- Email: [email protected]
A biomechanical model on muscle forces in the transfer of spinal load to the pelvis and legs
Journal of Biomechanics 32 (1999) 927}933
A biomechanical model on muscle forces in the transfer of spinal load to the pelvis and legs Gilbert A. Hoek van Dijke *, Chris J. Snijders , Rob Stoeckart, Henk J. Stam Department of Biomedical Physics and Technology, Faculty of Medicine and Allied Health Sciences, Erasmus University, PO BO 1738, 3000DR, Rotterdam, The Netherlands Department of Anatomy, Faculty of Medicine and Allied Health Sciences, Erasmus University, PO BO 1738, 3000DR, Rotterdam, The Netherlands Department of Rehabilitation, Faculty of Medicine and Allied Health Sciences, Erasmus University, PO BO 1738, 3000DR, Rotterdam, The Netherlands Received 15 April 1999
Abstract Based on musculoskeletal anatomy of the lower back, abdominal wall, pelvis and upper legs, a biomechanical model has been developed on forces in the load transfer through the pelvis. The aim of this model is to obtain a tool for analyzing the relations between forces in muscles, ligaments and joints in the transfer of gravitational and external load from the upper body via the sacroiliac joints to the legs in normal situations and pathology. The study of the relation between muscle coordination patterns and forces in pelvic structures, in particular the sacroiliac joints, is relevant for a better understanding of the aetiology of low back pain and pelvic pain. The model comprises 94 muscle parts, 6 ligaments and 6 joints. It enables the calculation of forces in pelvic structures in various postures. The calculations are based on a linear/non-linear optimization scheme. To gain a better understanding of the function of individual muscles and ligaments, deviant properties of these structures can be preset. The model is validated by comparing calculations with EMG data from the literature. For agonistic muscles, good agreement is found between model calculations and EMG data. Antagonistic muscle activity is underestimated by the model. Imposed activity of modelled antagonistic muscles has a minor e!ect on the mutual proportions of agonistic muscle activities. Simulation of asymmetric muscle weakness shows higher activity of especially abdominal muscles. 1999 Elsevier Science Ltd. All rights reserved. Keywords: Biomechanics; Sacroiliac joint; Pelvis; Skeletal muscle; Anatomical model
1. Introduction Epidemiology indicates that low back pain involves both mechanical and non-mechanical factors. Both dynamic movements and static postures are reported to provoke low back pain (Andersson, 1981; Burdorf and Sorock, 1997). Patients with low back pain indicate especially static postures as painful (Deursen and Patijn, 1993). To identify mechanical factors that play a role in the aetiology of injuries or pain, several biomechanical models have been developed on joint and muscle forces, e.g. Anderson et al., 1985; Brand et al., 1982; Cholewicki and McGill, 1996; Crowninshield et al., 1978; Goel and Svensson, 1977; Gracovetsky et al., 1977; Granata and Marras, 1993; Koopman, 1989; Moroney et al., 1988; Plamondon et al., 1995; Scholten et al., 1988; Schultz and Andersson, 1981; Snijders et al., 1991; Tracy, 1990. More complete * Corresponding author. E-mail address: [email protected] (G.A. Hoek van Dijke)
surveys of biomechanical models can be found in review articles (e.g. Adams and Dolan, 1995; Cha$n, 1988; King, 1984). A recent concept emphasizes static postures (Snijders et al., 1993a; Snijders et al., 1993b). Violating speci"c requirements concerning the role of muscles in the stability of the sacroiliac (SI) joints may lead to the overload of structures in the low back region. No models are known to us that focus on the stability of the pelvis in relation to the load transfer from the upper body to the legs. The aim of this study is to develop a biomechanical model that can serve as a tool for further analysis of this load transfer by investigation of the separate contributions of muscles, ligaments and joints in di!erent postures.
2. Methods To develop a model suitable for analysis of forces in the pelvis, it was essential to represent the pelvis as three
0021-9290/99/$ - see front matter 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 1 - 9 2 9 0 ( 9 9 ) 0 0 0 8 5 - 8
928
G.A. Hoek van Dijke et al. / Journal of Biomechanics 32 (1999) 927}933
separate links (left and right hip bone and sacrum). The left and right femur, "ve lumbar vertebrae and the lowest thoracic vertebra were included as rigid links as well (see Fig. 1). Muscles and ligaments were selected on the basis of the expected importance for the load in the SI joints. Much attention is paid to muscles that cross the SI joints. Forces in these muscles necessarily cause reaction forces in the SI joints (Snijders et al., 1993a,b). Special attention is paid to di!erent parts of the erector spinae, connected either to the sacrum or to the hip bone. Muscles and ligaments are modelled as vector forces. Loads are balanced by muscle forces, ligament forces and joint reaction forces. Several sources had to be combined to complete the geometrical part of the model. A set of 164 transversal MRI-scans of a healthy young male adult (from T12 to the knee) functioned as a guide to combine the separate data sources. Of great help were the data of a model of the lumbar spine as listed by Cholewicki and McGill (1996) and the data made available by Koopman (1989) on his model of the human leg. This model is based on the model by Brand et al. (1982). The MRI-slices enabled the quanti"cation of 3D coordinates of muscle attachments when data were restricted to qualitative descriptions or directions of muscle "bres, as of the iliocostalis (Macintosh and Bogduk, 1987; Macintosh and Bogduk, 1991). Other data sources were Dostal and Andrews (1981), Dumas et al. (1988,1991), Granata and Marras (1993), McGill et al. (1988), Nemeth and Ohlsen (1985,1986), Sobotta (1994) and Visser et al. (1990). For muscles with broad attachments, more than one line of action was de"ned. Special attention is paid to the gluteus maximus. At the cranial side, this muscle has attachments to the sacrum (partially via the sacrotuberous ligament), the hip bone and the thoracolumbar fascia (Sobotta, 1994). Since forces in these three parts have fundamental di!erent e!ects on the load in the SI joints (Snijders et al., 1993a,b), this muscle had to be modelled as at least three separate force vectors. Since no literature was found that describes the anatomy of the gluteus maximus according to this point of view, in one embalmed specimen the left- and right-gluteus maximus were dissected. According to their area of attachment, the muscle "bers were distinguished in three groups. The three lines of action that represent the gluteus maximus were given a relative crosssectional area according to the mass proportions of these groups. The psoas major muscle has been modelled as two serial lines of action representing the upper and lower part. The force in both lines of action is required to be equal. In this way, the psoas major is modelled as a rope that runs over a pulley connected to the ramus superior. A similar procedure has been used for the superior part of the gluteus maximus that originates from the thoracolumbar fascia and inserts on the femur.
