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Effect of intervertebral translational flexibilities on estimations of trunk muscle forces, kinematics, loads, and stability a

a

b

Farshid Ghezelbash , Navid Arjmand & Aboulfazl Shirazi-Adl a

Department of Mechanical Engineering, Sharif University of Technology, Azadi Avenue, Tehran 11155-9567, Iran b

Division of Applied Mechanics, Department of Mechanical Engineering, École Polytechnique, C.P. 6079, Succ. Centre-ville, Montréal, Québec, Canada H3C 3A7 Published online: 17 Sep 2014.

To cite this article: Farshid Ghezelbash, Navid Arjmand & Aboulfazl Shirazi-Adl (2014): Effect of intervertebral translational flexibilities on estimations of trunk muscle forces, kinematics, loads, and stability, Computer Methods in Biomechanics and Biomedical Engineering, DOI: 10.1080/10255842.2014.961440 To link to this article: http://dx.doi.org/10.1080/10255842.2014.961440

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Computer Methods in Biomechanics and Biomedical Engineering, 2014 http://dx.doi.org/10.1080/10255842.2014.961440

Effect of intervertebral translational flexibilities on estimations of trunk muscle forces, kinematics, loads, and stability Farshid Ghezelbasha1, Navid Arjmanda* and Aboulfazl Shirazi-Adlb2 a

Department of Mechanical Engineering, Sharif University of Technology, Azadi Avenue, Tehran 11155-9567, Iran; b Division of Applied Mechanics, Department of Mechanical Engineering, Ećole Polytechnique, C.P. 6079, Succ. Centre-ville, Montreál, Que´bec, Canada H3C 3A7

Downloaded by [Ecole Polytechnique Montreal] at 10:15 20 September 2014

(Received 25 May 2014; accepted 1 September 2014) Due to the complexity of the human spinal motion segments, the intervertebral joints are often simulated in the musculoskeletal trunk models as pivots thus allowing no translational degrees of freedom (DOFs). This work aims to investigate, for the first time, the effect of such widely used assumption on trunk muscle forces, spinal loads, kinematics, and stability during a number of static activities. To address this, the shear deformable beam elements used in our nonlinear finite element (OFE) musculoskeletal model of the trunk were either substantially stiffened in translational directions (SFE model) or replaced by hinge joints interconnected through rotational springs (HFE model). Results indicated that ignoring intervertebral translational DOFs had in general low to moderate impact on model predictions. Compared with the OFE model, the SFE and HFE models predicted generally larger L4 – L5 and L5 – S1 compression and shear loads, especially for tasks with greater trunk angles; differences reached , 15% for the L4 –L5 compression, , 36% for the L4 –L5 shear and ,18% for the L5– S1 shear loads. Such differences increased, as location of the hinge joints in the HFE model moved from the mid-disc height to either the lower or upper endplates. Stability analyses of these models for some select activities revealed small changes in predicted margin of stability. Model studies dealing exclusively with the estimation of spinal loads and/or stability may, hence with small loss of accuracy, neglect intervertebral translational DOFs at smaller trunk flexion angles for the sake of computational simplicity. Keywords: spine; model; optimization; intervertebral joint; degrees of freedom

Introduction Proper estimation of trunk muscle forces, spinal loads, and stability during physical activities is essential to evaluate risk of workplace injury, design effective prevention programs, and improve existing rehabilitation and performance enhancement programs. As a consequence and in absence of non-invasive in vivo approaches, biomechanical models are extensively used as the only viable means available. Due to the complexity of spinal motion segments, simplifying assumptions are often made when simulating the lumbar spine. Two common assumptions include the enforcement of the moment equilibrium requirements at only a single lumbar level (typically the L3 –L4 or L4 –L5 joint) with no consideration for the equilibrium at the remaining lumbar joints (van Dieen et al. 2003; Granata et al. 2005; Marras et al. 2006) and the simulation of intervertebral joints as pivots with no translational degrees of freedom (DOFs) (Cholewicki and McGill 1996; Granata and Wilson 2001). We have previously demonstrated that satisfying the equilibrium at only a single lumbar joint results in erroneous estimation of trunk muscle forces that violate moment equilibriums at the remaining lumbar joints (Arjmand et al. 2007). As for the latter assumption, i.e., neglecting

