Mechanical stability of the in vivo lumbar spine: implications for injury and chronic low back pain

Cbkxd Biomechnics Vol. 11, No. 1. pp. l-15,

1996 Copyright 0 1995 Elsevier Science Limited Printed in Great Britain. All rights reserved 02684033196 $15.00 + 0.00

ELSEVIER

New Concepts and Hypotheses

Mechanical stability of the in viva lumbar spine: implications for injury and chronic low back pain J Cholewicki

PhD,

S M McGill Pm

Occupational Biomechanics and Safety Laboratories, Department of Kinesiology, Applied Health Sciences, University of Waterloo, Ontario, Canada

Faculty of

Summary-One important mechanical function of the lumbar spine is to support the upper body by transmitting compressive and shearing forces to the lower body during the performance of everyday activities. To enable the successful transmission of these forces, mechanical stability of the spinal system must be assured. The purpose of this study was to develop a method and to quantify the mechanical stability of the lumbar spine in viva during various three-dimensional dynamic tasks. A lumbar spine model, one that is sensitive to the various ways that individuals utilize their muscles and ligaments, was used to estimate the lumbar spine stability index three times per second throughout the duration of each trial. Anatomically, this model included a rigid pelvis, ribcage, five vertebrae, 90 muscle fascicles and lumped parameter discs, ligaments and facets. The method consisted of three sub-models: a cross-bridge bond distribution-moment muscle model for estimating muscle force and stiffness from the electromyogram, a rigid link segment body model for estimating external forces and moments acting on the lumbar vertebrae, and an 18 degrees of freedom lumbar spine model for estimating moments produced by 90 muscle fascicles and lumped passive tissues. Individual muscle forces and their associated stiffness estimated from the EMG-assisted optimization algorithm, along with external forces were used for calculating the relative stability index of the lumbar spine for three subjects. It appears that there is an ample stability safety margin during tasks that demand a high muscular effort. However, lighter tasks present a potential hazard of spine buckling, especially if some reduction in passive joint stiffness is present. Several hypotheses on the mechanism of injury associated with low loads and aetiology of chronic back pain are presented in the context of lumbar spine stability. Relevance-This method allows one to analyse the overall stability of the multi-degree-offreedom in viva lumbar spine under a wide variety of dynamic, 3-D loads and postures. Such a method is necessary to test new hypotheses implicating the dysfunction of the spine stabilizing system (muscles, ligaments, and central nervous system) as a cause of certain low back pain and injury cases. Under this type of analysis, a scenario is proposed to explain injury that could occur during a light task - perhaps picking up a pencil from the floor. Key words:

Lumbar

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spine, stability model

Vol. 11, No. 1, l-l

5, 1996

Introduction One important mechanical function of the lumbar spine is to support the upper body by transmitting compressive and shearing forces to the lower body during the performance of everyday activities. However, the isolated thoracolumbar spine buckles under comReceived: 8 November 1994; Accepted: 25 May 1995 Correspondence and reprint requests to: Jacek Cholewicki, Biomechanics Research Laboratory, Department of Orthopaedics, Yale University School of Medicine, PO Box 208071, New Haven, CT 06520-8071, USA

pressive loads exceeding 20 N’ and the lumbar part of the spine buckles under approximately 90 N*. But in viva a spine may experience compressive loads ranging from about 6000 N for more demanding everyday tasks3 and up to 18000 N during competitive powerlifting4. Clearly, as an issue quite apart from injury related to loading in excess of tissue tolerance limits, mechanical stability of the spinal system must be of concern at all levels of loading. The crucial role of the trunks musculature to support the spine similar to guy wires spanning a bending mast is well documented5-‘. In addition some have

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suggested that intra-abdominal pressure contributes to the overall stability of the spine6.*. These stabilizing mechanisms are controlled by the neural system. It is sensible to expect that in order to prevent buckling, both the motor control system and the osteoligamentous spinal linkage will operate within the range of mechanical stability, with some safety margin over and above the critical load. On the other hand there is evidence of cocontraction of antagonistic trunk muscles during most daily activities, which is energetically and mechanically costiyc’. Marc interestingly, such cocontractions increase when people prepare for unexpected and sudden loading such as an unknown weight dropping in a hand-held bucket’” I’. Any such cocontractions increase joint Load and muscular energy requirement for moment support. A plausible explanation is that a certain level of muscle cocontraction may be necessary to maintain the spine within a margin of safe, stable equilibrium. Indeed, new concepts were recently outlined that link the mechanical instability of the lumbar spine to the aetiology of clinical spine disorders. Panjabi’“,‘” postulated that the dysfunction of the spine stabilizing system (muscles. ligaments, and central nervous system) may lead to low back pain and injury. Cholewicki’” hypothesized that the sudden need to regain spine stability may result in excessive muscle activity and tissue overload. Prolonged muscular compensation to maintain the mechanical stability of the spine may lead to chronic low back pain’“~‘“. However, further progress in understanding the aetiology of lumbar spine injury or pain in relation to its mechanical stability requires an in viva quantification of spine stability. Previous research efforts have studied spine stability using either optimization models to distribute muscle forces’.“.“’ or considered only the general effect of muscle forcesi “. These authors demonstrated that, theoretically, regions of spine instability can occur under certain conditions. However, current optimization models of the lumbar spine that minimize some cost functions to obtain a unique equilibrium solution and the resultant tissue loads are not suitable for in viva stability analysis. The optimization approach inherently lacks the biological sensitivity to various possible muscle recruitment and cocontraction patterns. It provides the same solution set of muscle forces for everyone executing the same task, regardless if one individual used his muscles in quite a different way’“. The purpose of this study was twofold. Firstly we developed a model overcoming the above difficulties. Its features include anatomical detail (90 muscle fascicles), together with a method that combines the EMG-assisted approach with optimization to predict muscle forces. A method for quantifying the mechanical stability of the in viva lumbar spine is outlined. Secondly we compared the mechanical stability of the spine, in the context of clinical implications, in three healthy subjects performing several three-dimensional, dynamic tasks. It was hypothesized that the stability of

the spine is maintained by the neural system to a constant level, regardless of various moment or joint force demands and postures. Methods

Three male subjects (age 24.3 years; SD, 2.6; height 1.77 m; SD, 0.03; body mass 82.2 kg; SD, 2.0) performed two trials for each of seven slow dynamic (to minimize inertial interactions), three-dimensional tasks (14 trials in total for each subject), with a load applied to the hands via a dead weight routed through cables and a pulley system. The tasks consisted of a twohanded lift off the floor (33.56 kg), trunk lateral bending with the load in a right hand (20 kg), trunk twist while holding the cable handle in front with outstretched arms (10 kg), one-handed push (10 kg), onehanded pull (10 kg), one-handed sweeping motion from the floor up with a twist (5 kg), and upright standing while the load held in both hands was increased in three steps (0, 20, 40 and 60 kg total) (Figure 1). Each task lasted 10 s with the exception of upright standing, which was collected over a 12-s period. In addition, all subjects performed four maximum voluntary contraction tasks designed to activate each muscle group maximally for normalization of the myoelectric signals20. The subjects were videotaped with two synchronized cameras, arranged approximately at right-angles to each other. The joint centres of the subjects were digitized 15 frames per second and low-pass filtered at a cut-off frequency of 2 Hz. Residual analysis showed that such movements were slow, smooth, and contain virtually no signal above 2 Hz21. A calibration frame (1.1 x 0.9 x 0.9 m) was used to calibrate a volume in which the subjects were tested, yielding an average error of 5 mm of marker reconstruction in any direction. The joint centres of the body segments were digitized directly from both video images, producing the threedimensional (3-D) coordinates of the actual joint centres and not surface skin markers. These data were input to a 3-D linked segment model (adapted from the 2-D version described in McGill and Norman2’), from which the upper body segment weights were obtained together with the external forces acting on the ribcage and in turn, on the lumbar spine (see flow chart Figure 2).

Fourteen channels of EMG were recorded using bipolar, Ag-AgCl surface, disposable electrodes, placed with a centre-to-centre spacing of 3 cm over the following muscles on each side of the body: rectus abdominis (3 cm lateral to the umbilicus), external oblique (approximately 15 cm lateral to the umbilicus), internal oblique (approximately midway between the anterior superior iliac spine and symphysis pubis, above the inguinal ligament), latissimus dorsi (lateral to Tg over the muscle belly), thoracic erector spinae (5 cm lateral to Tg spinous process), lumbar erector spinae (3 cm lateral to L3 spinous process) and multifidus (2 cm lateral to L4-5 spinous processes). The signals were A/D converted at a sample rate of 1200 Hz,

Cholewicki

and McGill:

Mechanical

stability

of the

in vivo

lumbar

spine

3

4

B

/’ I, I ,’

_*-**’

I’ ,’ I

c) SWEEP

d) STAND

e) TWIST

f) PULL

Figure 1. Dynamic tasks considered in this study: (a) two-handed lift, (b) lateral trunk bend, (c)one-handed load in the hands, (e) trunk twist, (f) one-handed pull, and (g) one-handed push.