Table 1 Lines of action (unilateral) for structures incorporated in the model Name structure Longissimus Iliocostalis Multi"dus Quadratus lumborum Rectus abdominis Internal oblique External oblique Transversus abdominis Psoas Gluteus maximus Gluteus medius Gluteus minimus Biceps Semimembranosus Semitendinosus Gracilis Rectus femoris Iliacus Sartorius Pectineus Adductor magnus Adductor brevis Adductor longus Piriformis Gemellus inferior Obturatorius externus Obturatorius internus Quadratus femoris Gemellus superior Tensor fascia latae Sacrotuberous ligament Sacrospinal ligament Posterior SI ligament Hip joint SI joint Pubic symphysis L5-S1 joint
Lines of action 1 1 1 5 1 2 2 1 2 4 3 3 1 1 1 1 1 1 1 1 3 2 1 1 1 1 1 1 1 1 1 1 1 3 3 3 3
In the model, 94 vectors for muscle forces, 6 vectors for ligament forces and 18 force vectors for the 6 joints (three perpendicular vector components for each joint) are incorporated, see Table 1. A detailed description of the model will be provided on request. The kinematic algorithms that describe relative motions of the links and the positions of mass centres are based on data that was made available by De Looze et al. (1992). The mutual proportions of rotation of vertebrae, which are assumed to be constant during motion, are taken from Frobin et al. (1996) and Pearcy et al. (1984). The vertebrae are included to derive in various positions the direction of the lines of action of muscles that are connected to the vertebrae. For the calculation of forces, the upper body is regarded as one link. The load in the model consists of the weight of the upper body and, if wanted, an external load (e.g. a weight in the hand). For each load situation, the equilibrium of
G.A. Hoek van Dijke et al. / Journal of Biomechanics 32 (1999) 927}933
the links has to be calculated. Since the model is statically indeterminate, additional criteria had to be formulated to select a speci"c combination of muscle forces, ligament forces and joint reaction forces. This study especially concerns static postures. Therefore, an optimization technique was chosen that minimizes the highest value of the occurring muscle contraction intensities. In this way, the upper limit for the muscle stress valid for all muscles is minimized, which may be considered as maximizing the endurance time. However, this criterion does not necessarily lead to a unique solution: it is possible that a trunk muscle, e.g. the erector spinae, determines the lowest possible value for the maximal muscle stress, while still an in"nite number of solutions exists for the muscle forces around the hip. To come to a unique solution, a supplementary, quadratic criterion minimizes the sum of squared muscle stresses, with the additional constraint that no muscle stress may exceed the value that follows from the "rst criterion. This second criterion distributes the load (within the boundaries according to the "rst criterion) over synergetic muscles. However, strictly upholding this second criterion leads to unrealistic activity of muscles when a small decrease of the maximum muscle stress can be reached by recruiting many muscles. To avoid these situations, this second criterion was facilitated somewhat by allowing the maximum muscle stress to exceed this theoretical minimum value with a certain factor. For this factor, the value 0.1 has been chosen since this appeared to yield plausible results. A custom made computer program describes the positions of all points of the model in di!erent postures. The program GAMS (Brooke et al., 1996) is used for the determination of the optimal solution for the forces. Both programs run on a personal computer. The reliability of the model is tested by comparing calculated muscle stresses with measured EMG activities that are described in the literature. To be suitable for model simulation, the experiments had to be de"ned accurately. Mutual proportions of EMG activities had to be given, either the activity of various muscles in at least one load situation, or the activity of a single muscle in a variety of load situations. Due to these strict requirements, only a limited number of experiments could be simulated. The activity as a percentage of the maximum voluntary contraction (MVC) is regarded as a measure for the muscle stress. To test the functioning of the algorithms that handle additional constraints and to show the e!ect of deviant coordination patterns, a simulation was performed of limited strength of the superior part of the gluteus maximus that origins from the thoracolumbar fascia. In the concept of load transfer as described by Snijders et al. (1993a,b), the contribution of the pubic symphysis is assumed to be of minor importance. To investigate the consequences of this aspect, the forces in the symphysis are set to zero.
929
Fig. 1. Ventral view in the upright position and lateral view at 303 trunk #exion of the model. The origin of the coordinate system is located in the midsagittal plane between the rotation centres of the hip joints. In order not to hide them, all lines of actions are drawn on top of images of the bony structures.
3. Results A ventral view and a lateral view of the model are shown in Fig. 1. The contours of the bony structures, as derived from the MRI scans, serve as a reference only. The relationship between the hip #exion angle with the trunk held straight and the normalized EMG values of back muscles, as published by Andersson et al. (1996), is given in Fig. 2. In addition, the corresponding model calculated muscle stresses are drawn. The posture with 903 hip #exion is omitted since this posture is assumed to be too close to the end of the range of motion of the hip joint to neglect the force contribution of joint capsules. With the scaling factor 1% MVC"0.0037 N/mm, the calculated muscle stresses are within the reported standard error of the EMG measurements, with two exceptions at 603 hip #exion (see the arrows in the graph). Compared to the model calculations, here a slightly higher activity of the quadratus lumborum muscle is found whereas a signi"cant deviation is found for the super"cial erector spinae. Except for this latter point, good mutual proportions of muscle stresses are found (coe$cient of correlation is 0.904).
930
G.A. Hoek van Dijke et al. / Journal of Biomechanics 32 (1999) 927}933
Fig. 2. Normalized EMG values of low back muscles at the L level at 0, 30 and 603 hip #exion with the trunk held straight (Andersson et al., 1996) (vertical bars) and calculated muscle stresses obtained from a model simulation of this experiment (solid lines). The vertical bars indicate the standard error of the measurements. Fig. 4. Relationship between the registered EMG activities during isometric axial trunk torque (McGill, 1991) and the model calculated muscle stresses in a simulation of this experiment. Here, the antagonistic internal oblique abdominal and antagonistic erector spinae are imposed to generate tension comparable with the in vivo observed activity. See also Fig. 3.
Fig. 3. Relationship between the registered EMG activities during isometric axial trunk torque (McGill, 1991) and the model calculated muscle stresses in a simulation of this experiment, expressed as a percentage of the highest observed activity (here in the internal oblique abdominal). The abbreviations refer to the internal oblique (IO), external oblique (EO), erector spinae (ES) and rectus abdominis (RA). The vertical bars indicate the standard deviation of the EMG measurements.
Fig. 3 shows the relationship between the registered EMG activities of trunk muscles during isometric axial trunk torque (McGill, 1991) and a model simulation. The index &ag' (agonistic) refers to muscles at that side of the body where they are assumed to support axial torque generation. For rotating to the right (attempting to push the left shoulder forward), these agonists are the left external oblique, the right internal oblique and the right erector spinae. The index &ant' (antagonistic) refers to muscles at the opposite side. For the rectus abdominis muscles, no consistent relation is found between the direction of the applied torque and the activity at the right or left side (McGill, 1991). Therefore, the data of the left and right rectus abdominis are combined. For the internal and external oblique, the EMG activities are compared with the mean value of the model calculated
stresses in the separate lines of action of these muscles. The activity of the erector spinae is compared with the mean calculated stress in the longissimus and iliocostalis muscles. For agonistic muscles and the rectus abdominis, a good correlation is found between the calculated muscle stress and the EMG activity (correlation coe$cient 0.984). Antagonistic muscle activity is not predicted by the model. The activities of the antagonistic internal and external obliques appear to be coupled: adding a supplementary condition to the model that imposes the antagonistic internal oblique to generate tension causes also the antagonistic external oblique to generate tension (see Fig. 4). The mutual proportions of agonistic muscle activities remain roughly the same. A comparable result is found when the modelled antagonistic longissimus and iliocostalis are forced to generate tension. Fig. 5 shows the relation between the EMG activity of the erector spinae as measured by Seroussi and Pope (1987) and calculated muscle stresses in asymmetric load situations. In the upright position, moments in the sagittal and frontal plane were varied systematically by holding a weight in 28 asymmetric positions. The correlation coe$cient is 0.869. In the same load situations, the activity of the external oblique muscle was analyzed. Fig. 6 shows the relationship between the frontal plane moment arm and the activity of the external oblique muscle, for both the EMG measurements (Seroussi and Pope, 1987) and the model simulation. For four di!erent sagittal plane moment arms, Seroussi and Pope found almost the same curve. This "nding is con"rmed by our model: the di!erence between the calculated muscle stresses in the highest and lowest sagittal plane load situations was only 3%. Again,
G.A. Hoek van Dijke et al. / Journal of Biomechanics 32 (1999) 927}933
Fig. 5. Relationship between registered EMG signals of the erector spinae under 28 varying asymmetric sagittal and frontal plane loading conditions (Seroussi and Pope, 1987) and the corresponding model calculated muscle stresses in a simulation of this experiment (*). The line represents a linear "t of this relationship.