*Corresponding author. Email: [email protected] q 2014 Taylor & Francis

translational DOFs of intervertebral joints, lumbar motion segments are usually modeled as ball-and-socket (hinge or spherical) joints supported via rotational springs (e.g., Cholewicki and McGill 1996; Granata and Wilson 2001; Hajihosseinali et al. 2014; AnyBody Modelling System). These springs exclusively represent rotational stiffness of the passive motion segments (intervertebral discs, posterior elements, and ligaments) neglecting thereby the shear and axial compliance of the joints. In vivo imaging studies demonstrate, albeit small, axial, and transverse movements between the lumbar vertebrae during physical activities (Hayes et al. 1989; Frobin et al. 1996; McGregor et al. 2002; Li et al. 2009; Aiyangar et al. 2014). While this assumption is widely used in both single- and multi-joint models of the spine as well as commercial software (e.g., AnyBody Modelling System; the University of Michigan’s 3D Static Strength Prediction Programe, 3DSSPP), its effect on predictions of muscle forces, spinal loads, and stability remains yet unknown. The present study, therefore, aims to quantify, for the first time, the likely effect of neglecting intervertebral lumbar translational DOFs on trunk muscle forces as well as spinal loads, kinematics, and stability for a number of physical static activities. To address this, our kinematics-


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F. Ghezelbash et al.

driven, nonlinear finite element (FE), musculoskeletal model of the trunk is used (Arjmand and Shirazi-Adl 2005, 2006a; Arjmand et al. 2006, 2009, 2010). This is the only multi-joint FE model of the spine that simultaneously accounts for all rotational/translational DOFs of the thoracolumbar T12 – S1 joints and thus associated moment/force equilibrium requirements, passive nonlinear material properties of the ligamentous thoracolumbar motion segments in different directions, detailed passive – active trunk systems, complex anatomy of muscles, and wrapping of thoracic muscles around bony structures. To evaluate the relative effect of translational DOFs on predictions, shear deformable beam elements used in this model to represent the overall nonlinear stiffness of the T12 – S1 motion segments are either substantially stiffened in translational directions or replaced with hinge joints interconnected through rotational springs. Due to the likely effects of kinematics on load position, muscle wrapping/ length, and passive ligamentous properties, we hypothesize that alterations in shear and axial flexibilities of motion segments influence the trunk active and passive response depending on the loading and posture considered. Such investigations are essential to assess the accuracy of the simulation models/software and hence their applicability in ergonomic, rehabilitation, and biomechanical applications. Method To evaluate the role of translational DOFs on model predictions, three distinct models were used: (1) the original kinematics-driven FE model as the reference one (OFE), (2)

the model with stiffened shear and axial rigidities along the segmental translational DOFs (SFE), and (3) the hinge model in which the translational DOFs are totally ignored (HFE). A number of tasks were simulated by each model to calculate and compare spinal kinematics, net moments at different levels, muscle forces, L4 – L5 and L5 – S1 compression, and shear loads as well as stability margins of the spine. Simulated tasks included the static symmetric trunk flexion from the neutral upright to forward flexion of 908 (total forward trunk rotation as the sum of pelvis and lumbar rotations) with an interval of 108 while holding no load or 180 N in hands (Arjmand et al. 2009, 2010).