3” (1

ANAIXSIS

18 DP

LUMBAR

SlfINE

MODEL

a MUSCLE

1

1

standing

with increasing

Lumbar spine model formulation

a

1 DM

sweep motion,(d)

rectified, low-pass filtered, and normalized to the maximum voluntary contraction values. Low-pass filtering (second-order Butterworth filter) was performed using a cutoff frequency of 6 Hz to represent muscle activation dynamics (calcium release and diffusion process) with a time constant of about 25 ms23-25.Subsequently these EMG data served as an input to the distribution-moment muscle contraction dynamics mode126-28,40.

STABILITY INDEX STABILTY

g) PUSH

ricrixl

D

DA3.A COLLECHON Figure 2. Block diagram of the modelling procedure for the spine stability analysis. This procedure consisted of three sub-models: the cross-bridge bond distribution-moment muscle model for estimating muscle force and stiffness from EMG”, the rigid link segment body model for -- estimating external forces and moments acting on the and the 18 degrees of freedom lumbar spine model for ribcage”, estimating moments produced by muscles and passive tissues described here in detail. Estimated muscle forces were then adjusted with EMG assisted optimization42 to achieve moment balance between the external load acting on the ribcage and the muscle and passive tissue forces. Adjusted muscle forces and stiffness together with the external forces were needed for the stability analysis.

The anatomical representation and the calculation of tissue loads were based extensively on the lumbar spine model developed by McGill and Norman3 and refined for 3-D analysis by McGi1129. However, substantial extensions to this model were necessary, including full anatomical development over the entire length of the lumbar spine, to obtain viable input into the stability analysis module. Original muscle cross-sectional areas and attachments, obtained from cadaver dissections and CT and MRI scans, were supplemented with data from the recent work of Bogduk et a1.30.For details see Appendix A. Anatomical representation The anatomical model consisted of a rigid pelvis and sacrum, five lumbar vertebrae separated by a lumped parameter disc and ligament equivalent for rotational stiffness about the three axes, rigid ribcage and 90 muscle fascicles (Figure 3). Three axes of rotations were assigned to each intervertebral joint for a total of

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18 degrees of freedom (df) (6 joints x 3 df each). Assuming a normal spine with no pre-existing translational laxity, joint translations were constrained to be zero in the current version. Muscles were represented with the centroid line approach; however, many of them were forced to pass through several nodes attached to the various vertebrae to account for the curved lines of action. Each muscle consisted of an active contractile part, a passive parallel elastic element, and a passive non-linear tendon. The amounts of lumbar lordosis (or lack of it) and kinematics of the spine were derived from the relative rotations between the ribcage and the pelvis measured with an electromagnetic device 3SPACE ISOTRAK (Polhemus Inc., Vermont, USA). In this way, the model spine kinematics and range of motion were calibrated to match each subject’s spine. The assumption was made that each intervertebral joint contributes a constant proportion of the total rotation angle between Si and Ti2 in all three directions: flexion/ extension, lateral bending, and axial twist (after White and Panjabi31 and McGi112’). Given the rotation angles between the ribcage and pelvis, the new coordinates of all lumbar vertebrae were obtained by rotating them in the sequence about 2: X, and Y local (vertebrae) axes (see Figure 3 for axes convention). External load

The external forces acting on the ribcage and the points of their application were obtained from a quasidynamic, three-dimensional rigid linked-segment model of a human body. Nineteen digitized joint centres, reconstructed from a direct linear transformation algorithm contained in the 3-D video system (Peak Performance, Colorado, USA), defined the 15 body segments (hands, forearms, arms, head, trunk, pelvis, thighs, shanks, and feet). Segment lengths, massesand centres of mass were taken from Winter32, although parameters for the trunk and pelvis were obtained from Zatsiorsky and Seluyanov”“. Two additional markers indicated the direction of force applied to the hands by means of a steel cable and a

6 F&we 3. Schematic diagram of the analysis of spine stability. Forces acting onthe hands and torso (a) aretransmitted to the ribcage and to the lumbar spine and must be supported by torsional (passive tissue) and linear springs (muscles) (b). The model (c)consists of a rigid ribcage (0). 5 lumbar vertebrae Iwgments 1 to 5) and pelvis (6). Euler angles (rotations about Z, X and Y axes) served as the generalized coordinates.

system of pulleys. The tasks were performed slowly and smoothly, which permitted a static analysis of the moments resulting from the body segment masses.The hand force was measured directly with a transducer mounted in series between the handle and a cable to produce a quasidynamic analysis. Forces arising from the mass of the hands, forearms, arms, head, and trunk, their centre of mass coordinates, and the hand forces were transformed into a local coordinate system embedded in the pelvis. Together with the rotated coordinates of the vertebral bodies, these forces were summed to calculate moments about the disc centroids at each intervertebral level. Subsequently passive tissue and muscle forces were determined to balance the moments calculated at every lumbar joint. Passive tissue forces

The restorative moments contributed by passive tissues (discs, ligaments, and other tissues) were calculated first. Here the knowledge of intervertebral joint stiffness and a coupling of rotations throughout the entire range of motion is essential. Unfortunately the data in the literature are both sparse and scattered. A few 6 x 6 stiffness matrices for the isolated motion segments have been reported, but pertain only to the stiffness about the neutral joint position34,35.However, McGill et al.36 measured the passive trunk stiffness with respect to the L4-5 joint, throughout the entire range of motion for young males and females. These data incorporate the passive contribution of fascia, skin, viscera, and other tissues. The exponential shape of the positive joint displacement/load relationship36,‘7, and a coupling between the bending and twisting motion34 were assumed. Given the absolute Euler angles as the generalized coordinates, such a relationship for the ‘j’th joint was expressed by: M,j=uxj[ebo(~~--“‘+‘)-

I] +K(I/I~--I+~~+,)

Myj=uyj[ebyl(~~--~+I)- l] +K(4j-~j+ Mzj= aZj[eb”(‘~-‘~+f)-11

J (14

Where: Mij moment about ‘i’th axis of a ‘j’th joint, a,b coefficients (they become negative for the negative angles), K coupling coefficient In the case of trunk flexion, correction of the formula was necessary. The data reported by McGill et a1.36 were collected while the subjects maintained bent knees and a semi-seated position. According to Andersson et a1.38,such a posture produces an average of 28.4” of lumbar flexion when compared with a standing posture. Subtraction of that angle in Equation (1b) yielded a more realistic range of motion in the case of trunk flexion: Mzj=

-a,i{e[-bz~(%+B,

1,-28.41-c},

c = e( - 28.4bzJ (1’3

Cholewicki

and McGill:

The coupling coefficient K was set between -8 N m/ rad34 and -46 N m/rad35 for all intervertebral joints. The function ‘shape’ coefficients (‘b’ Equation 1) were taken from McGill et a1.36and the ‘magnitude’ coefficients (‘a’ Equation 1) were adjusted to fit the individual trunk flexibility of each subject in flexion and lateral bending. This adjustment was accomplished by measuring the lumbar flexion and lateral bending angles and moments at the point where the trunk was supported by passive tissues. Coefficients ‘a’ were then calculated to limit the range of motion at those angles and to accommodate differences in subjects’ flexibility. Coefficients for twisting and trunk extension remained as the average values given by McGill et a1.36for all subjects. Muscle force and stiffness