931
Fig. 7. Model calculated muscle forces during quasi-static trunk #exion with simulated weakness of the superior part of the left gluteus maximus. Here, the force in this muscle part is limited at 250 N (horizontal line). The solid lines, which concern muscle forces with simulated muscle weakness, show increased activity of abdominal muscles. The dotted lines re#ect forces in the situation without muscle weakness.
is raised, and the left external oblique abdominal becomes active.
4. Discussion
Fig. 6. Relationship between the frontal plane moment arm and the EMG activity of the external oblique muscle (Seroussi and Pope, 1987), and the model calculated results of a simulation of this experiment. The right half of the graph concerns the load situation in which the muscle functions as an agonist.
antagonistic activity (left half of the graph) is not predicted by the model. The correlation coe$cient for this relationship is 0.961. The use of the model as a tool for analyzing the role of individual structures and the e!ects of deviant coordination patterns is shown on the basis of a model simulation of quasi-static trunk #exion. In this simulation, asymmetric loss of muscle strength is introduced. The main results are shown in Fig. 7. At about 303 trunk #exion, the force in the concerning part of the left gluteus maximus reaches the given upper limit, which simulates muscle weakness. Beyond this angle, the decreased contribution of the left gluteus maximus in balancing the trunk is partly taken over by the right gluteus maximus. Abdominal muscles must contribute to restore the equilibrium of the links: the activity of the right internal oblique abdominal
The presented biomechanical model enables the analysis of the load transfer from the upper body to the pelvis and legs in static postures, which is an area of interest in the study of the aetiology of low back pain. The SI joint load is of special interest. A more detailed description of the SI joints is needed before conclusions can be drawn about safe or suspicious loading situations. For example, compression forces are considered as 'safe', where shear forces are considered as &unsafe' (Snijders et al., 1993a,b). However, due to the irregular surface of this joint, the e!ective direction of a force that must be considered as a shear force may di!er from the orientation of this joint surface itself (see Fig. 8). Also shear forces in various directions could have di!erent impacts. The number of passive structures in the model is small. No joint capsules are incorporated into the model yet, therefore, the model can be valid only for postures that are not close to the ends of the range of motion of the joints. At 603 hip #exion, the model prediction for the activities of the longissimus and iliocostalis deviates from the EMG measurements (see Fig. 2). Andersson et al. (1996) observed no increase of activity of these muscles during the latter half of the trunk motion. They assume thoracic muscles to contribute to the lumbar extensor moment transmitted via the thoracolumbar fascia. These muscles are not incorporated into our model, but additional equations and constraints could be formulated to meet
932
G.A. Hoek van Dijke et al. / Journal of Biomechanics 32 (1999) 927}933
Fig. 8. Schematic drawing of the irregular surface of the sacroiliac joint. Due to this irregularity, the e!ective shear direction may deviate considerably from the global joint surface direction.
such a mechanism. The predicted activities of the three involved back muscle groups (the longissimus and iliocostalis, the multi"dus and the quadratus lumborum) are in good proportions with respect to each other. The simulation of the asymmetric load experiments shows a good correlation between measured EMG data and model calculations for agonistic muscles (see Figs. 3}6). Antagonistic muscle activity is not predicted by the model (see Figs. 3 and 6). The force patterns in vivo can be seen as the sum of two components. The "rst component concerns the minimum forces required for equilibrium of the links, the second component concerns the contributions of both agonistic and antagonistic muscles for reasons as mechanical stability. This second component has no net e!ect. As a consequence of the present optimization criterion, only the "rst component of the forces is predicted by the model. EMG assisted mechanical models, such as the model by Granata and Marras (1993), have better capabilities to describe co-contraction of antagonistic muscles and are capable to estimate accurate numerical values for individuals. The aim of our model, however, is primarily to serve as a tool for analyzing the general relations between forces, rather than to focus on individuals. Therefore, it is desirable to simulate all kinds of "ctitious situations and muscle con"gurations (e.g. Fig. 7). Obviously, no EMG data are available for these deviant situations. To enable the use of EMG data as input for the model, the facility is made to de"ne upper and lower limits for each force. In this way, the e!ect of co-contraction can be investigated. This is shown in the simulation of the torsion experiments by McGill (1991) (see Fig. 4). The in#uence of the choice of optimization criteria and cross-sectional areas of muscles on calculated forces in the leg is investigated by Pedersen et al. (1987) and Brand et al. (1986). In general, various choices led to di!erent calculated forces, but obtained solutions were qualitatively similar. Again, this "nding emphasizes that the
most relevant use of this model is to determine general relations between forces. The use of the model as a tool to investigate the relations between muscle forces is shown by a simulation with a weakened part of the left-gluteus maximus muscle (see Fig. 7). The diminished contribution of this muscle can be compensated for by other muscles without inordinate forces. However, the increase of the forces in the compensating muscles, in this example the abdominals, is not neglectible. This implies that inadequate coordination patterns of muscles may exist without acute consequences, but they may lead to disorders at long terms. In spite of the fact that the forces in the pubic symphysis are set to zero in the model calculations, plausible results are obtained. This supports the supposition that the contribution of the symphysis to the transfer of gravitational loads is of minor importance. This study leads to the following conclusions: E The agonistic muscle activity patterns predicted by the model correspond well with EMG recorded activities in the literature; E In the model, antagonistic muscle activity can be imposed; E Pathology as weakened muscles can be simulated; E The contribution of the pubic symphysis to the transfer of gravitational loads seems to be of minor importance; E Deviant situations such as (asymmetric) muscle weakness require higher abdominal muscle activity.
Acknowledgements Authors thank the National Prevention Fund and the Anna Fund for their "nancial support, and B. Koopman and I. Kingma for sharing their data with us.
