed by incorporating the muscular fiber deformation, inter-fiber stiffness, and volume constancy. This element was then utilized in discretizing the diaphragm and transversus abdominis, forming an IAP–muscle coupling system of the abdominal cavity. Based on this methodology, a forward dynamic simulation of spinal flexion was presented to examine the unloading effect of abdominal breathing. The results confirm that core muscle contraction during the abdominal breathing cycle can substantially reduce the forces of spinal compression together with trunk extensor muscles, and this effect is more pronounced when the IAP increase is produced by contraction of the transversus abdominis. This unloading effect still holds even with the co-activation of other abdominal muscles, providing a potential choice when designing trunk movements during weight-lifting tasks.

Appendices

Appendix A: Validation of the proposed muscular membrane element

Fig. 15

Benchmark problem of the muscle membrane under pressure loading. a Membrane element discretization under simply supported boundary condition; b cable element discretization (Guo et al. 2020a) under simply supported boundary condition; c membrane element discretization under fixed boundary condition

1.1 Pressure compression benchmark

As shown in Fig. 15, the muscle membrane is under sinusoidal pressure $p=(1 + 6.65 \sin {2 \pi t}){}{\hbox {kPa}}$, the amplitude of which represents the standing coughing state (Hoyte et al. 2004). A Cartesian coordinate system $O-xyz$ was created with its origin located at the mass center of the muscle membrane. The muscle fiber was set along the y-axis under activation $a = 0.05,~0.5,~1.0$. The muscular geometry and strength were adopted from the RA muscle in Bruno et al. (2015), and other parameters were taken from Pato and Areias (2010), as listed in Table 7.

Table 7 Mechanical parameters of the muscle membrane (Pato and Areias 2010; Bruno et al. 2015)

In anatomy, in addition to muscle/bone insertion, other boundaries of RA are constrained by extramuscular connective tissues. As shown in Fig. 15, four different models were built here to understand the role of boundary conditions and inter-fiber tissue stiffness:

• 1. The muscle was discretized by $7 \times 1$ cable elements (Guo et al. 2020a), and its origin/insertion points were connected to the ground by a spherical joint.

• 2. The muscle was discretized by $7 \times 7$ membrane elements, taking only $\varvec{\sigma }^{\rm {fiber}}$ into account, and its origin/insertion points were connected to the ground by a spherical joint.

• 3. The muscle was discretized by $7 \times 7$ membrane elements, taking $\varvec{\sigma }^{\rm {fiber}}$, $\varvec{\sigma }^{\rm {mat}}$, and $\varvec{\sigma }^{\rm {bulk}}$ into account, and its origin/insertion points were connected to the ground by a spherical joint.

• 4. The muscle was discretized by $7 \times 7$ membrane elements, taking $\varvec{\sigma }^{\rm {fiber}}$, $\varvec{\sigma }^{\rm {mat}}$, and $\varvec{\sigma }^{\rm {bulk}}$ into account, and all its boundaries were connected to the ground by a spherical joint.

Fig. 16

Comparison of the membrane deflections. a Muscle deflections with $a = 0.05,~p = {14.3}{\hbox {kPa}}$; b muscle deflections with $a = 1.0,~p = {14.3}{\hbox {kPa}}$; c time-dependent deflections with $a = 0.05$; d time-dependent deflections with $a = 1.0$

The membrane deflection profile is illustrated in Fig. 16. Under the maximum pressure, this membrane deformed to a catenary curve along the y-axis at $x=0$, and its maximum deflection occurred at the center of mass (Fig. 16a and b). When stimulated by minimal activation, the deflection of this muscle membrane behaves nonlinearly with the pressure change, yet the nonlinearity diminishes with the increase in the activation level (Fig. 16c and d). Moreover, the membrane deformation of Models 1 and 2 was identical, validating the proposed membrane element.

According to Fig. 16, different constraints and internal loading conditions strongly affected the membrane deformation distributions. To further analyze these effects, we further depicted the deformation distribution at different muscle activation levels in Fig. 17. As for Model 3, the compressive resistance of the muscle membrane originated from its contractile force at the maximum activation, producing the same results as Model 2. However, introducing $\varvec{\sigma }^{\rm {mat}}$ and $\varvec{\sigma }^{\rm {bulk}}$ increased the membrane stiffness when $a=0.05$. Furthermore, the magnitude of the membrane displacement in Model 4 can be reduced by constraining the lateral boundaries.

