Axial stiffness of human lumbar motion segments, force dependence

doi:10.1016/S0021-9290(98)00037-2

Journal of Biomechanics

Volume 31, Issue 6, 1 June 1998, Pages 503-509

https://doi.org/10.1016/S0021-9290(98)00037-2 Get rights and content

Abstract

This paper addresses the axial stiffness of human lumbar motion segments while subjected to moderate loads. Impacts in axial direction were applied to Functional Spinal Units while they were subjected to weights acting as static pre-load. Accelerations were recorded proximal and distal of the FSU. The transfer function and the resonant frequency were calculated from this data. The stiffness was calculated from the resonant frequency and the load. A simple non-linear model was fitted to the data and a linear relationship was found between stiffness squared and force.The non-linear component in the model strongly affected the stiffness within the chosen load range. The present model may allow in vivo dynamic force determination with improved accuracy, e.g. in experiments where accelerometers have been fixated to pins inserted into the spinous processes of lumbar vertebrae if the static force is known.

Introduction

The mechanical response of the spine to different loading situations is one of the keys to understanding, as well as predicting, its behaviour. The influence of factors like load history, exposure time, as well as total load has to be considered.

Measurements of axial deformation due to loading (Dieen and Toussaint, 1993; Kulak et al., 1975, Kulak et al., 1976) have been performed in vitro as well as in vivo in order to establish the relation between force and deformation. Stiffness, generally defined as the capability of a structure to withstand load, is one of the parameters that has been determined in a number of tests. Age, mineral content and strain rate are some of the factors that have been shown to influence stiffness (Kazarian and Graves, 1977; Koeller et al., 1986; Neumann et al., 1994). Most of the reported stiffness values have been derived during semi-static loading conditions. To understand better the dynamic properties of the functional spinal unit (FSU), it is essential to determine stiffness for physiological loads (i.e. loads that might be experienced in daily life). Typical force– deformation curves have traditionally been divided into three different regions: toe, elastic and yield. The toe region has been designated by Panjabi as the ‘Neutral zone’. In this region, even small forces can cause relatively large motions. Examination of force–deformation curves reveals that the stiffness is increasing with load (Fig. 1 shows the general form of the curve).

The large variation in reported stiffness values is to some extent explained by biological variations, but is also due to different definitions of stiffness: mean, ultimate (slope from origo to the failure point in the force– deformation curve), elastic and tangential stiffness. The functional spinal unit is composed of several tissues: ligaments, cartilage, intervertebral disc and bone structure, each structure having individual properties (Burstein and Frankel, 1968; Markolf and Morris, 1974; McNally and Arridge, 1995; Shah et al., 1977; Yoganandan et al., 1989aYoganandan et al., 1989b). The influence of each component in the overall performance of the FSU is hitherto not fully understood.Neumann et al. (1994) reported a non-linear behaviour of ligaments as did Viidik, 1968, Viidik, 1980 in studies of collagenous tissue. Isolated disc structures such as the annulus exhibit a predominantly non-linear behaviour as demonstrated by Wu and Yao (1976) and Skaggs et al. (1994). Typical force–deformation curves from intact vertebrae as well as vertebral trabecular bone show a progressively increasing stiffness (Hansson et al., 1980; Hansson et al., 1986).Linde and Hvid (1987) stated in a study of trabecular bone stiffness that ‘it is doubtful if a linear part exist at all’. They believed that for practical reasons this specific interval could be regarded as linear but it is only a matter of resolution and number of specimens to demonstrate that this part is also non-linear.

In the present study, data have been fitted to a simple non-linear model. The aim has been to determine the axial stiffness of the FSU and its relation to the applied load. In order to analyse various physiological loading conditions, a simple model is proposed that, with a reasonably good accuracy, permits prediction of spinal stiffness over a range where the maximum is about three times the minimum load. The relevance of this model is in situations where the FSU can be treated as a ‘black box’ and its internal mechanisms are not the main concern (e.g. modelling of the whole spine).

We have also tried to relate the parameters from the load-stiffness fit to the following factors:

•
Age.
•
Amount of bone mineral (BMC).
•
Degree of disc degeneration.
•
Morphometry (the geometry of the FSU).
•
Position in the time–deflection curve (degree of creep).