References Adams, M.A., Dolan, P., 1995. Recent advances in lumbar spinal mechanics, their clinical signi"cance. Clinical Biomechanics 10, 3}19. Anderson, C.K., Cha$n, D.B., Herrin, G.D., Matthews, L.S., 1985. A biomechanical model of the lumbosacral joint during lifting activities. Journal of Biomechanics 18, 571}584. Andersson, E.A., Oddsson, L.I.E., Grundstrom, H., Nilsson, J., Thorstensson, A., 1996. EMG activities of the quadratus lumborum and erector spinae muscles during #exion-relaxation and other motor tasks. Clinical Biomechanics 11, 392}400. Andersson, G.B., 1981. Epidemiologic aspects on low-back pain in industry. Spine 6, 53}60. Brand, R.A., Crowninshield, R.D., Wittstock, C.E., Pedersen, D.R., Clark, C.R., van Krieken, F.M., 1982. A model of lower extremity muscular anatomy. Journal of Biomechanical Engineering 104, 304}310. Brand, R.A., Pedersen, D.R., Friederich, J.A., 1986. The sensitivity of muscle force predictions to changes in physiologic cross-sectional area. Journal of Biomechanics 19, 589}596.
G.A. Hoek van Dijke et al. / Journal of Biomechanics 32 (1999) 927}933 Brooke, A., Kendrick, D., Meeraus, A., 1996. GAMS (General Algebraic Modeling System). Burdorf, A., Sorock, G., 1997. Positive and negative evidence of risk factors for back disorders. Scandinavian Journal of Work and Environmental Health 23, 243}256. Cha$n, D.B., 1988. Biomechanical modelling of the low back during load lifting. Ergonomics 31, 685}697. Cholewicki, J., McGill, S.M., 1996. Mechanical stability of the in vivo lumbar spine: implications for injury and chronic low back pain. Clinical Biomechanics 11, 1}15. Crowninshield, R.D., Johnston, R.C., Andrews, J.G., Brand, R.A., 1978. A biomechanical investigation of the human hip. Journal of Biomechanics 11, 75}85. Deursen, L.L.J.M.v., Patijn, J., 1993. Aufwertung der ligamentaK re Kreuzschmerzen. Manuelle Medizin 31, 108}110. Dostal, W.F., Andrews, J.G., 1981. A three-dimensional biomechanical model of hip musculature. Journal of Biomechanics 14, 803}812. Dumas, G.A., Poulin, M.J., Roy, B., Gagnon, M., Jovanovic, M., 1988. A three-dimensional digitization method to measure trunk muscle lines of action. Spine 13, 532}541. Dumas, G.A., Poulin, M.J., Roy, B., Gagnon, M., Jovanovic, M., 1991. Orientation and moment arms of some trunk muscles. Spine 16, 293}303. Frobin, W., Brinckmann, P., Leivseth, G., Biggemann, M., Reikeras, O., 1996. Precision measurement of segmental motion from #exionextension radiographs of the lumbar spine. Clinical Biomechanics 11, 457}465. Goel, V.K., Svensson, N.L., 1977. Forces on the pelvis. Journal of Biomechanics 10, 195}200. Gracovetsky, S., Farfan, H.F., Lamy, C., 1977. A mathematical model of the lumbar spine using an optimized system to control muscles and ligaments. Orthop Clin North Am 8, 135}153. Granata, K.P., Marras, W.S., 1993. An EMG-assisted model of loads on the lumbar spine during asymmetric trunk extensions. Journal of Biomechanics 26, 1429}1438. King, A.I., 1984. A review of biomechanical models. Journal of Biomechanics Engineering 106, 97}104. Koopman, B., 1989. The three-dimensional analysis and prediction of human walking. PhD. thesis, Twente University. Looze, M.P. de, Kingma, I., Bussmann, J.B.J., Toussaint, H.M., 1992. Validation of a dynamic linked segment model to calculate joint moments in lifting. Clinical Biomechanics 7, 161}169. Macintosh, J.E., Bogduk, N., 1987. 1987 Volvo award in basic science. The morphology of the lumbar erector spinae. Spine 12, 658}668. Macintosh, J.E., Bogduk, N., 1991. The attachments of the lumbar erector spinae. Spine 16, 783}792. McGill, S.M., 1991. Electromyographic activity of the abdominal and low back musculature during the generation of isometric and dy-
933
namic axial trunk torque: implications for lumbar mechanics. Journal of Orthopedic Research 9, 91}103. McGill, S.M., Patt, N., Norman, R.W., 1988. Measurement of the trunk musculature of active males using CT scan radiography: implications for force, moment generating capacity about the L4/L5 joint. Journal of Biomechanics 21, 329}341. Moroney, S.P., Schultz, A.B., Miller, J.A., 1988. Analysis and measurement of neck loads. Journal of Orthopedic Research 6, 713}720. Nemeth, G., Ohlsen, H., 1985. In vivo moment arm lengths for hip extensor muscles at di!erent angles of hip #exion. Journal of Biomechanics 18, 129}140. Nemeth, G., Ohlsen, H., 1986. Moment arm lengths of trunk muscles to the lumbosacral joint obtained in vivo with computed tomography. Spine 11, 158}160. Pearcy, M., Portek, I., Shepherd, J., 1984. Three-dimensional X-ray analysis of normal movement in the lumbar spine. Spine 9, 294}297. Pedersen, D.R., Brand, R.A., Cheng, C., Arora, J.S., 1987. Direct comparison of muscle force predictions using linear and nonlinear programming. Journal of Biomechanical Engineering 109, 192}199. Plamondon, A., Gagnon, M., Gravel, D., 1995. Moments at the L5/S1 joint during asymmetrical lifting: e!ect of di!erent load trajectories and initial load positions. Clinical Biomechanics 10, 128}136. Scholten, P.J., Schultz, A.B., Luchies, C.W., Miller, J.A., 1988. Motions and loads within the human pelvis: a biomechanical model study. Journal of Orthopedic Research 6, 840}850. Schultz, A.B., Andersson, G.B., 1981. Analysis of loads on the lumbar spine. Spine 6, 76}82. Seroussi, R.E., Pope, M.H., 1987. The relationship between trunk muscle electromyography and lifting moments in the sagittal and frontal planes. Journal of Biomechanics 20, 135}146. Snijders, C.J., Hoek van Dijke, G.A., Roosch, E.R., 1991. A biomechanical model for the analysis of the cervical spine in static postures. Journal of Biomechanics 24, 783}792. Snijders, C.J., Vleeming, A., Stoeckart, R., 1993a. Transfer of lumbodorsal load to iliac bones and legs part 1: Biomechanics of self-bracing of the sacroiliac joints and its signi"cance for treatment and exercise. Clinical Biomechanics 8, 285}294. Snijders, C.J., Vleeming, A., Stoeckart, R., 1993b. Transfer of lumbosacral load to iliac bones and legs part 2: Loading of the sacroiliac joints when lifting in a stooped posture. Clinical Biomechanics 8, 295}301. Sobotta, 1994. Atlas of human anatomy, eds Putz, R., Pabst, R. Urban and Schwarzenberg, Munich-Vienna-Baltimore. Tracy, M.F., 1990. Three-dimensional force model of the low-back for simple computer programming. Clinical Biomechanics 5, 175}179. Visser, J.J., Hoogkamer, J.E., Bobbert, M.F., Huijing, P.A., 1990. Length and moment arm of human leg muscles as a function of knee and hip-joint angles. European Journal of Applied Physiology 61, 453}460.