Fig. 17

Deflection distribution of the muscle membrane under the maximum pressure. a Model 2; b Model 3; c Model 4

1.2 Musculotendon benchmark

Another validation benchmark is based on the simulations performed by Millard et al. (2013). As shown in Fig. 18a, the musculotendon actuator consisted of a pennated muscle belly and a long tendon. The initial length of this whole actuator was $l^{\rm {fiber}}_0 \cos {\alpha _0} + l^{\rm {tend}}_0$, where $\alpha _0 = 30^\circ$ denotes the pennation angle. The left end of this actuator was fixed to the ground, and a sinusoidal displacement curve $u(t) = l^{\rm {fiber}}_0 {\rm {sin}} 2 \pi t$ was applied to its free end. Meanwhile, the muscle was stimulated by constant activations ranged from 0.1 to 1. In the numerical model, each half of the muscle belly was discretized by a membrane element, and the $\varvec{\sigma }^{\rm {mat}}$ and $\varvec{\sigma }^{\rm {bulk}}$ were neglected within the membrane element to mimic the Hill-type nonlinear spring. Moreover, according to Günther et al. (2007), a linear damper was introduced and positioned in parallel with the tendon element to stabilize the forward dynamics integration. The musculotendon geometry and strength parameters were adopted from Millard et al. (2013), and its density values were taken from Talmadge et al. (2002), as listed in Table 8.

Fig. 18

Flexible musculotendon model. a A pennated muscle belly in series connection with a long tendon; b flexible multibody model based on the proposed membrane element

Table 8 Mechanical parameters of the musculotendon actuator (Talmadge et al. 2002; Millard et al. 2013)

As shown in Fig. 19a, the fiber length change predicted by our membrane model is consistent with Millard et al. (2013). During the tendon lengthening, its passive tension tracts the muscle fibers toward the longitudinal direction, decreasing the pennation angle to $\approx 20^\circ$. Afterward, the diminishing of tendon traction sharply increased the pennation angle to $\approx 90^\circ$ regardless of the activation level. Compared with Millard et al. (2013), the flexible multibody model presented almost the same pennation results except for the slack stage (Fig. 19b).

The musculotendon force results are further depicted in Fig. 19c and d. Compared with Millard et al. (2013), our membrane model performed a quantitatively different muscle force configuration (Fig. 19c) because of the differences of the Hill-type relation. Note that the equilibrium equations of the proposed flexible element were cast in weak form. As a result, the summation of fiber forces projected to the longitudinal direction was not identical to the tendon one (Fig. 19d). Moreover, by including a damper in series (Günther et al. 2007), the membrane model with distributed mass did not perform any oscillation even stimulated, improving the performances compared with Millard and Delp (2012).

Fig. 19

Simulation results of the musculotendon model of different activation levels ($a = 0.1,~0.5,~1.0$). Here, the solid line indicates our results, and the dashed line indicates the results performed by Millard et al. (2013); a muscle fiber length; b pennation angle; c muscle stress normalized by $\sigma ^{\rm {fiber}}_0$; d tendon stress normalized by $\sigma ^{\rm {fiber}}_0$. Norm. $=$ normalized

Appendix B: Adjustment of diaphragmatic thickness using in vivo measurements

To obtain the diaphragmatic thickness in vivo, we performed an ultrasound-based measurement based on seven male participants ($19.9 \pm 1.0$ years, $m=61.6 \pm {2.3}{\hbox {kg}}$, $h=170.9 \pm {1.2}{\hbox {cm}}$). The study protocol was approved by the Ethical Committee of Air Force Medical Center, PLA (2020-157-PJ01), and the written informed consent was obtained from each subject. Following the procedure presented by Goligher et al. (2015), a linear array transducer was placed in the ninth or tenth intercostal place near the mid-auxiliary line. The diaphragmatic thickness was measured at end-expiration and peak-inspiration instants (Table 9), and the observed data were then averaged to obtain the subject-specific thickness value.

Table 9 Diaphragmatic and PS thickness

Meanwhile, to ensure the physiologic consistency between the measured thickness and the other PCSA values, we also measured the PS thickness at the L4-L5 level by B-mode ultrasound. According to Takai et al. (2011), the PS cross-sectional area was then obtained using linear interpolation. Finally, a linear scaling law (Rasmussen et al. 2005) is adopted to obtain the diaphragmatic thickness (${4.46}{\hbox {mm}}$) in this model.

Appendix C: Sensitivity results of different modeling parameters

Fig. 20

Sensitivity study of muscle activations and IVD compression to muscle stiffness. a Activation with different $\sigma ^{\rm {fiber}}_0$; b activation under different stages of muscle deterioration; c IVD compression at L3–L4 level with different $\sigma ^{\rm {fiber}}_0$; (d) IVD compression at L3–L4 level under muscle deterioration

Fig. 21

Sensitivity study of muscle activations and IVD compression to the core muscle cross-sections. a Activation with different TA PCSA; b activation with different diaphragm PCSA; c IVD compression at L3–L4 level with different TA PCSA; d IVD compression at L3–L4 level with different diaphragm PCSA