Section snippets

Materials and methods

Six lumbar functional spinal units (FSU) obtained from four subjects were tested. Each FSU consisted of one intact disc, ligaments and two adjacent vertebrae. Data for each FSU can be found in Table 1. The degeneration of the discs was classified according to the visual four-graded scale by Friberg and Hirsch (1949).

The bone mineral in each vertebra was determined with dual photon absorptiometry (DPA) technique and expressed in g cm^-1. After excision of the soft tissue except the disc and

Results

The relationship between k² and force for all six tests can be seen in Fig. 5. A close linear relationship is suggested for each FSU. An overall fit of k² as a function of force (r²=0.59) gives the following values: k_l= 0.81 kN mm^-1 and k_n=2.7 kN mm^-2. A relationship between k²_l and BMC was also found: $k^{2}_{l} =−1.5+0.58 · BMC$ (significant at the 5% level, r²=0.73).

Age and degeneration had no significant influence upon stiffness. This is in agreement with findings of dynamic stiffness determined during

Discussion

The non-linear behaviour of the motion segment within the load levels studied has been reported by a number of authors (Asano et al., 1992; Brown et al., 1957; Hirsch, 1955; Hutton et al., 1979; Kasra et al., 1992; Koreska et al., 1977; Kulak et al., 1976, Kulak et al., 1975; Yoganandan et al., 1989a, Yoganandan et al., 1989b). So far, no mathematical expression accounting for the non-linearity has been generally accepted. The proposed model describes, in agreement with Panjabi’s neutral zone

Conclusions

The method of using impacts superimposed to a static load allows for evaluation of short time as well as long time viscoelastic properties and has proven useful. A non-linear behaviour of the FSU stiffness is apparent. This implies that linear Maxwell and Kelvin units are not appropriate in modelling, except in narrow force range.

Acknowledgements

The authors wish to acknowledge the support from the Swedish Council for Work Life Research and Ingabritt and Arne Lundberg’s Research Foundation. We also wish to thank Allison M. Kaigle, Ph.D., for valuable comments on the manuscript.

References (33)

K.B. Broberg
Slow deformation of intervertebral discs
Journal of Biomechanics
(1993)
M. Frisen et al.
Rheological analysis of soft collageneus tissue, Part 1theoretical considerations
Journal of Biomechanics
(1969)
M. Frisen et al.
Rheological analysis of soft collagenous tissue, Part IIExperimental evaluation and verifications
Journal of Biomechanics
(1969)
W. Koeller et al.
Biomechanical properties of human intervertebral disks subjected to axial dynamic compression — influence on age and degeneration
Journal of Biomechanics
(1986)
J. Koreska et al.
Biomechanics of the lumbar spine and its clinical significance
Orthopaedic Clinic of North America
(1977)
R.F. Kulak et al.
Biomechanical characteristics of vertebral motion segments and intervertebral discs
Orthopaedic Clinic of North America
(1975)
R.F. Kulak et al.
Nonlinear behavior of the human intervertebral disc under axial load
Journal of Biomechanics
(1976)
F Linde et al.
Stiffness behaviour of trabecular bone specimens
Journal of Biomechanics
(1987)
D.S. McNally et al.
An analytic model of intervertebral disc mechanics
Journal of Biomechanics
(1995)
M.M. Panjabi
Experimental determination of spinal motion segment behaviour
Orthopaedic Clinic of North America
(1977)

A. Viidik

A rheological model for uncalcified parallel-fibred collagenous tissues

Journal of Biomechanics

(1968)

H. Wu et al.

Mechanical behavior of the human annulus fibrosus

Journal of Biomechanics

(1976)

N. Yoganandan et al.

Stiffness and strain energy criteria to evaluate the threshold of injury to an intervertebral joint

Journal of Biomechanics

(1989)

S. Asano et al.

The mechanical properties of the human L4-5 functional spinal unit during cyclic loading; The structural effects of the posterior elements

Spine

(1992)

K.B. Broberg et al.