A biomechanical model on muscle forces in the transfer of spinal load to the pelvis and legs Gilbert A. Hoek van Dijke *, Chris J. Snijders , Rob Stoeckart, Henk J. Stam Department of Biomedical Physics and Technology, Faculty of Medicine and Allied Health Sciences, Erasmus University, PO BO 1738, 3000DR, Rotterdam, The Netherlands Department of Anatomy, Faculty of Medicine and Allied Health Sciences, Erasmus University, PO BO 1738, 3000DR, Rotterdam, The Netherlands Department of Rehabilitation, Faculty of Medicine and Allied Health Sciences, Erasmus University, PO BO 1738, 3000DR, Rotterdam, The Netherlands Received 15 April 1999
Abstract Based on musculoskeletal anatomy of the lower back, abdominal wall, pelvis and upper legs, a biomechanical model has been developed on forces in the load transfer through the pelvis. The aim of this model is to obtain a tool for analyzing the relations between forces in muscles, ligaments and joints in the transfer of gravitational and external load from the upper body via the sacroiliac joints to the legs in normal situations and pathology. The study of the relation between muscle coordination patterns and forces in pelvic structures, in particular the sacroiliac joints, is relevant for a better understanding of the aetiology of low back pain and pelvic pain. The model comprises 94 muscle parts, 6 ligaments and 6 joints. It enables the calculation of forces in pelvic structures in various postures. The calculations are based on a linear/non-linear optimization scheme. To gain a better understanding of the function of individual muscles and ligaments, deviant properties of these structures can be preset. The model is validated by comparing calculations with EMG data from the literature. For agonistic muscles, good agreement is found between model calculations and EMG data. Antagonistic muscle activity is underestimated by the model. Imposed activity of modelled antagonistic muscles has a minor e!ect on the mutual proportions of agonistic muscle activities. Simulation of asymmetric muscle weakness shows higher activity of especially abdominal muscles. 1999 Elsevier Science Ltd. All rights reserved. Keywords: Biomechanics; Sacroiliac joint; Pelvis; Skeletal muscle; Anatomical model
1. Introduction Epidemiology indicates that low back pain involves both mechanical and non-mechanical factors. Both dynamic movements and static postures are reported to provoke low back pain (Andersson, 1981; Burdorf and Sorock, 1997). Patients with low back pain indicate especially static postures as painful (Deursen and Patijn, 1993). To identify mechanical factors that play a role in the aetiology of injuries or pain, several biomechanical models have been developed on joint and muscle forces, e.g. Anderson et al., 1985; Brand et al., 1982; Cholewicki and McGill, 1996; Crowninshield et al., 1978; Goel and Svensson, 1977; Gracovetsky et al., 1977; Granata and Marras, 1993; Koopman, 1989; Moroney et al., 1988; Plamondon et al., 1995; Scholten et al., 1988; Schultz and Andersson, 1981; Snijders et al., 1991; Tracy, 1990. More complete * Corresponding author. E-mail address: [email protected] (G.A. Hoek van Dijke)
surveys of biomechanical models can be found in review articles (e.g. Adams and Dolan, 1995; Cha$n, 1988; King, 1984). A recent concept emphasizes static postures (Snijders et al., 1993a; Snijders et al., 1993b). Violating speci"c requirements concerning the role of muscles in the stability of the sacroiliac (SI) joints may lead to the overload of structures in the low back region. No models are known to us that focus on the stability of the pelvis in relation to the load transfer from the upper body to the legs. The aim of this study is to develop a biomechanical model that can serve as a tool for further analysis of this load transfer by investigation of the separate contributions of muscles, ligaments and joints in di!erent postures.
2. Methods To develop a model suitable for analysis of forces in the pelvis, it was essential to represent the pelvis as three
0021-9290/99/$ - see front matter 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 1 - 9 2 9 0 ( 9 9 ) 0 0 0 8 5 - 8
928
G.A. Hoek van Dijke et al. / Journal of Biomechanics 32 (1999) 927}933
separate links (left and right hip bone and sacrum). The left and right femur, "ve lumbar vertebrae and the lowest thoracic vertebra were included as rigid links as well (see Fig. 1). Muscles and ligaments were selected on the basis of the expected importance for the load in the SI joints. Much attention is paid to muscles that cross the SI joints. Forces in these muscles necessarily cause reaction forces in the SI joints (Snijders et al., 1993a,b). Special attention is paid to di!erent parts of the erector spinae, connected either to the sacrum or to the hip bone. Muscles and ligaments are modelled as vector forces. Loads are balanced by muscle forces, ligament forces and joint reaction forces. Several sources had to be combined to complete the geometrical part of the model. A set of 164 transversal MRI-scans of a healthy young male adult (from T12 to the knee) functioned as a guide to combine the separate data sources. Of great help were the data of a model of the lumbar spine as listed by Cholewicki and McGill (1996) and the data made available by Koopman (1989) on his model of the human leg. This model is based on the model by Brand et al. (1982). The MRI-slices enabled the quanti"cation of 3D coordinates of muscle attachments when data were restricted to qualitative descriptions or directions of muscle "bres, as of the iliocostalis (Macintosh and Bogduk, 1987; Macintosh and Bogduk, 1991). Other data sources were Dostal and Andrews (1981), Dumas et al. (1988,1991), Granata and Marras (1993), McGill et al. (1988), Nemeth and Ohlsen (1985,1986), Sobotta (1994) and Visser et al. (1990). For muscles with broad attachments, more than one line of action was de"ned. Special attention is paid to the gluteus maximus. At the cranial side, this muscle has attachments to the sacrum (partially via the sacrotuberous ligament), the hip bone and the thoracolumbar fascia (Sobotta, 1994). Since forces in these three parts have fundamental di!erent e!ects on the load in the SI joints (Snijders et al., 1993a,b), this muscle had to be modelled as at least three separate force vectors. Since no literature was found that describes the anatomy of the gluteus maximus according to this point of view, in one embalmed specimen the left- and right-gluteus maximus were dissected. According to their area of attachment, the muscle "bers were distinguished in three groups. The three lines of action that represent the gluteus maximus were given a relative crosssectional area according to the mass proportions of these groups. The psoas major muscle has been modelled as two serial lines of action representing the upper and lower part. The force in both lines of action is required to be equal. In this way, the psoas major is modelled as a rope that runs over a pulley connected to the ramus superior. A similar procedure has been used for the superior part of the gluteus maximus that originates from the thoracolumbar fascia and inserts on the femur.