Original FE model (OFE) The T1 – S1 nonlinear FE model (ABAQUS, Simulia Inc., Providence, RI, USA) of the thoracolumbar spine along with the kinematics-driven algorithm (see below) were employed to predict trunk muscle forces, L4 –L5 and L5– S1 compressive and shear loads and spinal stability for the simulated tasks (Arjmand and Shirazi-Adl 2005, 2006a; Arjmand et al. 2006, 2009, 2010). Six quadratic shear deformable beams with nonlinear properties to represent T12 –S1 segments and seven rigid elements to represent T1 –T12 (as a single body) and lumbosacral vertebrae (L1 – S1) were used (Figure 1(a)). The deformable beams represented the overall nonlinear stiffness of T12 – S1 motion segments (i.e., vertebrae, disc, facets, and ligaments) at different directions and levels thus modeling all rotational and translational DOFs. Deformable beams

Figure 1. A schematic of (a) the original finite element (OFE) model, its global and local musculatures in (b) the frontal and (c) sagittal planes (only fascicles on one side are shown) as well as (d) the hinge-joint finite element (HFE) model (with identical musculatures). ICPL: iliocostalis, lumborum pars lumborum, ICPT, iliocostalis lumborum pars thoracic; IP, iliopsoas; LGPL, longissimus thoracis pars lumborum; LGPT, longissimus thoracis pars thoracic; MF, multifidus; QL, quadratus lumborum; IO, internal oblique; EO, external oblique; RA, rectus abdominus.


Computer Methods in Biomechanics and Biomedical Engineering were shifted posteriorly from the end-plate centers by 4 mm to account for the posterior movement in the disc axis of rotation observed under loads in different directions (Shirazi-Adl et al. 1986). The nonlinear force – displacement and moment –rotation responses of the deformable beams were considered based on numerical and measured results of previous studies (Arjmand and Shirazi-Adl 2006a; Shirazi-Adl 2006). For a total body weight of 68 kg and based on available anthropometric data, an upper body weight of 344.4 N was considered in the model (Arjmand et al. 2009, 2010). Weights of upper arms (35.6 N), forearms/hands (29.3 N) and head (46 N) were estimated and applied in each loading case at their mass centers estimated based on in vivo measurements (Arjmand et al. 2009, 2010). The remaining upper body weight (i.e., 344.4 2 35.6 2 29.3 2 46 ¼ 233.4 N) was distributed eccentrically along the entire length of the spine from the T1 to the L5 (Pearsall 1994; Arjmand and Shirazi-Adl 2006a). In the FE model, the partitioned gravity forces were applied off center on each vertebra via rigid links. The weight in hands was applied at its location measured in vivo via a rigid element attached to the T3 vertebra. Position of weight in hands (180 N) as well as centers of mass of arm segments with respect to the T3 (shoulder) were taken identical for the three models. A sagittally symmetric muscle architecture with 46 local (attached to lumbar vertebrae) and 10 global (attached to the thoracic cage) muscle fascicles was considered (Stokes and Gardner-Morse 1999) (Figure 1(b), (c)). To simulate curved paths of back muscles in forward flexion tasks, wrapping of global extensor muscles around lumbar vertebrae along with generated contact forces were considered (Arjmand et al. 2006). Inputs of the model included the thorax sagittal rotation (T), pelvis sagittal rotation (P) as well as concentrated external and distributed gravity loads. These inputs were measured in earlier in vivo studies (Arjmand et al. 2009, 2010) for different activities. Total lumbar rotation (L), calculated as the difference between the foregoing two rotations (L ¼ T 2 P), was subsequently partitioned between individual thoracolumbar vertebrae (T12 to L5) in accordance with the proportions reported in earlier investigations (11% at T12 – L1, 15% at L1– L2, 14% at L2 –L3, 18% at L3 – L4, 22% at L4 –L5, and 20% at L5 – S1 level) (Arjmand and Shirazi-Adl 2006a; Arjmand et al. 2009, 2010). A prescribed rotation at each vertebral level (T12 –L5) generates a moment equilibrium equation in the form of Sr £ f ¼ M where r, f, and M are lever arm of muscles with respect to the vertebra to which they are attached, unknown total forces in muscles attached to the level under consideration and required net moment, respectively. The net moment at each level is the reaction moment predicted by the FE model due to the prescribed rotation and is balanced by muscles attached to the same level. To resolve this redundant equation at each spinal