Modeling of intervertebral discs

Spine

(1980)

T. Brown et al.

Some mechanical tests on the lumbosacral spine with particular reference to the intervertebral discs

Journal of Bone Joint Surgery

(1957)

Cited by (20)

Feedforward backpropagation artificial neural networks for predicting mechanical responses in complex nonlinear structures: A study on a long bone
2022, Journal of the Mechanical Behavior of Biomedical Materials
Citation Excerpt :
Bones are highly complex biological materials that possess exotic mechanical properties, and therefore are characterized as complex engineering structures (Mouloodi et al., 2020a) and are also categorized as nanocomposite and anisotropic solids (Rahmanpanah et al., 2020; Keaveny et al., 2001). The mechanical properties and responses of these advanced engineering structures have been shown, for example, to be density-dependent (Carter and Hayes, 1977; Rice et al., 1988), site- and length-dependent (Fradinho et al., 2015; Nobakhti et al., 2017), force- and loading-rate-dependent (Rostedt et al., 1998; Edwards et al., 1987; Kulin et al., 2011), time-dependent (viscoelasticity) (Xie et al., 2017; Manda et al., 2017; Mouloodi et al., 2020b), strain-rate-dependent (exhibiting both viscoelastic and viscoplastic behavior) (Johnson et al., 2010), size- and scale-dependent (Mirzaali et al., 2016; Frame et al., 2017), shape-dependent (Davies, 2001a; Mouloodi et al., 2019a), age-dependent (Mouloodi et al., 2020a; Kulin et al., 2011; Patton et al., 2019), sex-dependent (Patton et al., 2019). It would be difficult to find another material that possesses such multivariable and exceptional mechanical properties.
Feedforward backpropagation artificial neural networks (ANNs) have been increasingly employed in many engineering practices concerning materials modeling. Despite their extensive applications, how to achieve successfully trained ANNs is not thoroughly explained in the literature, nor are there lucid discussions to delineate influential parameters obtained from analyses. Long bones are composite materials possessing nonhomogeneous and anisotropic properties, and their mechanical responses exhibit dependency on numerous variables. Material complexity hinders researchers from arriving at a consensus in implementing an optimal constitutive model or encourages them to adopt a simple constitutive model including many simplifying assumptions. However, such exceptional features and engineering challenges make long bones materials worth investigating, enriching our comprehension of complex engineering structures using novel techniques where traditional methods may present limitations. This paper reports on the prediction of loading, displacement, load and displacement simultaneously, and strains using feedforward backpropagation ANNs trained with experimental recordings. The technique was used to find optimum network structures (architectures) that encompass the best prediction ability. To enhance predictions, the influence of several elements such as a network training algorithm, injecting noise to datasets prior to training, the level of injected noise which directly affects model fitting and regularization, and data normalization prior to training were investigated and discussed. Essential parameters influencing decision making in identifying well-trained and well-generalized ANNs were elaborated. A considerable emphasis in this study was placed on examining the generalization ability of the already trained and tested ANNs, thus guaranteeing unbiased models that avoided overfitting. Gaining favorable outcomes in this study required three years of performing experiments and data collection before establishing the networks. The subsequent training, testing, and determination of the generalization of more than 60,000 ANNs are promising and will assist researchers in comprehending mechanical responses of complicated engineering structures that exhibit peculiar nonlinear properties.
Design of a new intervertebral disc prosthesis
2019, Materials Today: Proceedings
In the degenerative disc disease, the traditional treatment consists in a removal of the intervertebral disc followed by the fusion of the two adjacent vertebral bodies (arthrodesis). An alternative treatment is the use of an artificial intervertebral disc. The advantage of an artificial intervertebral disc is that the mobility of the vertebral column will be restored.
Aim of this paper consists in the definition of a prosthetic device for insertion in the intervertebral zone.
Different types of artificial bone prostheses have been proposed, seldom based on similarity with physiological discs, then made with a central part (nucleus) consisting of hydrogel and an outer containment belt (annulus) consisting of a plastic material of greater stiffness. Upper and lower containment plates allow realizing a closed structure to be inserted in place of the original disc. Then, in our solution, the external parts (annulus and plates) were modeled by HDPE and the inner part (nucleus) by hydrogel. Both the materials are highly biocompatible. Both materials were modeled by the Ogden parameters related to elastomers. In fact, under the action of near-bending and twisting forces, the disc is subject to large deformations that the materials are able to withstand in elastic conditions. In order to assign the correct Ogden parameters to the materials, a procedure was used to fit the curves acquired by literature. The discs were subjected to compressive load as well as to forward and back bending (flexion and extension), lateral bending, torsion and combined loading.