Table 1 Lines of action (unilateral) for structures incorporated in the model Name structure Longissimus Iliocostalis Multi"dus Quadratus lumborum Rectus abdominis Internal oblique External oblique Transversus abdominis Psoas Gluteus maximus Gluteus medius Gluteus minimus Biceps Semimembranosus Semitendinosus Gracilis Rectus femoris Iliacus Sartorius Pectineus Adductor magnus Adductor brevis Adductor longus Piriformis Gemellus inferior Obturatorius externus Obturatorius internus Quadratus femoris Gemellus superior Tensor fascia latae Sacrotuberous ligament Sacrospinal ligament Posterior SI ligament Hip joint SI joint Pubic symphysis L5-S1 joint
Lines of action 1 1 1 5 1 2 2 1 2 4 3 3 1 1 1 1 1 1 1 1 3 2 1 1 1 1 1 1 1 1 1 1 1 3 3 3 3
In the model, 94 vectors for muscle forces, 6 vectors for ligament forces and 18 force vectors for the 6 joints (three perpendicular vector components for each joint) are incorporated, see Table 1. A detailed description of the model will be provided on request. The kinematic algorithms that describe relative motions of the links and the positions of mass centres are based on data that was made available by De Looze et al. (1992). The mutual proportions of rotation of vertebrae, which are assumed to be constant during motion, are taken from Frobin et al. (1996) and Pearcy et al. (1984). The vertebrae are included to derive in various positions the direction of the lines of action of muscles that are connected to the vertebrae. For the calculation of forces, the upper body is regarded as one link. The load in the model consists of the weight of the upper body and, if wanted, an external load (e.g. a weight in the hand). For each load situation, the equilibrium of
G.A. Hoek van Dijke et al. / Journal of Biomechanics 32 (1999) 927}933
the links has to be calculated. Since the model is statically indeterminate, additional criteria had to be formulated to select a speci"c combination of muscle forces, ligament forces and joint reaction forces. This study especially concerns static postures. Therefore, an optimization technique was chosen that minimizes the highest value of the occurring muscle contraction intensities. In this way, the upper limit for the muscle stress valid for all muscles is minimized, which may be considered as maximizing the endurance time. However, this criterion does not necessarily lead to a unique solution: it is possible that a trunk muscle, e.g. the erector spinae, determines the lowest possible value for the maximal muscle stress, while still an in"nite number of solutions exists for the muscle forces around the hip. To come to a unique solution, a supplementary, quadratic criterion minimizes the sum of squared muscle stresses, with the additional constraint that no muscle stress may exceed the value that follows from the "rst criterion. This second criterion distributes the load (within the boundaries according to the "rst criterion) over synergetic muscles. However, strictly upholding this second criterion leads to unrealistic activity of muscles when a small decrease of the maximum muscle stress can be reached by recruiting many muscles. To avoid these situations, this second criterion was facilitated somewhat by allowing the maximum muscle stress to exceed this theoretical minimum value with a certain factor. For this factor, the value 0.1 has been chosen since this appeared to yield plausible results. A custom made computer program describes the positions of all points of the model in di!erent postures. The program GAMS (Brooke et al., 1996) is used for the determination of the optimal solution for the forces. Both programs run on a personal computer. The reliability of the model is tested by comparing calculated muscle stresses with measured EMG activities that are described in the literature. To be suitable for model simulation, the experiments had to be de"ned accurately. Mutual proportions of EMG activities had to be given, either the activity of various muscles in at least one load situation, or the activity of a single muscle in a variety of load situations. Due to these strict requirements, only a limited number of experiments could be simulated. The activity as a percentage of the maximum voluntary contraction (MVC) is regarded as a measure for the muscle stress. To test the functioning of the algorithms that handle additional constraints and to show the e!ect of deviant coordination patterns, a simulation was performed of limited strength of the superior part of the gluteus maximus that origins from the thoracolumbar fascia. In the concept of load transfer as described by Snijders et al. (1993a,b), the contribution of the pubic symphysis is assumed to be of minor importance. To investigate the consequences of this aspect, the forces in the symphysis are set to zero.
929
Fig. 1. Ventral view in the upright position and lateral view at 303 trunk #exion of the model. The origin of the coordinate system is located in the midsagittal plane between the rotation centres of the hip joints. In order not to hide them, all lines of actions are drawn on top of images of the bony structures.
3. Results A ventral view and a lateral view of the model are shown in Fig. 1. The contours of the bony structures, as derived from the MRI scans, serve as a reference only. The relationship between the hip #exion angle with the trunk held straight and the normalized EMG values of back muscles, as published by Andersson et al. (1996), is given in Fig. 2. In addition, the corresponding model calculated muscle stresses are drawn. The posture with 903 hip #exion is omitted since this posture is assumed to be too close to the end of the range of motion of the hip joint to neglect the force contribution of joint capsules. With the scaling factor 1% MVC"0.0037 N/mm, the calculated muscle stresses are within the reported standard error of the EMG measurements, with two exceptions at 603 hip #exion (see the arrows in the graph). Compared to the model calculations, here a slightly higher activity of the quadratus lumborum muscle is found whereas a signi"cant deviation is found for the super"cial erector spinae. Except for this latter point, good mutual proportions of muscle stresses are found (coe$cient of correlation is 0.904).
930
G.A. Hoek van Dijke et al. / Journal of Biomechanics 32 (1999) 927}933
Fig. 2. Normalized EMG values of low back muscles at the L level at 0, 30 and 603 hip #exion with the trunk held straight (Andersson et al., 1996) (vertical bars) and calculated muscle stresses obtained from a model simulation of this experiment (solid lines). The vertical bars indicate the standard error of the measurements. Fig. 4. Relationship between the registered EMG activities during isometric axial trunk torque (McGill, 1991) and the model calculated muscle stresses in a simulation of this experiment. Here, the antagonistic internal oblique abdominal and antagonistic erector spinae are imposed to generate tension comparable with the in vivo observed activity. See also Fig. 3.
Fig. 3. Relationship between the registered EMG activities during isometric axial trunk torque (McGill, 1991) and the model calculated muscle stresses in a simulation of this experiment, expressed as a percentage of the highest observed activity (here in the internal oblique abdominal). The abbreviations refer to the internal oblique (IO), external oblique (EO), erector spinae (ES) and rectus abdominis (RA). The vertical bars indicate the standard deviation of the EMG measurements.
Fig. 3 shows the relationship between the registered EMG activities of trunk muscles during isometric axial trunk torque (McGill, 1991) and a model simulation. The index &ag' (agonistic) refers to muscles at that side of the body where they are assumed to support axial torque generation. For rotating to the right (attempting to push the left shoulder forward), these agonists are the left external oblique, the right internal oblique and the right erector spinae. The index &ant' (antagonistic) refers to muscles at the opposite side. For the rectus abdominis muscles, no consistent relation is found between the direction of the applied torque and the activity at the right or left side (McGill, 1991). Therefore, the data of the left and right rectus abdominis are combined. For the internal and external oblique, the EMG activities are compared with the mean value of the model calculated
stresses in the separate lines of action of these muscles. The activity of the erector spinae is compared with the mean calculated stress in the longissimus and iliocostalis muscles. For agonistic muscles and the rectus abdominis, a good correlation is found between the calculated muscle stress and the EMG activity (correlation coe$cient 0.984). Antagonistic muscle activity is not predicted by the model. The activities of the antagonistic internal and external obliques appear to be coupled: adding a supplementary condition to the model that imposes the antagonistic internal oblique to generate tension causes also the antagonistic external oblique to generate tension (see Fig. 4). The mutual proportions of agonistic muscle activities remain roughly the same. A comparable result is found when the modelled antagonistic longissimus and iliocostalis are forced to generate tension. Fig. 5 shows the relation between the EMG activity of the erector spinae as measured by Seroussi and Pope (1987) and calculated muscle stresses in asymmetric load situations. In the upright position, moments in the sagittal and frontal plane were varied systematically by holding a weight in 28 asymmetric positions. The correlation coe$cient is 0.869. In the same load situations, the activity of the external oblique muscle was analyzed. Fig. 6 shows the relationship between the frontal plane moment arm and the activity of the external oblique muscle, for both the EMG measurements (Seroussi and Pope, 1987) and the model simulation. For four di!erent sagittal plane moment arms, Seroussi and Pope found almost the same curve. This "nding is con"rmed by our model: the di!erence between the calculated muscle stresses in the highest and lowest sagittal plane load situations was only 3%. Again,
G.A. Hoek van Dijke et al. / Journal of Biomechanics 32 (1999) 927}933
Fig. 5. Relationship between registered EMG signals of the erector spinae under 28 varying asymmetric sagittal and frontal plane loading conditions (Seroussi and Pope, 1987) and the corresponding model calculated muscle stresses in a simulation of this experiment (*). The line represents a linear "t of this relationship.