The remaining moment (after accounting for passive tissue contribution) necessary to balance the external moment resulting from the hand-held load and upper body weight, was partitioned between all 90 muscle fascicles using EMG. Anatomically or functionally similar muscle fascicles (muscle group) were assigned the same EMG level. A previously documented EMGto-force processing approach3,29 has the disadvantage that it does not yield muscle stiffness - a key parameter in analysis of stability. Instead, a crossbridge bond distribution moment (DM) model was employed here to obtain muscle force and stiffness simultaneously. The DM is a mathematical approximation to Huxley’s original two-state muscle contraction dynamics39. The additional advantage of this model over Hill’s traditional equations is its ability to mimic the muscle response to various dynamic stretchshortening input cycles. The most recent version of the DM model was described by Ma and Zahalak28 and it was enhanced here by adding a non-linear tendon and a passive elastic component (see Cholewicki and McGil14’ for detailed description). The rectified filtered and normalized EMG represented calcium release and diffusion (activation dynamics), while the contraction dynamics were modulated with the instantaneous muscle length and velocity. The tendon slack length, contractile tissue optimum (resting) length and the maximum isometric muscle force of 35 N cm* constituted the input constants. This maximum stress was selected only as the initial value, since all the muscle forces were later adjusted with individual gains. Several assumptions had to be made regarding the neural activation of muscles not accessible for surface EMG recording. Psoas and quadratus lumborum were driven with the EMG signals of their synergists (internal obliques and lower erector spinae respectively). Lafortune et a1.41 observed such synergy patterns from many EMG electrode sites over the lumbar spine during unconstrained three-dimensional lifting tasks. These similarities could not be attributed to electrical cross-talk. A more detailed discussion of the above assumption can be found in Cholewicki et al . l9. The activation patterns of small transversospinalis

Mechanical

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of the in vivo lumbar

spine

5

and rotatores muscles are unknown, but the small spine stabilizing potential of these muscles could be simulated with. an increased passive joint stiffness5,7’16”7. Finally, each muscle force predicted from the EMG was adjusted with an optimization algorithm to satisfy the external moment requirements (within 10 N m tolerance level). The objective of this method was to balance all three moments acting about a given joint by applying the least possible adjustment to the individual muscle forces. This EMG-assisted optimization (EMGAO) combines the advantage of both EMGassisted and optimization methods for estimating muscle forces in an indeterminate biomechanical model. It preserves physiologically observed muscle recruitment patterns with an added advantage of the balance of moment equationsi’. The EMGAO procedure, discussed in detail in Cholewicki and McGi114*, was expanded here to encompass 18 df and 90 muscle fascicles. The individual muscle gains obtained from this algorithm were applied to ajdust muscle forces as well as their stiffness coefficients. Stability analysis

The minimum potential energy method5,7,‘6,17,37was carried to the first order (no post-buckling analysis) considering three frames per second in every trial. Conceptually, upon mathematical perturbation of the spine in all directions, it is then explored whether it has the ability to return to its original position. Mathematically a complete relative minimum of the potential energy (V) of the system is a necessary and sufficient condition to satisfy the mechanical stability criteria43. It is equivalent to stating that the second variation of the potential V must be positive definite44. Therefore the determinant as well as the principal minors of the Hessian matrix (second partial derivatives of the potential V with respect to each of the generalized coordinates Qi) must be positive (Equation 2). The determinant D is called the stability determinant.

[ 1 a*v

D=det -

aQiaQj

a2v a2v . . . a2v aQ: aQlaQ2 am a2v

= det

aQ2aQl

a2v

.

aQ:

* ...

a*v

aQnaQ1

-

>

OADij>O

:

.

a2v

.

aQ,z

(2)

Each data frame was ‘frozen’ in time, while the instantaneous spine geometry, external load, muscle forces and their instantaneous stiffness constituted the input to the analysis of stability (Figure 2). Upon perturbation the potential of the system consisted of

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the elastic energy stored in the linear springs (U,) (muscles and tendons), elastic energy stored in the torsionat springs (Ur) (lumped intervertebral joint discs, ligaments and other passive tissues) minus the work performed on the external load (W): v=C’~,tC’,--W

(3)

52 3

800-

$

u

600

5 3

Partiai derivatives of the potential V were calculated (detailed calculations are presented in Appendix B) and inserted to Equation 2. Once the entire 18 x 18 Hessian matrix of the partial derivatives of the system’s potential energy was compiled, it was diagonalized and its determinant calculated. Elements of the diagonalized Hessian matrix represent the curvature of the potential energy surface in the direction of a given generalized coordinate; the larger the curvature, the more stable the system is. If any value is less than or equal to zero, instability of the spine is indicated. While the first mode of buckling will possibly occur in the direction of the lowest potential curvature, the stability determinant describes better the ‘depth’ of the entire 18dimensional surface in all directions. Therefore, the 18th root of the Hessian’s determinant served as a relative (root average) stability index (SI). Figures 2 and 3 summarize the modelling procedure. Resntts The average maximum moments about the L4-s joint, generated by the subjects, were 217 (SD, 28) Nm of trunk extension in the lifting tasks, 114 (SD, 9) Nm of lateral bend in the bending trial, and 54 (SD, 4) Nm of twist in the twisting trial. Maximum Lhm5 joint compression forces occurred during the lifting trials and averaged 3911 (SD, 211) N across the three subjects. The intra-abdominal pressure (IAP), measured in one subject, reached the highest values during the lifting trials (189 mmHg). The lowest IAP peaks occurred during the standing trials (44 mmHg). The ‘sweeping’ trials constituted a good test for the

lm!-

-+

-m- FLEXIEXT

LAT. BEND

! I

I

2400

-2

0

2

4

Figure 5. Stability index (9) for all three subjects were positive and high in magnitude.

6

8

Time (s) Figure 4. Three-dimensional moments (flexion/extension, lateral bend, axialtwist) about the L4-s joint during a ‘sweep’ trial of subject no. 1, demonstrating the loading challenge to thespine.

10

10

during

‘sweep’

trials

model performance in complex 3-D tasks. There was a mixture of all three moments acting on the lumbar spine, which had to be balanced by the musculature (Figure 4). Forces for the selected muscles and their corresponding stiffness estimates in a ‘sweep’ trial are listed in Table 1. All major muscle groups were active, indicating the high degree of muscle cocontraction. The resultant Lde5 joint compression force varied between 2 and 2.5 kN, while the stability index (SI) characterized the lumbar spine as a stable structure under such conditions (Figure 5). Passive joint properties (disc and ligaments) contributed little to the overall moment requirements to balance the external load in all trials. However, these passive tissues played a crucial role in stabilizing the spine during tasks requiring little demand on muscular forces. This phenomenon will be discussed in greater detail later in the paper. Sensitivity unalysis

Sensitivity analysis was performed on several parameters used in the model to determine their influence on the stability index (SI). For example, normalizing (or not) the adjustments according to the muscle size in the EMGAO algorithm designed to balance the moments had little effect on the SI (see Cholewicki and McGi114’ for the discussion of EMGAO) (Figure 6a). Another sensitivity test was concerned with the site for external load application. The hands holding the weight were assumed to be rigidly attached to the ribcage in this model, which is not the case in real life. When the

Muscle

4

8

6

Time (s)

Table 1. Estimated force and stiffness second of the ‘sweep’ trial of subject

2

--ff

A

R rect. abdominis L rect. abdominis R ext. oblique 1 L ext. oblique 1 R int. oblique 1 L int. oblique 1 R iliocost. lumb. L iliocost. lumb. R lat. dorsi L3 L lat. dorsi L3

of the selected muscles, no.1 (Figure 4) Force (NJ 74 39 66 6 18 48 109 67 55 81

Stiffness

for the 5th

IN cm 147 108 232 56 55 421 291 249 144 226

‘I

Cholewicki

-

SI

f-

SI (so1 2)

-

SI (2.7)

-++

SI (min)

and McGill:

7-Y

4

6

8

10

Time (s)

20 b

2

60 Lg

40 ke

4

6 Time (s)

a

10

12

Figure 6a. The effects of the stability index (SI) calculated as the 18th root of the determinant of the stability matrix, the passive joint stiffness increase (2.7-fold), the alternate form of the objective function in the EMGAO algorithm (Sol 2). and the alternate definition of the stability index (SI min), where S1 was taken as the lowest value in the diagonalized stability matrix, during a lifting trial for subject no. 1.6b. The effect of passive joint stiffness increase (4.7 times) around a neutral spine position and the effect of increasing some of the small intrinsic muscle activity (multifidi and longissimus thoracis/iliocostalis lumborum pars lumborum) from zero to 3% maximum. The activation level of these muscles, as indicated by their EMG, never exceeded 3%. However, EMGAO algorithm would sometimes reduce these forces to zero in order to balance the moments. STAND2 trial for subject no. 1. Zero values of SI indicate spine instability (SI
simulated load was instead applied to the shoulder region, the difference in the SI was smaller than 1%. A more thorough analysis was performed on the assumptions pertaining to the passive properties of intervertebral joints. First, the effect of a lateral bending and axial twisting motion coupling coefficient (when varied from 0 to -46 in Equation la) on the SI was determined to be less than lo%, even at the extreme ranges of motion. Second, the experiments of Crisco and Panjabi37 and Crisco et al.* on lateral stability of the osteoligamentous spine were simulated. The average buckling load of 97 (SD, 21) N obtained from the 3 subjects corresponded well with the experimental value of 88 N (Crisco et al., 1992). However, the objective of the lumped passive stiffness parameter in this study was to encompass the effects of fascia, viscera, skin, etc., in addition to the discs and ligaments. Crisco and Panjabi37 found that the exponential joint stiffness model underestimated the joint stiffness in a neutral position. In fact, they found it necessary to increase the resting joint stiffness by an average of 2.7