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level, an optimization algorithm with the cost function of sum of cubed muscle stresses was used along with inequality equations of unknown muscle forces remaining greater than their passive-force components (calculated based on muscle strain and a tension – length relationship (Davis et al. 2003)) but smaller than the maximum physiological active forces (i.e., 0.6 MPa £ physiological cross-sectional areas) plus the passive-force components. The maximum allowable muscle stress of 0.6 MPa is in the mid-range of those reported in the literature (0.3 – 1.0 MPa) (Arjmand and Shirazi-Adl 2006a). In each iteration, muscle forces were fed back as updated external loads onto the nonlinear FE model at the vertebral level to which they were attached, and the iteration was repeated till convergence was reached. Once muscle forces were calculated, the equilibrium of forces was considered to determine local axial compression and anterior – posterior shear loads at the L4 – L5 and L5 – S1 joints. Stability analyses were subsequently performed while modeling muscles with uniaxial elements each having stiffness, k, assigned using the linear stiffness– force relation k ¼ qf/l in which the muscle stiffness was proportional to the muscle force, f, and inversely proportional to its instantaneous length, l, with q as a non-dimensional muscle stiffness coefficient taken the same for all muscles (Arjmand and Shirazi-Adl 2005, 2006a). Nonlinear analyses were performed for different q values thus identifying the minimum (critical) q value above which a convergent solution in a force-controlled loading environment existed. In addition to nonlinear analyses, linear perturbation analyses at loaded, deformed, configurations were also carried out to further estimate stability margin in sagittal and frontal planes.

Stiffened FE model (SFE) This model is identical to the OFE model except that the axial and transverse force –displacement properties of the flexible beams were stiffened by 1000 times to practically eliminate the translational DOFs. Identical inputs and procedures were used to estimate muscle forces, spinal loads, and stability.

Hinge-joint FE model (HFE) In this model, the flexible beam elements in the OFE model were replaced by hinge (spherical) joints located at the mid-heights of the respective beams (Figure 1(d)). At each joint, three nonlinear springs with distinct passive stiffness were added to connect adjacent rigid vertebrae. Moment – rotation relationship of springs was subsequently evaluated in a manner to have identical moment – rotation response in individual motion segments of these two models, OFE and HFE. To do so for each


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F. Ghezelbash et al. Results The SFE and HFE models predicted very similar spinal deformed configurations but slightly different from those predicted by the OFE model especially so under more demanding activities at larger flexion angles with 180 N in hands (Figure 2). The maximal difference between the HFE and OFE models for the predicted position of the head remained smaller than 12 mm for the most demanding activity, i.e., lifting 180 N at trunk flexion of 908. Correlation analyses indicated that the three models had pattern-wise close predictions in T12 – L5 net moments, sum of global/local muscle forces, as well as the L4 –L5 and L5 –S1 spinal loads (everywhere larger than 0.98 for SFE or HFE vs OFE models). Differences in the predicted net moments were slightly more pronounced between the HFE and OFE models as compared with those between the SFE and OFE models (Table 1). The peak difference remained, however, smaller than 20% (less than 5 N m) at any level for any given activity simulated in this study. Compared with the OFE model, the SFE and HFE models predicted larger global muscle forces (peak of ,45%) although the differences in all simulated tasks but three (flexing forward by 708, 808, and 908 without hand load) remained smaller than 8% (Table 2). Also, the SFE and HFE models predicted slightly different local muscle forces compared with the OFE model (peak difference of ,8%) (Table 2). The SFE and HFE models predicted generally larger compression loads on both the L4–L5 and L5–S1 joints as compared to the OFE model, especially for tasks with greater trunk angles without load in hands that reached ,15% (differences in all simulated tasks but three, i.e., flexing forward by 708, 808, and 908 without hand load,

Figure 2. Spine configuration under load holding of 180 N at 908 trunk flexion predicted by the OFE, SFE, and HFE models in the sagittal plane.

spring at each level and plane, the corresponding motion segment (i.e., two rigid vertebral beams and the flexible intervertebral disc beam in-between) was separated from the OFE model, the lower rigid vertebra was fully fixed, and rotations up to 6 158 were applied to the upper rigid vertebra (separately in each plane) while recording the reaction moment at the lower fixed end. Moreover, to investigate the effect of changes in the location of the hinge along the disc height on the model predictions of the HFE model, the center of hinge joints were also placed once on the lower (LHFE model) and once on the upper (UHFE model) end-plates, and the analyses were repeated. Identical inputs and procedures were used to estimate muscle forces, spinal loads and stability.