The degenerative state of the intervertebral disk independently predicts the failure of human lumbar spine to high rate loading: An experimental study
2015, Clinical Biomechanics
Citation Excerpt :
In this study, the dynamic stiffness computed for the lumbar spine (having two disks in series) can be compared to previous experiments using a single functional spinal unit by doubling the stiffness and damping values measured in this study. Ranging from 17.9 to 755 kN/m, our results are in agreement with that reported for lumbar FSU under cyclic loading (Kasra et al., 1992) and with the observed effect of preload on the stiffness (Rostedt et al., 1998). However, these stiffness values are higher than the “instantaneous” stiffness values reported for creep response of spine (Keller and Spengler, 1987) while being generally lower than those reported for axial stiffness of spine segments (Rostedt et al., 1998).
In the elderly, 30%–50% of patients report a fall event to precede the onset of vertebral fractures. The dynamic characteristics of the spine determine the peak forces on the vertebrae in a fall. However, we know little about the effect of intervertebral disk degeneration on the failure of human spines under the high loading rates associated with such falls. We hypothesized that MR estimates of disk hydration and viscoelastic properties will provide better estimates of failure strength than bone density alone.
Seventeen L1–L3 human spine segments were imaged (magnetic resonance imaging, dual-energy X-ray absorptiometry), their dynamic responses quantified using pendulum based impact, and the spines tested to failure under high rate loading simulating a fall event. The spines' stiffness and damping constants were computed (Kelvin–Voigt model) with disk hydration and geometry assessed from T2 and proton density images.
Under impact, the spines exhibited a second-order underdamped response with stiffness and damping ranging (17.9–754.5) kN/m and (133.6–905.3) Ns/m respectively. Damping, but not stiffness, was negatively correlated with higher ultimate strength (P < 0.05). Higher bone mineral density and MR-estimated disk hydration correlated with higher ultimate strength (P < 0.01 for both). No such correlations were observed for the T2 values. Adding disk hydration yielded a 20% increase in the model's association with failure load compared to bone density alone (MANOVA, P < 0.001).
The strong correlation between disk viscoelastic properties and MR-estimated hydration with the spine segments' ultimate strength clearly demonstrates the need to include disk degeneration as part of fracture risk assessment in the elderly spine.
Models incorporating pin joints are suitable for simulating performance but unsuitable for simulating internal loading
2012, Journal of Biomechanics
Citation Excerpt :
Despite this, bone bending may make up a non-negligible component of the force-attenuating properties of the body during extremely high loading. Rostedt et al. (1998) investigated the axial stiffness of functional spinal units (FSUs) from the lumbar region. Although obtained from a lower range of loading levels, the stiffness relationship they observed predicts that, in order to compress the unit by 1 mm, a load of approximately 3500 N would have to be applied.
Simulation models of human movement comprising pin-linked segments have a potential weakness for reproducing accurate ground reaction forces during high impact activities. While the human body contains many compliant structures such a model only has compliance in wobbling masses and in the foot–ground interface. In order to determine whether accurate GRFs can be produced by allowing additional compliance in the foot–ground interface, a subject-specific angle-driven computer simulation model of triple jumping with 13 pin-linked segments was developed, with wobbling masses included within the shank, thigh, and trunk segments. The foot–ground interface was represented by spring-dampers at three points on each foot: the toe, ball, and heel. The parameters of the spring-dampers were varied by a genetic algorithm in order to minimise the differences between simulated GRFs, and those measured from the three phases of a triple jump in three conditions: (a) foot spring compression limited to 20 mm; (b) this compression limited to 40 mm; (c) no restrictions. Differences of 47.9%, 15.7%, and 12.4% between simulation and recorded forces were obtained for the 20 mm, 40 mm, and unrestricted conditions, respectively. In the unrestricted condition maximum compressions of between 43 mm and 56 mm were obtained in the three phases and the mass centre position was within 4 mm of the actual position at these times. It is concluded that the unrestricted model is appropriate for simulating performance whereas the accurate calculation of internal forces would require a model that incorporates compliance elsewhere in the link system.
Changes in axial stiffness of the trunk as a function of walking speed
2006, Journal of Biomechanics