Mechanical

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7

times to predict a reasonable critical load. As in Crisco’s experiments, we presently found that the stiffness of the local stabilizing system (muscles attaching to the individual vertebrae and passive joint stiffness5Y16)around the neutral joint position, was insufficient to maintain spine stability during tasks requiring minimal muscle forces. According to the initial model output, spines of all the subjects appeared unstable during some periods of the standing trial, particularly under the heaviest load. The pushing tasks were also characterized by low levels of stability, with two subjects’ spines buckling during these trials. The instances in which the subjects were unstable were associated with the neutral postures and the near zero muscle forces in the multifidi and the lumbar erector spinae muscles (longissimus thoracis and iliocostalis lumborum pars lumborum) at the mid-lumbar level (L&. If the activity of these small muscles was increased to at least l-3% MVC, the stability of the whole system was restored (Figure 6b). Similar to Crisco and Panjabi’s37 findings, the three- to five-fold increase in the passive stiffness of the intervertebral joints about their neutral position, assured stability of the whole spine in all cases (Figure 6b). Perhaps this may be an equivalent to the stiffening effect of the large preload present in the in vivo lumbar spine46. However, the most important outcome from the sensitivity analysis was that while the stability index was affected by the passive joint stiffness, its relative magnitude was always preserved (Figure 6). Relative stability index (SZ) The stability index (SI) exhibited two trends across all subjects and trials. First, the stability increased with the increased moment demand or the joint compression force during the task (Figure 7). The lowest level of stability was present when there was no demand on high muscular forces, such as upright standing just prior to the beginning of the lifting trial (Figure 7). In such cases, the stability of the whole spinal system relied predominantly on the passive stiffness of the inter-

,e 6 i

1

-

SI (Nmhdhad)

+

MOMENT

e +

L4/LS

COMP.

(N)

(Nm)

4 $ 2 y2

0

0

2

6

4 Time

8

10

(8)

Figure 7. Stability index (SI) followed a similar pattern in magnitude with the L4-s joint compression force and the trunk extensor moment (LIFT1 trial of subject no. 1).

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vertebral joints. Second, in the asymmetrical tasks, such as bending or twisting, the SI was also related to posture (Figu-fe 8). The highest level of stability was reached when the subjects were bent or twisted in such a way that the external load placed on the lumbar spine was acting thrtiugh the shortest moment arm. If the load was-held in an outstretched arm, the sttibility level was lower (Figure 8). Discussion The objective of this work was to develop a method for quantifying the relative spine stability in viva. The need for such a method arose from new hypotheses implicating the dysfunction of the spine stabilizing system (mechanical stability) in the aetiology of some lumbar spine disorders’3,1”. The method presented here can be used to test these hypotheses and to study the clinical issues related to the mechanical stability of the spine. The stability index (SI) introduced in this study has the units of stiffness per radian (Nm/rad/rad). It was calculated as the 18th root of the product of 18 diagonal elements in the diagonalized Hessian matrix. Each element corresponds to the increase in stiffness arising from rotating the spinal column in the direction of the corresponding generalized coordinate. The SI then has the physicai interpretation of a ‘root average’ spine stiffness slope in all directions. Since the spine is most likely to buckle in the direction of the coordinate with the lowest stiffness slope, it may be appropriate to also define the SI as the lowest value in the diagonalized Hessian matrix. The SI was calculated both ways; however, as the relative measure, the SI defined either way, follows a similar pattern (Figure 6a). The most important finding of this study was that the stability of thehuman lumbar spine seemed to increase during the most demanding tasks as defined by the joint compression force, and diminished during the periods of low muscular activity. Thus the hypothesis that the spine system maintains a constant stability safety margin, regardless of the moment demands or postures, was not supported by the results of this work.

Time (8) Fire 8. Stability index‘(SI) and the lateral rotation of the ribcage in reletion to the pelvis, demonstrating that the SI is higher when the externai toadvector acts closer to the lumbar spine (BEND2 trial of subject no. 1)

However, the control of spine stability suggested by the present study is plausible, because the error of the central nervous system and spine buckling at high loads can be detrimental. On the other hand the stability during very light tasks such as standing may not be as critical, and perhaps is handled to a large extent by the passive tissues to prevent the expenditure of energy. While the large muscles spanning between the pelvis and ribcage provide the bulk of stiffness to the spinal column7,‘7, activity of the short, intrinsic muscles that span only one or two joints (such as the multifidi and the rotatores) was necessary to maintain stability of the whole lumbar spine. Although the small transversospinalis and rotatores muscles were not modelled in this study, their effect was simulated by increasing passive joint stiffness. It was found that the spine will buckle if the activity of multifidi and lumbar erector spinae is zero, even when the forces in large muscles are substantial. Instability can be prevented by increasing the activity (stiffness) of small muscles or by increasing passive joint stiffness. It highlights the importance of motor control to coordinate muscle recruitment between the large musculature and the small intrinsic spine muscles when handling small loads. These observations are consistent with Bergmark5,16, who lumped all passive joint properties and intrinsic muscles into the ‘local stabilizing system’. He found that there was an upper limit on the possible activation of the large musculature at a given activation level of the local stabilizing system, beyond which the spine buckles. This work contributes to the explanation of spine injury occurrence during activities requiring low muscular forces (it is already quite easy to show how injury occurs at high loads, where the load tolerance of an individual tissue is exceeded). A hypothetical relationship can be proposed, by which the injury risk due to the loss of stability increases with the decreased demand on muscular effort (Figure 9). For example, this may explain how a person could work all day at a demanding job and ‘throw out his back’ picking up a

TASK DEMAND (JOINT COMPRESSION) Fiire 8. Hypothetical model for injury risk to the spine due to tissue failure and spine instability. While high loads can cause injury by tissue disruption, instability at low loads may allow sufficient local joint movement to overload or irritate soft tissues.

Cholewicki

and McGill:

pencil at the end of the day. The exact mechanism of the injury is not known at this point, but it is hypothesized that the momentary loss of stability that would lead to unexpected displacements, irritate nociceptors in connective and/or soft tissues or nerve roots. Conversely the sudden need to regain spine stability may result in muscle spasm and overload of a single tissue. If the motor control system faces the danger of lumbar spine buckling, a plausible response would be to activate a small intrinsic muscle(s) that crosses a particularly unstable joint to counteract the large displacements. In turn, this muscle is subjected to a greater risk of tissue overload and injury. Larger muscles spanning several joints are not suitable for such action, as this would increase the load borne by the inferior intervertebral joints, magnifying the effects of buckling. We observed single vertebrae displacement, confined only to rotation at L2-s, when we filmed a powerlifter in vivo with videofluoroscope45. This study confirmed that buckling behaviour can be limited to a single lumbar level from inappropriate activation of muscles. The results of this study can also permit the formulation of several hypotheses on the aetiology of chronic low back pain and support the theory proposed by Panjabi13’i4. Following the initial trauma, a decrease in intervertebral joint stiffness or an increase in the neutral zone may occur. The motor control system can compensate to a certain degree with the additional cocontractions of the intrinsic muscles, but this may lead to muscle fatigue. Healthy people may be able to rely on passive joint properties for spine stability during standing or sitting, while clinically deficient patients need to sustain prolonged static muscle contractions. Perhaps the intrinsic spine musculature may itself be deficient in providing sufficient stiffness to the spine following nerve injury, muscle injury, or fatigue. Another possibility is that people lacking motor control skills commit errors during light activities and repeat injuries to the same tissues. Clinicians need to explore the effectiveness of motor control training as an adjunct to the muscle strength improvement for reducing low back pain episodes. There are several limitations of the presented model. The overall stability of the spine may be underestimated, because no effect of intra-abdominal pressure was considered together with omission of the transversus abdominis and small spinal rotator muscles. On the other hand, the assumption that the spine is a conservative system, overestimates the stability index due to some losses of potential energy stored in viscoelastic structures after the perturbation. A set skeletal geometry and muscle cross-sectional areas were used for all subjects. It was not possible to record the activity of intrinsic spine muscles such as psoas and quadratus lumborum, although surface electrodes were strategically placed to act as surrogates. Our recent experience with indwelling electrodes indicates that such choices seem reasonably valid. No translations between the intervertebral joints were allowed and the centres of joint rotation did not migrate. We assumed a normal