931
Fig. 7. Model calculated muscle forces during quasi-static trunk #exion with simulated weakness of the superior part of the left gluteus maximus. Here, the force in this muscle part is limited at 250 N (horizontal line). The solid lines, which concern muscle forces with simulated muscle weakness, show increased activity of abdominal muscles. The dotted lines re#ect forces in the situation without muscle weakness.
is raised, and the left external oblique abdominal becomes active.
4. Discussion
Fig. 6. Relationship between the frontal plane moment arm and the EMG activity of the external oblique muscle (Seroussi and Pope, 1987), and the model calculated results of a simulation of this experiment. The right half of the graph concerns the load situation in which the muscle functions as an agonist.
antagonistic activity (left half of the graph) is not predicted by the model. The correlation coe$cient for this relationship is 0.961. The use of the model as a tool for analyzing the role of individual structures and the e!ects of deviant coordination patterns is shown on the basis of a model simulation of quasi-static trunk #exion. In this simulation, asymmetric loss of muscle strength is introduced. The main results are shown in Fig. 7. At about 303 trunk #exion, the force in the concerning part of the left gluteus maximus reaches the given upper limit, which simulates muscle weakness. Beyond this angle, the decreased contribution of the left gluteus maximus in balancing the trunk is partly taken over by the right gluteus maximus. Abdominal muscles must contribute to restore the equilibrium of the links: the activity of the right internal oblique abdominal
The presented biomechanical model enables the analysis of the load transfer from the upper body to the pelvis and legs in static postures, which is an area of interest in the study of the aetiology of low back pain. The SI joint load is of special interest. A more detailed description of the SI joints is needed before conclusions can be drawn about safe or suspicious loading situations. For example, compression forces are considered as 'safe', where shear forces are considered as &unsafe' (Snijders et al., 1993a,b). However, due to the irregular surface of this joint, the e!ective direction of a force that must be considered as a shear force may di!er from the orientation of this joint surface itself (see Fig. 8). Also shear forces in various directions could have di!erent impacts. The number of passive structures in the model is small. No joint capsules are incorporated into the model yet, therefore, the model can be valid only for postures that are not close to the ends of the range of motion of the joints. At 603 hip #exion, the model prediction for the activities of the longissimus and iliocostalis deviates from the EMG measurements (see Fig. 2). Andersson et al. (1996) observed no increase of activity of these muscles during the latter half of the trunk motion. They assume thoracic muscles to contribute to the lumbar extensor moment transmitted via the thoracolumbar fascia. These muscles are not incorporated into our model, but additional equations and constraints could be formulated to meet
932
G.A. Hoek van Dijke et al. / Journal of Biomechanics 32 (1999) 927}933
Fig. 8. Schematic drawing of the irregular surface of the sacroiliac joint. Due to this irregularity, the e!ective shear direction may deviate considerably from the global joint surface direction.
such a mechanism. The predicted activities of the three involved back muscle groups (the longissimus and iliocostalis, the multi"dus and the quadratus lumborum) are in good proportions with respect to each other. The simulation of the asymmetric load experiments shows a good correlation between measured EMG data and model calculations for agonistic muscles (see Figs. 3}6). Antagonistic muscle activity is not predicted by the model (see Figs. 3 and 6). The force patterns in vivo can be seen as the sum of two components. The "rst component concerns the minimum forces required for equilibrium of the links, the second component concerns the contributions of both agonistic and antagonistic muscles for reasons as mechanical stability. This second component has no net e!ect. As a consequence of the present optimization criterion, only the "rst component of the forces is predicted by the model. EMG assisted mechanical models, such as the model by Granata and Marras (1993), have better capabilities to describe co-contraction of antagonistic muscles and are capable to estimate accurate numerical values for individuals. The aim of our model, however, is primarily to serve as a tool for analyzing the general relations between forces, rather than to focus on individuals. Therefore, it is desirable to simulate all kinds of "ctitious situations and muscle con"gurations (e.g. Fig. 7). Obviously, no EMG data are available for these deviant situations. To enable the use of EMG data as input for the model, the facility is made to de"ne upper and lower limits for each force. In this way, the e!ect of co-contraction can be investigated. This is shown in the simulation of the torsion experiments by McGill (1991) (see Fig. 4). The in#uence of the choice of optimization criteria and cross-sectional areas of muscles on calculated forces in the leg is investigated by Pedersen et al. (1987) and Brand et al. (1986). In general, various choices led to di!erent calculated forces, but obtained solutions were qualitatively similar. Again, this "nding emphasizes that the
most relevant use of this model is to determine general relations between forces. The use of the model as a tool to investigate the relations between muscle forces is shown by a simulation with a weakened part of the left-gluteus maximus muscle (see Fig. 7). The diminished contribution of this muscle can be compensated for by other muscles without inordinate forces. However, the increase of the forces in the compensating muscles, in this example the abdominals, is not neglectible. This implies that inadequate coordination patterns of muscles may exist without acute consequences, but they may lead to disorders at long terms. In spite of the fact that the forces in the pubic symphysis are set to zero in the model calculations, plausible results are obtained. This supports the supposition that the contribution of the symphysis to the transfer of gravitational loads is of minor importance. This study leads to the following conclusions: E The agonistic muscle activity patterns predicted by the model correspond well with EMG recorded activities in the literature; E In the model, antagonistic muscle activity can be imposed; E Pathology as weakened muscles can be simulated; E The contribution of the pubic symphysis to the transfer of gravitational loads seems to be of minor importance; E Deviant situations such as (asymmetric) muscle weakness require higher abdominal muscle activity.
Acknowledgements Authors thank the National Prevention Fund and the Anna Fund for their "nancial support, and B. Koopman and I. Kingma for sharing their data with us.