Table 1. Predicted net moments (N m) by the OFE, SFE, and HFE models at different levels (T12 through S1) under various activities in upright and flexed postures with and without 180 N in hands. Flexion (8) without hand load Level T12 L1 L2 L3 L4 L5

Flexion (8) with 180 N in hands

Model

0

10

20

30

40

50

60

70

80

90

0

10

20

30

40

50

60

70

80

90

OFE SFE HFE OFE SFE HFE OFE SFE HFE OFE SFE HFE OFE SFE HFE OFE SFE HFE

3 4 – 1 1 – 0 0 – 0 0 – 0 0 – 0 0 –

10 11 11 1 1 1 1 1 1 2 2 2 1 1 1 2 2 2

16 17 17 3 3 3 3 3 3 4 3 3 2 2 2 4 4 4

20 22 22 4 4 4 5 5 5 5 5 5 4 4 4 4 4 4

23 25 25 5 6 6 7 7 7 8 7 8 5 6 6 3 4 4

24 26 26 6 7 7 9 9 9 11 11 11 7 7 7 4 4 3

23 26 26 7 8 8 10 11 11 13 13 13 8 8 9 5 4 4

23 25 25 8 9 9 11 12 12 15 14 15 9 10 10 6 5 5

21 23 23 8 10 10 12 14 14 16 16 17 10 11 11 7 6 6

18 20 20 9 10 10 13 15 15 18 18 18 11 12 12 7 7 6

31 31 – 0 0 – 0 0 – 0 0 – 1 0 – 5 5 –

39 40 40 1 0 0 0 0 0 2 2 2 3 4 4 7 7 7

49 51 51 3 3 3 3 3 2 4 4 4 6 6 7 10 10 10

56 59 59 6 6 6 6 6 6 7 6 6 9 9 9 11 12 12

62 65 66 8 8 9 9 9 9 11 10 10 11 12 12 13 12 12

67 71 71 10 11 11 11 12 12 14 14 14 14 14 14 15 13 13

66 70 71 12 13 14 14 15 15 17 17 17 16 16 17 16 14 13

65 69 70 14 15 16 17 18 18 20 20 20 18 19 19 18 15 14

61 65 65 16 17 17 19 20 21 23 23 23 19 20 21 18 16 15

55 59 59 16 18 19 21 22 23 26 25 26 21 22 22 18 16 15

Note: The HFE model did not converge for activities performed in upright standing posture.

Computer Methods in Biomechanics and Biomedical Engineering

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Table 2. Predicted sum of all global and sum of all local muscle forces (total forces on both left and right sides) by the OFE, SFE, and HFE models under various activities in upright and flexed postures with and without 180 N in hands. Flexion angle (8)


Sum of muscle forces (N)

Hand load (N)

Global muscles

0

Local muscles

0

Global muscles

180

Local muscles

180

Model

0

10

20

30

40

50

60

70

80

90

OFE SFE HFE OFE SFE HFE OFE SFE HFE OFE SFE HFE

61 69 – 49 42 – 572 578 – 183 169 –

183 192 194 212 209 208 715 736 739 396 376 376

298 312 313 438 437 439 904 932 936 779 758 762

375 397 399 628 630 634 1036 1081 1085 1112 1091 1099

422 456 458 808 818 819 1138 1204 1209 1457 1422 1429

445 482 484 976 992 1001 1230 1303 1310 1775 1742 1755

430 465 468 1130 1167 1178 1216 1296 1303 2049 2018 2038

417 607 598 1299 1304 1321 1192 1274 1281 2328 2302 2326

537 772 761 1384 1386 1406 1118 1197 1204 2542 2530 2556

672 930 920 1408 1420 1436 1003 1085 1082 2677 2692 2715


remained smaller than 7%) (Table 3 and Figure 3(a)). As for the anterior–posterior shear forces, the SFE and HFE models also predicted larger loads on both the L4–L5 and L5–S1 joints as compared with the OFE model (Table 3 and Figure 3 (b)) with differences reaching ,18% at the critical L5–S1 level and ,36% at the L4–L5 level (Figure 3(b)). As the location of the hinge joints moved from the mid-disc height to either the lower (LHFE) or upper (UHFE) endplates, such differences with the OFE model for the shear forces