Research suggests that abnormal coordination patterns between the thorax and pelvis in the transverse plane observed in patients with Parkinson's disease and the elderly might be due to alteration in axial trunk stiffness. The purpose of this study was to develop a tool to estimate axial trunk stiffness during walking and to investigate its functional role. Fourteen healthy young subjects participated in this study. They were instructed to walk on the treadmill and kinematic data was collected by 3D motion analysis system. Axial trunk stiffness was estimated from the angular displacement between trunk segments and the amount of torque around vertical axis of rotation.
The torque due to arm swing cancelled out the torque due to the axial trunk stiffness during walking and the thoracic rotation was of low amplitude independent of changes in walking speeds within the range used in this study (0.85–1.52 m/s). Estimated axial trunk stiffness increased with increasing walking speed. Functionally, the suppression of axial rotation of thorax may have a positive influence on head stability as well as allowing recoil between trunk segments. Furthermore, the increased stiffness at increased walking speed would facilitate the higher frequency rotation of the trunk in the transverse plane required at the higher walking speeds.
A rigid body model of the dynamic posteroanterior motion response of the human lumbar spine
2002, Journal of Manipulative and Physiological Therapeutics
Background: Clinicians apply posteroanterior (PA) forces to the spine for both mobility assessment and certain spinal mobilization and manipulation treatments. Commonly applied forces include low-frequency sinusoidal oscillations (<2 Hz) as used in mobilization, single haversine thrusts (<0.5 seconds) as imparted in high-velocity, low-amplitude (HVLA) manipulation, or very rapid impulsive thrusts (<5 ms) such as those delivered in mechanical-force, manually-assisted (MFMA) manipulation. Little is known about the mechanics of these procedures. Reliable methods are sought to obtain an adequate understanding of the force-induced displacement response of the lumbar spine to PA forces. Objective: The objective of this study was to investigate the kinematic response of the lumbar spine to static and dynamic PA forces. Design: A 2-dimensional modal analysis was performed to predict the dynamic motion response of the lumbar spine. Methods: A 5-degree-of-freedom, lumped equivalent model was developed to predict the PA motion of the lumbar spine. Lumbar vertebrae were modeled as masses, massless-spring, and dampers, and the resulting equations of motion were solved by using a modal analysis approach. The sensitivity of the model to variations in the spring stiffness and damping coefficients was examined, and the model validity was determined by comparing the results to oscillatory and impulsive force measurements of vertebral motion associated with spine mobilization and 2 forms of spinal manipulation. Results: Model predictions, based on a damping ratio of 0.15 (moderate damping) and PA spring stiffness coefficient ranging from 25 to 60 kN/m, showed good agreement with in vivo human studies. Quasi-static and low-frequency (<2.0 Hz) forces at L3 produced L3 segmental and L3-L4 intersegmental displacements up to 8.1 mm and 3.0 mm, respectively. PA oscillatory motions were over 2.5-fold greater for oscillatory forces applied at the natural frequency. Impulsive forces produced much lower segmental displacements in comparison to static and oscillatory forces. Differences in intersegmental displacements resulting from impulsive, static, and oscillatory forces were much less remarkable. The latter suggests that intersegmental motions produced by spinal manipulation may play a prominent role in eliciting therapeutic responses. Conclusions: The simple analytical model presented in this study can be used to predict the static, cyclic, and impulsive force PA displacement response of the lumbar spine. The model provides data on lumbar segmental and intersegmental motion patterns that are otherwise difficult to obtain experimentally. Modeling of the PA motion response of the lumbar spine to PA forces assists in the understanding the biomechanics of therapeutic PA forces applied to the lumbar spine and may ultimately be used to validate chiropractic technique procedures and minimize risk to patients receiving spinal manipulative therapy. (J Manipulative Physiol Ther 2002;25:485-96)

View all citing articles on Scopus

View full text

Axial stiffness of human lumbar motion segments, force dependence

Abstract

Introduction

Section snippets

Materials and methods

Results

Discussion

Conclusions

Acknowledgements

Journal of Biomechanics

Journal of Biomechanics

Journal of Biomechanics

Journal of Biomechanics

Orthopaedic Clinic of North America

Orthopaedic Clinic of North America

Journal of Biomechanics

Journal of Biomechanics

Journal of Biomechanics

Orthopaedic Clinic of North America

Journal of Biomechanics

Journal of Biomechanics

Journal of Biomechanics

The mechanical properties of the human L4-5 functional spinal unit during cyclic loading; The structural effects of the posterior elements

Spine

Modeling of intervertebral discs

Spine

Some mechanical tests on the lumbosacral spine with particular reference to the intervertebral discs

Journal of Bone Joint Surgery