Mechanical

stability

of the in vivo lumbar

spine

9

spine in our model, with no pre-existing translational laxity. Although translations can be included in the future, the addition of 18 degrees of freedom at present would obscure the mathematical traceability of this first attempt for quantifying the spine stability in vivo. Finally, slow and smooth dynamic tasks (not ballistic) justified application of static analysis of stability. However, accelerations were considered in calculation of the hand-held loads. Model validity

It is not yet possible to validate directly human spine models that predict muscle and other tissue forces. It would be even more difficult to validate the model quantifying spine stability in vivo. The present model belongs to a certain category of models in science, for which there are no tools for model validation47. According to Lewandowski47, attributes of the present work can be classified as a ‘hard’ (built on the basis of well-established natural laws), ‘causal’ (discriminating between cause and effect), and ‘deterministic’ (as opposed to probabilistic) modelling approach. The modelled system exists in the real world (human spine), but it is not possible to conduct direct experiments. For this categorgy of models, Lewandowski47 proposes a validation process consisting of component validation, internal validity checks, sensitivity analysis and a judgemental evaluation. The component validation is based on the premise that models built out of well validated submodels will probably be valid47. This approach was applied to the most critical variable in the calculation of spine stability - muscle stiffness. Stiffness estimates obtained from the sub-model of a muscle were consistent with the experimental data in the literature4’, which adds confidence to the stability values. Internal validity in ‘hard’ models pertains to the preservation of basic physical laws on which such a model was built. For example, the advantage of using Huxley’s muscle model in the present work was that each variable had a physical interpretation, allowing the results to be inspected for consistency with the initial assumptions4o. Energy balance and variational mechanical principles were the main elements in the stability analysis. Fulfilment of these laws was monitored in the model output by comparing the moment balance with the first partial derivatives of the system’s potential, which should have been (and was) equivalent. Energy conservation surfaced in the output as a symmetry in a Hessian matrix. A very significant check for the internal validity of the whole modelling process arose from the two independent estimates of moments acting on the lumbar spine. The first was obtained from the video image and a rigid linked body model and the second came from the estimates of muscle forces and a lumbar spine model (see Figure 2). It required an average of 48% (SD, 27) adjustment to the individual muscle forces to bring these two estimates to unanimity. Considering that the dynamic, three-dimensional tasks were studied, such a number indicates a good agree-

10

Chin. Biomech.

Vol. 11, No. 1, 1996

merit. In addition, since we initially assumed the maximum muscle stress of 35 N cm-2, the adjusted muscle forces will result in less than 61 N cme2 of muscle stress {mean plus one standard deviation adjustment) even for maximally activated muscles. Such values fall well within the physiological capacity of muscle force generating potential quoted by various authors48.4y. The anatomical representation of a lumbar spine and its musculature also has a bearing on the model output”‘. The subjects were selected to fit an average morphology assumed in this study. However, the ultimate evidence of the model’s inaccuracy would be a disagreement between the lumbar spine moments

healthy subjects. Clinical instability can be simulated in the model as the decreased intervertebral joint stiffness and necessary adaptations in the neuromuscular system can be studied. Acknowledgements

The helpful suggestions of Dr J Koorda regarding structural stability analysis, and the constructive comments of Drs K Norman and K Wells are greatly appreciated. The financial assistance of the Natural Science and Engineering Research Council, Canada (NSEKC) is also acknowledged.

estimated hy the two independent procedures, discussed earlier. Sensitivity analysis

was conducted

on the most

extensive assumptions regarding the passive properties of intervertebral joints. Fortunately the motion segment stiffness had very little effect on the relative values of SI, although the absolute values altered (Figure 6). The relative spine stability index was also unchanged by load placement or the moment balance algorithm (Figure 6). Therefore conclusions regarding spine stability. if based on the relative magnitude of SI,

References 1 Lucas DB, Bresler B. Stability of the Ligamentous Spine. Report no. 40 from the Biomechanics Laboratory, University of California, San Francisco, Berkeley, 1961 2 Crisco JJ, Panjabi MM, Yamamoto I, Oxland TR. Euler stability of the human ligamentous lumbar spine: Part II experiment. Clin Biomech 1992; 7: 27-32 3 McGill SM, Norman RW. Partitioning of the Lj-s dynamic moment into disc, ligamentous and muscular components during lifting. Spine 1986; 11: 666-78 4 Cholewicki J, McGill SM, Norman RW. Lumbar spine load during the lifting of extremely heavy weights. Med

can be considered valid. Finally the results obtained from this model, such as the l-,4-j joint compression forces, can be considered reasonable in view of similar results in literature.

SciSportsExerc 1991;23:1179-86 5 Bergmark A. Stability of the lumbar spine: a study in mechanical engineering. Acta Orthop Stand 1989; 60

Although

6 Dietrich M, Kedzior K, Zagrejek T. Modelling of muscle

none of the previous

spine models was

directly validated. such a comparison may serve as a ,judgemental evaluation in the validation process outlined by Lewandowski”‘. Optimization models generally underestimate joint compression force by 23-43% when compared with EMG-assisted models, because they do not account for antagonist muscle cocontraction’“. However, other EMG-assisted models simulating similar tasks. result in similar spine compression forces as in the present study. For example, Granata and Marras” reported 2-3.4 kN L&r joint compression

during

asymmetric

trunk

extensions

generating somewhat similar three-dimensional spine moments as our ‘sweep’ trial (Figure 4). In another model. Mirka and Marras” performed a stochastic sensitivity analysis of various trunk muscle coactivation patterns. They found that the lumbar spine compression force varied between 1.5 and 2.2 kN for dynamic, 40 Nm of trunk extension moment. We obtained between 2 and 2.5 kN of the L4m5 joint compression force in the ‘sweep’ trial, which compares well with the values cited above. We followed the validation process suggested by Lewandowski”’ for a category of models encompassing the one presented here. There is no single validation study for such models. However, the extensive testing process which we followed, failed to demonstrate that our approach to obtain a relative stability index is not valid. This model can be used in the future to study issues related to the mechanical stability of the spine. The~most immediate question arising from this study pertains

to differences

between

patients having motion

clinically

diagnosed and

segment ‘instability’

[Suppl230]: l-54 action and stability of the human spine. In: Winters JM, Woo SL-Y, eds. Multiple Muscle Systems: Biomechanics and Movement Organization. Springer-Verlag, New York, 1990; 451-60 7 Crisco JJ, Panjabi MM. The intersegmental and multisegmental muscles of the lumbar spine: a biomechanical model comparing lateral stabilizing potential. Spine 1991; 16: 793-9 8 Tesh KM, Shaw-Dunn J, Evans JH. The abdominal muscles and vertebral stability. Spine 1987; 12: 501-8 9 Marras WS, Mirka GA. Muscle activations during asymmetric trunk angular accelerations. J Orthop Res 1990;8:824-32 10 Marras WS, Rangarajulu SL, Lavender SA. Trunk loading and expectation. Ergonomics 1987; 30: 551-62 11 Lavender SA, Mirka GA, Schoenmarklin et al. The

effects of preview and task symmetry on trunk muscle response to sudden loading. Hum Factors 1989; 31: 101-1s 12 Lavender SA. Preparatory response strategies seen prior to sudden loading of the torso while in constrained posture. In: Karwowski W, Yates JW, eds. Advances in Industrial Ergonomics and Safety III. Taylor and Francis, 1991; 239-46 13 Panjabi MM. The stabilizing system of the spine. Part I. Function, dysfunction, adaptation and enhancement. 1 Spinal Disord 1992; 5: 383-9 14 Panjabi MM. The stabilizing system of the spine. Part II. Neutral zone and instability hypothesis. J Spinal Disord 1992;5: 390-7 1.5 Cholewicki J. Mechanical stability of the in vivo lumbar spine. [PhD Dissertation] Dept. of Kinesiology,

University of Waterloo, Waterloo, Ontario, Canada, 1993 16 Bergmark A. Mechanical Stability of the Human Lumbar Spine. [Doctoral Dissertation] Dept. of Solid Mechanics, Lund University, Lund, Sweden, 1987 17 Crisco JJ, Panjabi MM. Postural biomechanical stability and gross muscular architecture in the spine. In: Winters

Cholewicki

and McGill:

JM, Woo SL-Y, eds. Multiple Muscle Systems: Biomechanics and Movement Organization.