increased to ,30% at the critical L5–S1 level (Figure 4). Stability analyses for some select activities revealed small differences in critical muscle stiffness q between the OFE, SFE, and HFE models (Table 4). Discussion Neglecting intervertebral translational DOFs is a common assumption that significantly reduces the computational

Table 3. Predicted compressive and anterior – posterior shear loads at the L4 – L5 and L5 – S1 levels by the OFE, SFE, and HFE models under various activities in upright and flexed postures with and without 180 N in hands. Flexion angle (8) Force (N)

Hand load (N)

Model

0

10

20

30

40

50

60

70

80

90

L4 – L5 compression

0

L4 – L5 shear

0

L5 – S1 compression

0

L5 – S1 shear

0

OFE SFE HFE OFE SFE HFE OFE SFE HFE OFE SFE HFE OFE SFE HFE OFE SFE HFE OFE SFE HFE OFE SFE HFE

421 422 – 28 29 – 430 432 – 134 136 – 1085 1079 – 147 152 – 1183 1176 – 410 416 –

645 648 651 43 46 45 687 691 692 227 232 231 1378 1381 1386 127 137 134 1504 1501 1507 515 535 529

934 943 947 84 90 88 990 1001 1005 338 348 346 1847 1859 1868 189 211 205 2016 2016 2027 687 729 719

1181 1198 1205 110 121 118 1230 1250 1257 412 431 428 2259 2282 2297 227 267 257 2437 2447 2465 805 874 861

1404 1435 1448 129 146 142 1433 1472 1475 463 492 490 2645 2691 2716 252 315 303 2843 2856 2873 892 993 979

1563 1629 1643 141 161 158 1607 1653 1667 483 522 518 2977 3069 3100 268 348 339 3217 3237 3263 946 1079 1060

1659 1748 1769 149 171 168 1733 1798 1813 476 518 514 3172 3292 3334 273 366 356 3452 3482 3514 939 1091 1069

1760 1956 1975 167 192 188 1863 2032 2042 486 565 557 3342 3486 3534 286 396 384 3665 3701 3739 943 1116 1091

1893 2133 2154 178 207 202 2017 2225 2236 509 602 592 3414 3579 3631 296 416 402 3763 3812 3852 925 1106 1079

1983 2255 2276 183 216 210 2119 2357 2365 522 627 615 3388 3575 3618 306 427 412 3737 3820 3847 890 1069 1038

L4 – L5 compression

180

L4 – L5 shear

180

L5 – S1 compression

180

L5 – S1 shear

180


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F. Ghezelbash et al. Table 4. Critical muscle stiffness coefficient (q) predicted by the OFE, SFE, and HFE models in activities involving trunk flexion of 108, 208, 308, and 408 without hand load (the smaller the q is the more stable the spine is). Flexion angle (8) without hand load

Model


OFE SFE HFE

Figure 3. Relative differences (as percentage) between predictions of the HFE model with those of the OFE model for (a) compression and (b) anterior– posterior shear loads on the L4 – L5 and L5 –S1 joints under different activities in flexed postures with and without 180 N in hands.

Figure 4. Relative differences (as percentage) between predictions of the LHFE and UHFE models with those of the OFE model for the predicted L5 –S1 (a) compression and (b) anterior – posterior shear loads under different activities in flexed postures with and without 180 N in hands.