References Adams, M.A., Dolan, P., 1995. Recent advances in lumbar spinal mechanics, their clinical signi"cance. Clinical Biomechanics 10, 3}19. Anderson, C.K., Cha$n, D.B., Herrin, G.D., Matthews, L.S., 1985. A biomechanical model of the lumbosacral joint during lifting activities. Journal of Biomechanics 18, 571}584. Andersson, E.A., Oddsson, L.I.E., Grundstrom, H., Nilsson, J., Thorstensson, A., 1996. EMG activities of the quadratus lumborum and erector spinae muscles during #exion-relaxation and other motor tasks. Clinical Biomechanics 11, 392}400. Andersson, G.B., 1981. Epidemiologic aspects on low-back pain in industry. Spine 6, 53}60. Brand, R.A., Crowninshield, R.D., Wittstock, C.E., Pedersen, D.R., Clark, C.R., van Krieken, F.M., 1982. A model of lower extremity muscular anatomy. Journal of Biomechanical Engineering 104, 304}310. Brand, R.A., Pedersen, D.R., Friederich, J.A., 1986. The sensitivity of muscle force predictions to changes in physiologic cross-sectional area. Journal of Biomechanics 19, 589}596.
G.A. Hoek van Dijke et al. / Journal of Biomechanics 32 (1999) 927}933 Brooke, A., Kendrick, D., Meeraus, A., 1996. GAMS (General Algebraic Modeling System). Burdorf, A., Sorock, G., 1997. Positive and negative evidence of risk factors for back disorders. Scandinavian Journal of Work and Environmental Health 23, 243}256. Cha$n, D.B., 1988. Biomechanical modelling of the low back during load lifting. Ergonomics 31, 685}697. Cholewicki, J., McGill, S.M., 1996. Mechanical stability of the in vivo lumbar spine: implications for injury and chronic low back pain. Clinical Biomechanics 11, 1}15. Crowninshield, R.D., Johnston, R.C., Andrews, J.G., Brand, R.A., 1978. A biomechanical investigation of the human hip. Journal of Biomechanics 11, 75}85. Deursen, L.L.J.M.v., Patijn, J., 1993. Aufwertung der ligamentaK re Kreuzschmerzen. Manuelle Medizin 31, 108}110. Dostal, W.F., Andrews, J.G., 1981. A three-dimensional biomechanical model of hip musculature. Journal of Biomechanics 14, 803}812. Dumas, G.A., Poulin, M.J., Roy, B., Gagnon, M., Jovanovic, M., 1988. A three-dimensional digitization method to measure trunk muscle lines of action. Spine 13, 532}541. Dumas, G.A., Poulin, M.J., Roy, B., Gagnon, M., Jovanovic, M., 1991. Orientation and moment arms of some trunk muscles. Spine 16, 293}303. Frobin, W., Brinckmann, P., Leivseth, G., Biggemann, M., Reikeras, O., 1996. Precision measurement of segmental motion from #exionextension radiographs of the lumbar spine. Clinical Biomechanics 11, 457}465. Goel, V.K., Svensson, N.L., 1977. Forces on the pelvis. Journal of Biomechanics 10, 195}200. Gracovetsky, S., Farfan, H.F., Lamy, C., 1977. A mathematical model of the lumbar spine using an optimized system to control muscles and ligaments. Orthop Clin North Am 8, 135}153. Granata, K.P., Marras, W.S., 1993. An EMG-assisted model of loads on the lumbar spine during asymmetric trunk extensions. Journal of Biomechanics 26, 1429}1438. King, A.I., 1984. A review of biomechanical models. Journal of Biomechanics Engineering 106, 97}104. Koopman, B., 1989. The three-dimensional analysis and prediction of human walking. PhD. thesis, Twente University. Looze, M.P. de, Kingma, I., Bussmann, J.B.J., Toussaint, H.M., 1992. Validation of a dynamic linked segment model to calculate joint moments in lifting. Clinical Biomechanics 7, 161}169. Macintosh, J.E., Bogduk, N., 1987. 1987 Volvo award in basic science. The morphology of the lumbar erector spinae. Spine 12, 658}668. Macintosh, J.E., Bogduk, N., 1991. The attachments of the lumbar erector spinae. Spine 16, 783}792. McGill, S.M., 1991. Electromyographic activity of the abdominal and low back musculature during the generation of isometric and dy-
933
namic axial trunk torque: implications for lumbar mechanics. Journal of Orthopedic Research 9, 91}103. McGill, S.M., Patt, N., Norman, R.W., 1988. Measurement of the trunk musculature of active males using CT scan radiography: implications for force, moment generating capacity about the L4/L5 joint. Journal of Biomechanics 21, 329}341. Moroney, S.P., Schultz, A.B., Miller, J.A., 1988. Analysis and measurement of neck loads. Journal of Orthopedic Research 6, 713}720. Nemeth, G., Ohlsen, H., 1985. In vivo moment arm lengths for hip extensor muscles at di!erent angles of hip #exion. Journal of Biomechanics 18, 129}140. Nemeth, G., Ohlsen, H., 1986. Moment arm lengths of trunk muscles to the lumbosacral joint obtained in vivo with computed tomography. Spine 11, 158}160. Pearcy, M., Portek, I., Shepherd, J., 1984. Three-dimensional X-ray analysis of normal movement in the lumbar spine. Spine 9, 294}297. Pedersen, D.R., Brand, R.A., Cheng, C., Arora, J.S., 1987. Direct comparison of muscle force predictions using linear and nonlinear programming. Journal of Biomechanical Engineering 109, 192}199. Plamondon, A., Gagnon, M., Gravel, D., 1995. Moments at the L5/S1 joint during asymmetrical lifting: e!ect of di!erent load trajectories and initial load positions. Clinical Biomechanics 10, 128}136. Scholten, P.J., Schultz, A.B., Luchies, C.W., Miller, J.A., 1988. Motions and loads within the human pelvis: a biomechanical model study. Journal of Orthopedic Research 6, 840}850. Schultz, A.B., Andersson, G.B., 1981. Analysis of loads on the lumbar spine. Spine 6, 76}82. Seroussi, R.E., Pope, M.H., 1987. The relationship between trunk muscle electromyography and lifting moments in the sagittal and frontal planes. Journal of Biomechanics 20, 135}146. Snijders, C.J., Hoek van Dijke, G.A., Roosch, E.R., 1991. A biomechanical model for the analysis of the cervical spine in static postures. Journal of Biomechanics 24, 783}792. Snijders, C.J., Vleeming, A., Stoeckart, R., 1993a. Transfer of lumbodorsal load to iliac bones and legs part 1: Biomechanics of self-bracing of the sacroiliac joints and its signi"cance for treatment and exercise. Clinical Biomechanics 8, 285}294. Snijders, C.J., Vleeming, A., Stoeckart, R., 1993b. Transfer of lumbosacral load to iliac bones and legs part 2: Loading of the sacroiliac joints when lifting in a stooped posture. Clinical Biomechanics 8, 295}301. Sobotta, 1994. Atlas of human anatomy, eds Putz, R., Pabst, R. Urban and Schwarzenberg, Munich-Vienna-Baltimore. Tracy, M.F., 1990. Three-dimensional force model of the low-back for simple computer programming. Clinical Biomechanics 5, 175}179. Visser, J.J., Hoogkamer, J.E., Bobbert, M.F., Huijing, P.A., 1990. Length and moment arm of human leg muscles as a function of knee and hip-joint angles. European Journal of Applied Physiology 61, 453}460.