Springer-Verlag, New York, 1990; 438-50 18 Shirazi-Ad1 A, Parnianpour M. Nonlinear response analysis of the human ligamentous lumbar spine in compression. On mechanisms affecting the postural stability. Spine 1993; 18: 147-58 19 Cholewicki J, McGill SM, Norman RW. Comparison of muscle forces and joint load from an optimization and EMG assisted lumbar spine model: Towards development of a hybrid approach. J Biomech 1995; 28: 321-31 20 McGill SM. Electromyographic activity of the abdominal and low back musculature during the generation of isometric and dynamic axial trunk torque: implications for lumbar mechanics. J Orthop Res 1991; 9: 91-103 21 Wells RP, Winter DA. Assessment of signal and noise in the kinematics of normal, pathological and sporting gaits. Proc Canad Sot Biomech London, Ontario, Canada. 1980; 92-3 22 McGill SM, Norman RW. Statically and dynamically determined low back moments during lifting. J Biomech 1985; 18: 877-85 23 Jiibsis FF, O’Connor MJ. Calcium release and reabsorption in the sartorius muscle of the toad. Biomech Biophys Res Comm 1966; 25: 246-52 24 Bahler AS, Fales JT, Zierler KL. The active state of mammalian skeletal muscle. J Gen Physioll967; 50: 2239-53 25 Zajac FE. Muscle and tendon: properties, models, scaling, and application to biomechanics and motor control. Crit Rev Biomed Eng 1989; 17: 359-411 26 Zahalak GI. A distribution-moment approximation for kinetic theories of muscular contraction. Math Biosc 1981; 55: 89-114 27 Zahalak GI. A comparsion of the mechanical behaviour of the cat soleus muscle with a distribution-moment model. Biomech Eng 1986; 108: 131-40 28 Ma S-P, Zahalak GI. A distribution-moment model of energetics in skeletal muscle. J Biomech 1991; 24: 21-35 29 McGill SM. A myoelectrically based dynamic three-dimensional model to predict loads on lumbar spine tissues during lateral bending. J Biomech 1992; 25: 395-414 30 Bogduk N, Macintosh JE, Pearcy MJ. A universal model of the lumbar back muscles in the upright position. Spine 1992; 17: 897-913 31 White AA, Panjabi M. Clinical Biomechanics of the Spine. Lippincott, Philadelphia, 1990; 106-12 32 Winter DA. Biomechanics of Human Movement. John Wiley and Sons, New York, 1979; 151-2 33 Zatsiorsky V, Seluyanov V. The mass and inertia characteristics of the main segments of the human body. In: Matsui H, Kobayashi K, eds. Biomechanics VIII-B Champaign, Human Kinetics Publishers, 1983; 1152-9 34 Panjabi MM, Brand RA, White AA. Three dimensional flexibility and stiffness properties of human thoracic spine.

Mechanical

stability

of the in vivo lumbar

spine

11

J Biomech 1976; 9: 185-92 35 Patwardhan AG, Soni AH, Sullivan JA et al. Kinematic analysis and simulation of vertebral motion under static load - part II: simulation study. J Biomech Eng 1982; 104: 112-18 36 McGill SM, Seguin J, Bennett G. Passive stiffness of the lumbar torso in flexion, extension, lateral bending and axial rotation. Effect of belt wearing and breath holding. Spine 1994; 19: 696-704 37 Crisco JJ, Panjabi MM. Euler stability of the human ligamentous lumbar spine: Part I theory. Cfin Biomech 1992; 7: 19-26 38 Andersson GBJ, Murphy RW, Ortengren R, Nachemson AL. The influence of backrest inclination and lumbar support on lumbar lordosis. Spine 1979; 4: 52-8 39 Huxley AF. Muscle structure and theories of contraction. In: Butler JAV, Katz B, eds. Progress in Biophysics and Biophysical Chemistry. Pergamon Press, New York, The Macmillan Company, 1957; 6-318 40 Cholewicki J, McGill SM. Relationship between muscle force and stiffness in the whole mammalian muscle: a simulation study. J Biomech Eng 1995; 117: 339-42 41 Lafortune D, Norman RW, McGill SM. Ensemble average of linear envelope EMGs during lifting. Proc Canad Sot Biomech, Ottawa, Canada, 1988; 92-3 42 Cholewicki J, McGill SM. EMG assistedoptimization: a hybrid approach for estimating muscle forces in an interdeterminate biomechanical model. J Biomech 1994; 27: 1287-9 43 Thompson JMT, Hunt GW. Elastic Instability Phenomenon. John Wiley and Sons, New York, 1984 44 Langhaar HL. Energy Methods in Applied Mechanics. John Wiley and Sons, New York, 1962 45 Cholewicki J, McGill SM. Lumbar posterior ligament involvement during extremely heavy lifts estimated from fluoroscopic measurements. J Biomech 1992; 25: 17-28 46 Janevic J, Ashton-Miller JA, Schultz AB. Large compressive preloads decrease lumbar motion segment flexibility. J Orthop Res 1991; 9: 228-36 47 Lewandowski A. Issues in model validation, Angewandte Systemanalyse 3: 2-11. Reprinted in International Institute For Applied Analysis 1982; RR-82-37 48 Weis-Fogh T, Alexander RM. The sustained power output from striated muscle. In: Scale Effects in Animal Locomotion. Academic Press, London, 1977; 511-25 49 Reid JG, Costigan PA. Trunk muscle balance and muscular force. Spine 1987; 12: 783-6 50 McGill SM, Norman RW. Effects of an anatomically detailed erector spinae model on L4-s disc compression and shear. J Biomech 1987; 20: 591-600 51 Granata KP, Marras WS. An EMG-assisted model of loads on the lumbar spine during asymmetric trunk extensions. J Biomech 1993; 26: 1429-38 52 Mirka GA, Marras WS. A stochastic model of trunk muscle coactivation during trunk bending. Spine 1993; 18: 1396-409

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Vol. 11, No. 1, 1996

Appendix A: Anatomical Parameters

model

of the muscles included in the model are listed in Table Al and coordinates

Tw AI. Right body side muscles, their physiological nodes (see Table A2 for the skeleton coordinates)

cross-sectional

Muscle No. ___I___-.-1 2 3 4 5 6 7

8

9 10 ?l 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45

YO 10.0 10.0 9.0 9.0 8.0 5.6 6.0 5.7 5.0 4.0 11 .o 16.8 0.7 1.8 12 1.5 1.3 2.0 1.0 1.0 1.0 1 .o 2.0 2.0 4.0 4.0 4.0 4.0 1.4 2.9 2.4 I.5 0.9 0.8 0.6 0.6 0.6 0.6 0.6 0.6 4.4 4.4 4.4 4.4 4.4

Rect. Abdominis Ext. Oblique 1 Ext. Oblique 2 ht. Oblique 1 Int. Oblique 2 Pars Lumb. L;, Pars Lumb. LR Pars Lumb. Li Pars Lumb. LZ Pars Lumb. L, Iiiocost. Lumb. Long. Thor. P. Long. Thor. L: Long. Thor ii, Long. Thor. L, Long. Thor. L, Long. Thor. L, Quack. Lumb. P Quadr. Lumb. L! Quadr. Lumb. L2 Cluadr. Lumb. LJ Ouadr. Lumb L4 Lat. Dorsi P Lat. Dorsi L5 Lat. Dorsi L, Lat. Dorsi L3 Lat. Dorsi L, Lat. Dorsi L, M&if. P.LL, M&if. P.LJ Moltif. P.L? M&if. P.L3 Multif. P.L, M&if. &5.L3 M&if. L5.Ls Multif. L5.L, Multif. L4.LZ M&if. L4.L: Multif. L3.1, Multif. L2 TJ2 Psoas L5 Psoas L4 Psoas L3 Psoas L2 Psoas L!