10

20

30

40

17.3 17 17.1

9.5 9.5 8.8

6.8 7.1 7.4

5.6 5.8 6.2

complexity in the biomechanical models of the spine. No study has, however, addressed the likely effect of such simplification on the model predictions. This study therefore was conducted to investigate the effect of neglecting intervertebral translational DOFs on predictions of a musculoskeletal biomechanical model for spinal kinematics, net moments at different levels, trunk muscle forces, lower lumbar loads, and stability. Results indicated that neglecting these translational DOFs (or equivalently the shear and axial flexibilities) while simulating the intervertebral joints as hinge connections supported by nonlinear rotational springs had for the most part a low to moderate effects, depending on the simulated task, on the predicted net moments, muscle forces, spinal kinematics, loading, and stability. Changes in the position of the hinge joints from mid-disc height to lower and upper disc endplates somewhat accentuated these differences to the reference model predictions. Methodological issues As discussed in detail elsewhere (Arjmand and ShiraziAdl 2006a; Arjmand et al. 2009, 2010, 2011), the OFE model predictions for the L4 – L5 intradiscal pressure and back muscle forces for a wide range of lifting activities were found in agreement with in vivo measured intradiscal pressure values and qualitatively with electromyography data (Arjmand and Shirazi-Adl 2006a; Arjmand et al. 2009, 2010, 2011). Also, the OFE model of the lumbar spine (T12 –S1) devoid of muscles predicted a critical buckling load of , 100 N in agreement with in vitro studies on isolated osteo-ligamentous lumbar spine of cadaver specimens reporting mean critical compression loads of 90– 120 N (Crisco and Panjabi 1992; Crisco et al. 1992; Patwardhan et al. 2001). This model similar to its counterparts, however, has its own assumptions and simplifications that are discussed in details elsewhere (Arjmand and Shirazi-Adl 2005, 2006a; Arjmand et al. 2006, 2009, 2010). To name a few, the likely unloading and stabilizing effect of the intra-abdominal pressure (Arjmand and Shirazi-Adl 2006b), co-activities of antagonistic abdominal muscles (Arjmand et al. 2008), and the nonlinearity in the stiffness – force relation assumed for all muscles (Shadmehr and Arbib 1992)


Computer Methods in Biomechanics and Biomedical Engineering were not considered. Moreover, all models, though more sever and extreme in the HFE model, tend to constrain the segmental center of rotation despite the fact that its location alters as a function of disc properties and loading (Wachowski et al. 2009). As the only difference between the three models is in the representation of the intervertebral joints, however, modeling simplifications would equally affect the predictions of all models considered in this study. For the HFE model in the upright posture, the net moments in some local levels fell slightly below zero (i.e., negative) while no flexor (abdominal) muscles were attached to these vertebrae to balance such small flexion moments. As expected, the optimization algorithm did not converge in such cases. A slightly different initial spinal posture could remedy these cases. Analysis of data In agreement with in vivo studies (Frobin et al. 1996; McGregor et al. 2002; Li et al. 2009), the OFE model predicted , 2 mm net total relative translations in the sagittal plane (i.e., resultant of both axial and anterior – posterior translations after subtracting those due to rotation) between any two adjacent vertebrae (T12 and L1 through L5 and S1) under the most demanding task simulated in this study, i.e., lifting 180 N at 908 trunk flexion. For this task, for instance, these relative translations reached , 1.5, 1.9, 1.7, 1.3, 1.1, and 0.7 mm for L5 –S1, L4– L5, L3– L4, L2 – L3, L1 – L2, and T12– L1 vertebrae, respectively. These values are well within reported physiological limits (Hayes et al. 1989) and show similar trends as those measured in vivo under forward flexion activities in which intervertebral translations decreased from their maximal value at the L4 –L5 segment upward to the T12 – L1 level. Corresponding net intervertebral translations were exactly nil in the HFE model and negligible (, 0.15 mm) in the SFE model. Due to additional intervertebral translations predicted in the OFE model, the deformed spine configuration was slightly affected when eliminating the translational DOFs in the HFE model (Figure 2). Although the maximal difference between the HFE and OFE models for the predicted position of the head remained smaller than 12 mm (for the lifting of 180 N at trunk flexion of 908), the predicted passive force in back muscles were, however, greatly affected. This is because according to both in vivo and in vitro measurements (e.g., Davis et al. 2003), the muscle length – passive tension relationship is very stiff at larger muscle elongations (i.e., larger trunk flexion angles). A small difference in the predicted muscle length (, 10 mm larger muscle elongation in the SFE and HFE models as compared to the OFE model for the trunk flexion of 908), hence, resulted in significantly greater passive forces and thus total forces in back muscles in the SFE and HFE models (see Table 2 for tasks involving trunk flexion of 708,

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808, and 908 with no load in hands). As a consequence, flexion relaxation phenomenon in global muscles occurred at , 58 to 108 smaller trunk flexion angles in the SFE and HFE models as compared with the OFE model (, 608 vs 558 for ICPT muscle and , 708 vs 808 for the LGPT muscle). For the same reason, wrapping contact forces (between LGPT muscles, for instance, and the bony structures) were larger in the HFE (49 – 175 N) as compared with the OFE model (from 30 to 157 N) for the flexion of 908 with 180 N in hands. Had we excluded the lower limit on muscle forces (e.g., taken the lower limit equal to zero rather than the passive force in muscles) in the optimization algorithm the differences between the OFE and HFE models would have significantly reduced for these three tasks (to , 8% as in other tasks). On the other hand, in tasks with 180 N in hands, the lower limits on muscle forces set in the optimization algorithm played no role as the net moments and thus predicted muscle forces exceeded these lower bounds. Differences in the predicted muscle lengths and thus passive muscle forces between the OFE and HFE models had, therefore, no influence on the estimated total muscle forces (Table 2). Comparison of net moments at different levels (to be resisted by muscle forces attached to those levels) (Table 1) and muscle forces (Table 2) between the three models indicated a moderate (apart from the three aforementioned tasks) effect of neglecting translational DOFs on model predictions for muscle forces. Elimination of translational flexibilities in the SFE and HFE models resulted in generally larger contribution from muscles to balance external moments as compared with the OFE model (Table 2). Consequently and because a major portion of spinal loads is generated by muscle forces (Arjmand and Shirazi-Adl 2005), the predicted spinal joint loads were also moderately affected as translational DOFs were eliminated (Table 3 and Figure 3). On the other hand, changing the location of the hinge joints in the HFE model from the disc mid-height to adjacent endplates increased foregoing differences in shear forces at the critical level L5–S1 (Figure 4). Such differences between the OFE and HFE models could increase in presence of larger loads and trunk flexion angles. As stability is provided by stiffness from both activation of paraspinal muscles and the passive ligamentous spine (Arjmand and Shirazi-Adl 2006a), slight to moderate changes in the patterns of muscle activities and spinal load magnitudes as a result of elimination of translational flexibilities only slightly affected the margin of stability in flexed postures (Table 4). Also, in accordance with our previous analysis (Arjmand and Shirazi-Adl 2006a; Arjmand et al. 2006), all models predicted improved spinal stability, i.e. smaller critical q value, at larger trunk flexion angles associated with greater muscle and ligamentous passive properties. In conclusion, overlooking intervertebral translational DOFs in biomechanical models of the spine appeared to have generally slight to moderate impact on model

8

F. Ghezelbash et al.

predictions for muscle forces, spinal kinematics, loading, and stability when considering smaller trunk forward flexion angles. Modeling simulations dealing exclusively with the estimation of spinal loads and/or stability (with applications in ergonomics and occupational biomechanics) may, hence, neglect intervertebral translational DOFs under smaller trunk flexion angles for the sake of computational simplicity. In contrast, estimations of hinge joint models on the pattern of muscle activities or force in individual muscle fascicles (with application in rehabilitation engineering) should be interpreted with caution.


Funding This work is supported by grants from Sharif University of Technology (Tehran, Iran) and Institut de recherche Robert-Sauve´ en sante´ etensećurite´ du travail, IRSST (Montreál, Canada).

Notes 1. 2.

Email: [email protected] Email: [email protected]

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Computer Methods in Biomechanics and Biomedical ...

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