Table A2. Coordinates are zero)

defining

-

30.0 16.8 18.3 13.2 11.5 5.2 5.7 7,7 10.4 13.2 18.1 17.1 23.7 25.3 26.9 29.6 27.3 17.0 14.7 11.5 8.5 5.8 25.2 23.5 22.8 21.6 20.7 20.1 7.0 9.3 8.6 11,J 15.3 6.4 9.6 13.4 7.5 11.1 7.8 8.5 13.8 13.8 13.8 13.8 13.8

the skeletal

14.0 15.5 1.0 3.0 4.0 4.5 4.8 8.2 14.3 13.0 13.0 12.5 11.0 10.5 3.0 9.0 7.5 6.0 4.5 3.0 2.5 4.9 8.3 12.0 15.4

geometry

and muscle

of their origins, insertions

areas (no), resting

attachment

length (Xo), tendon

Origin

Insert

PEL (3) PEL (4) PEL (5) PEL (6) PEL (7) PEL (9) PEL (9) PEL (9) PEL (9) PEL (9) PEL (IO) PEL (II) L5 (4) L4 (4) L3 14) L2 (4) Ll (4) PEL (12) PEL (12) PEL (12) PEL (12) PEL (12) PEL (16) PEL (15) L4 (8) L3 (8) L2 (8) Ll (7) PEL (14) PEL (14) PEL (13) PEL (13) PEL (13) L5 (6) L5 (6) L5 (6) L4 (7) L4 (7) L3 (7) L2 (7) PEL (17) PEL (17) PEL (17) PEL (17) PEL (17)

RIB (3) RIB (4) RIB (5) RIB (6) RIB (7) L5 (3) L4 (3) L3 (3) L2 13) Ll (3) RIB (8) RIB (9) RIB (10) RIB (11) RIB (12) RIB (13) RIB (14) RIB (15) l-1 (5) L2 (5) L3 (5) L4 (5) RIB (17) RIB (17) RIB (17) RIB (17) RIB (17) RIB (I 7) L5 (5) L4 (6) L3 (6) L2 (6) Ll (6) L3 (6)

and nodal points are presented in Table A2. lengths

(Yo), points of origin,

insertion,

and

Nodal Points

L26)

54% RA length 64% RA length -

on L4 on L4

h (10) 4 (IO)

Lq (11) L3 (IO) L* (10) L, (9) i-4 (12)l-3(Ill Lz(11)L, (IO) La (13) L3 (12) L* (12) L1 (I 1) RIB L3 (12)L2(12)Ll (11) RIB L2 (12) L, (II) RIB L, (I 1) RIB

(23) (23) (23) (23)

RIB RIB RIB RIB RIB

-

(22) (21) (20) (19) (18)

-

Ll 16)

L2P-3 Ll 6) Ll (8) RIB (16) L5 (7) L4 (9) L3 (9) L2(9) Ll (8)

-

points

in a neutral

spine position

L,X(8) L,X(9) LSX(10) L,X(ll) LsX(12) t,X(13) L.X(l4) L,X(l5) L,X(16)

= 12 = 8.6 = 7.9 = 7.6 = 8 = 7.2 = 4.6 = 6 = 6.6

L,Y(8) LsY(9) L,Y(lO) L,Y(ll) L,Y(12) L,Y(13) LsY(14) L,Y(l5) L,Y(16)

L,,X(l) 4X(2) L‘&(3) L,&(4) L,,X(5) L,X(6) L,X(7) t&(8) L,X(9) L,X(lO) L4X(11) L&(12) L4x(I3) L4X(14) LhX(15) LdX(16)

= = = = = = = = =

La,Y(l) = 21.1 4Y(2) = 24.8 t,Y(3) = 23.4 tdY(4) = 22.4 LdY(5) = 23.8 L,Y(6) = 21.5 LdY(7) = 21.2 L4Y(8) = 22.4 L,Y(9) = 21.8 LdY(10) = 22.3 L,Y(ll) = 22.3 L.,Y(12) = 22.3 t,Y(13) = 22.3 4Y(l4) = 22.2 4Y(I5) = 22.3 LY(I8) = 22.6

(cm) (i.e. all relative

vertebral

rotations

PELVIS PELX(11 PELX(2) PELXI3) PELX14) PELX(5) PELXIG) PELX(7) PELX(8) PELX(9) PELXIIO) PELX(11) PEt.X(?21 PELX(T3) PELX(l4) PELX(15) PELX(16) PELX117) PELX(18) fJELXll9)

= 10.4 = 9.4 = 18.4 =: 12.8 = 19 = 9 = 16 = 12.8 = 2.4 =- 1.4 -. 1.4 = 6 = 2.6 = 2 = 3.6 = 4.8 =z 15 = 1.3 = 5.4

PELY(I) PELY(2) PELY(3) PELY(4) PELY(5) PELY(6) PELY(7) PEtY(8) PELY(9) PELY(10) PELY(l1) PELY(12) PELY(l3) PELY(l4) PELY(I5) PELY(16) PELY(17) PELY(l8) PELY(l9)

= = = = = = = = =

6.8 17.4 5 18.6 5 21.5 16 18.6 17.8 = 16.6 = 16.5 = 21.4 = 18 = 13.8 = 19.2 = 21.5 = 5 = 16.6 = 21

9.4 10.6 7.6 4 4 5.8 8.6

L,Y(l) L,Y(2) L,Y(3) L,Y(4) L,Y(6) LSY(6) L,Y17)

17.4 21.1 20.4 20.4 19.9 19.1 18.8

PEtZ(1) PELZ(2) PEtZ(3) PELZ(4) PELZI5) PELZiGj PELZ(7) PELZ(8) PELZ(9) PELZ(l0) PELZ(11) PELZ(I2) PELZ(I3) PELZ(l4) PELZ(15) PEtZ(16) PELZ(l7) PELZ(18) PELZ(19)

= = = = = = = = =

7.6 0 3 13 0 12.5 12 13 6 = 6.8 = 3.3 = 9 = 3.6 = 1.5 = 3 = 6 = 8.2 = 0 = 6.2

LsZ(1) L,Z(2) L,Zi3) L,Z(4) L,Z(5) L5Z(6) L,Z(7)

0 0 5 0.2 0.5 1.5 2.3

hip joint connect L5 rect. abd. ext. obl. 1 ext. obl. 2 int. obl. 1 int. obl. 2 transv. abd. pars. lumb. iliocost. lumb. long. thoracis quad. lumb. multifidus 01 multifidus 02 lat. dorsi sacr lat. dorsi iliu. psoas lig. suprasp. lumbod. fascia

L,X(l) L,X(2) L,X(31 La41 C&5) bXf6) L,XfJ)

= = = = = = =

= = = = = = =

= = = = = = =

= 19.2 = 20.2 = 20.9 = 20.4 = 21.4 = 22 = 20.8 = 21 = 21.2

L,Z(8) = 0 L,Z(9) = 0 L,Z(IO) = 0 L,Z(ll) = 3.6 L,Z(l2) = 2 L,Z(I3) = 2.4 L,Z(l4) = 0 L,Z(15) = 0 L,Z(16) = 0

lig. lig. lig. lig. lig. lig. lig. lig. lig.

ant. long. post. long. flavum intertransv. caps.1 (lat.) caps.2 (med.) interspin. interspin. interspin.

hZ(ll

connect L5 connect L3 pars. lumb. long. thoracis L,, quad. lumb. L4 multifidus (i.L.J multifidus (o.L,J lat. dorsi L4 psoas node PLl & PL2 node IL node LT node LT5 lig. ant. long. lig. post. long. lig. flavum

----J-4

L connect PEL connect L4 pars. lumb. long. thoracis L5 multifidus (i.L5) multifidus (o.Lg) psoas

10.6 10.6 7.4 3.8 7.2 4.1 6.7 3.6 9.2 = 2.5 = 2.5 = 2.5 = 2.5 = 12.4 = 8.7 = 8

= = = = = = = = =

0 La4 0 L,Z(3) 4 LJ(4) 0.2 4Z(5) 4.4 t‘,Z(6) 0.6 L4Z(7) 1.5 t,Z(8) 0.2 L,Z(9) 2.3 L‘$Z(lO) = 5 L,Z(ll) = 7.4 4Z(12) = 3.8 L,Z(13) = 1.5 LdZ(14) = 0 LaZ(15) = 0 4Z(16) = 0

Cholewicki

and McGill:

Mechanical

stability

of the in vivo lumbar

spine

13

Table A2 conri,

23.4 22.1 22 22 21.8 22.3 21.4 21.7 21.7 X X

LJ(17) L,Z(18) LJ(19) LJ(20) 42(21) LJ(22) LJ(23) L,Z(24) LJ(25) L,Z(26) L,Z(27)

10.6 9.8 6.9 3.2 6.2 4 6.3 3 8.8 = 2 = 2 = 2

L,V(l) = 24.8 LsVf2) -= 28.5 LsV(3) = 26.6 L,Vf4) = 25.2 LsY(5) = 26.8 LsV(6) = 24 LsV(7) = 24.3 LsY(8) = 25.2 LsV(9) = 25.3 LsV(l0) = 25.2 LsV(11) = 25.2 L,V(12) = 25.2

L,Zfl) LsZ(2) LsZf3) LsZ(4) LsZf5) L,Z(6) LsZ(7) LsZ(8) LsZ(9) LsZ(l0) LsZ(11) LsZ(12)

= = = = = = = = =

9.8 8.7 5.9 2.6 5.2 3.2 6 2.4 7.7 = 1.4 = 1.4 = 1.4

L,V(l) L,V(2) L,V(3) L,V(4) L*Y(5) LzV(6) LzY(7) LsV(8) L,V(9) L,V(lO) LsV(11) L,V(12)

L,Z(l) L,Z(2) L,Z(3) LsZ(4) LzZ(5) LsZf6) L,Z(7) L,Z(8) L*Z(9) L,Z(lO) L,Z(ll) L,Z(12)

= = = = = = = = =

L.J(l7) L&(18) L&(19) 4x120) LJ(21) L,&(22) LdX(23) L,X(24) LaX(25) LX(26) L&(27)

= = = = = = = = = = =

7.4 7.2 7.2 3.6 4.6 3.6 12.2 10.7 9 X X

L4Y(17) 4Vf18) L,rV(19) 4W20) L4yf21) L.,Y(22) LV(23) LV(24) bV(25) LV(26) L4V(27)

= = = = = = = = = = =

= = = = = = = = = = =

3.6 2 1.6 0 0 0 0 0 0 X X

lig. intertransv. lig. caps.1 (lat.) lig. caps.2 (med.) lig. interspin. lig. interspin.2&3 lig. supraspin. ant. end plate mid. end plate post. end plate talc. node for E02 talc. node for 102

0 0 3 0.2 3.8 0.5 1.5 0.2 2.1 = 7.6 = 3.9 = 1.5

connect L,r connect Lz pars. lumb. long. thoracis Ls quad. lumb. L3 multifidus (i.Ls) multifidus (o.LJ lat. dorsi Ls psoas node IL node LT nodeLT4and LT5

0 0 2.7 0.2 3.8 0.5 1.5 0.2 2 = 7.8 = 4.1 = 1.5

connect Ls connect L, pars. lumb. long. thoracis L2 quad. lumb. Lz multifidus (i.Ls) multifidus (o.LJ lat. dorsi L2 psoas node IL node LT node LT3, LT4 & LT5

L3 L,X(l) LsXf2) LsX(3) LaX(4) LsX(5) 4X(6) 4X(7) LsX(8) LsX(9) L,X(lO) L,X(ll) LsX(12)

= = = = = = = = =

L,X(l) L,X(2) L*X(3) L,X(4) LsX(5) L2X(6) L,X(7) LsX(8) L,X(9) L*X(lO) L,X(ll) LsX(12)

= = = = = = = = =

= = = = = = = = =

28.5 32.2 29.8 27.8 30 26.9 27.4 27.8 29 = 27.9 = 27.9 = 27.9

Appendix B: Stability analysis At any given frame, the potential of the spine system (V) is expressed as the sum of the elastic energy stored in the linear springs (U,) (muscles and tendons), elastic energy stored in the torsional springs (Ur) (lumped intervertebral joint discs, ligaments and other passive tissues) minus the work performed on the external load (W):

L1 L,X(l) = 8.7 L,X(2) = 7.2 L,Xf3) = 4.4 L,X(4) = 1.4 L,X(5) = 4 L,X(6) = 2.2 L,X(7) = 1.2 L,Xf8) = 6.6 L,X(9) = 0.2 L,X(lO) = 0.2 L,X(l 1) = 0.2

L,V(l) L,V(2) L,Y(3) L,Y(4) L,Yf5) L,Vf6) L,V(7) L,V(8) L,V(9) L,V(lO) L,Y(ll)

RIBX(1) RIBX(2) RIBX(3) RIBX(4) RIBX(5) RIBX(6) RIBX(7) RIBX(8) RIBX(9) RIBX(l0) RIBX(11) RIBX(12) RIBX(13) RIBX(14) RIBX(15) RIBX(16) RIBX(17) RIBX(18) RIBX(19) RIBX(20) RIBX(21) RIBX(22) RIBX(23) RIBX(24) RIBX(25) RIBX(26)

= 7.2 = 7.5 = 19 = 6 = 12.5 = 15 = 19 R 1.6 = 2 = 2 = 2 = 2.5 = 2.5 = 2.5 = 3.5 = 1.8 = 9 = 1.2 = 2.4 = 3 = 3.6 = 3.6 = 0.2 = 18.2 = 11.5 = 4.6

RIBV(1) RIBV(2) RIBV(3) RIBV(4) RIBV(5) RIBY(6) RIBX(7) RIBV(8) RIBV(9) RIBV(10) RIBY(11) RIBV(12) RIBV(13) RIBV(14) RIBV(15) RIBY(16) RIBV(17) RIBV(18) RIBV(19) RIBY(20) RIBV(21) RIBV(22) RIBY(23) RIBV(24) RIBV(25) RIBVf26)

s=

f [F,+K,(l,-l,,)] m=l

= 0 = 0 = 2.6 = 0.2 = 3.6 = 0.5 = 0.2 = 1.9 = 8 = 4.3 = 1.5

connect L2 connect ribcage pars. lumb. long. thoracis L, quad. lumb. L, multifidus (i.L,) lat. dorsi L, psoas node IL node LT node LT2.3.4 & 5

I

= 35.5 = 68 = 35 = 30 = 31.5 = 29 = 38 = 39 = 44 = 53.5 = 58 = 62.5 = 66.7 = 68.5 = 35.5 = 34.2 = 47 = 34.5 = 31.7 = 29 = 26.4 = 24.2 = 34 = 37.6 = 37.2 = 34.8

RIBZ(1) RIBZ(2) RIBZ(3) RIBZ(4) RIBZ(5) RIBZ(6) RIBZ(7) RIBZ(8) RIBZ(9) RIBZ(10) RIBZ(11) RIBZ(12) RIBZ(13) RIBZ(14) RIBZ(15) RIBZ(16) RIBZ(l7) RIBZ(18) RIBZ(19) RIBZ(20) RIBZ(21) RIBZ(22) RIBZ(23) RIBZ(24) RIBZ(25) RIBZf26)

= 0 = 0 = 7 = 12.5 = 10.5 = 7 = 0 = 8.4 = 5 = 2 = 2 = 2 = 2 = 2 = 7.2 = 0.5 = 12 = 6.5 = 6.5 = 6.5 = 6.5 = 6.5 = 1.5 = 0 = 0 = 0

connect L, G rect. abd. ext. obl. 1 ext. obl. 2 int. obl. 1 int. obl. 2 iliocost. lumb. long. thoracis long. thoracis Ls long. thoracis L4 long. thoracis L3 long. thoracis L2 long. thoracis L, quad. lumb. multifidus (i) lat. dorsi. node LDI node LD2 node LD3 node LD4 node LD (sacrum) node LTl, 2,3,4,5 ant. diaphragm mid. diaphragm post. diaphragm

!!fk da,

a2u, 90 -= c 1K.~~+CF,+Kdr,~-r3]~ &tiaaj m= J 1

’ ’ (B4)

Since the partial derivatives are evaluated at the unperturbed point of equilibrium, I,, -l,, = 0 and the Equations (B4) reduce to the following:

au, 90Fdl,, -= c maa, aa, Ill=1

av au, au, aw -=-.-.+---

ami au, au, au, a2v azu, --~ a9, a2w ~=aaiaa,+acliarj aaiaaj

L,Z(l) L,Z(2) L,Z(3) L,Z(4) L,Zf5) L,Zf6) L,Z(7) L,Z(8) L,Zf9) L,Z(lO) L,Z(ll)

-RIBCAGE

v=u,+u,-w Partial derivatives of the potential V were calculated separately for each component taking the Euler angles ai (3 rotation angles X 6 joints = 18 df) as the generalized coordinates:

= 32.2 = 35.5 = 32.8 = 31 = 33 = 30.2 = 31 = 32.3 = 31 = 31 = 31

032) If the muscle length is represented with a sum of n sections (when the muscle passes through the nodal point), its potential energy derivatives consist of a sum of its sections with some additional terms. Thus, if l,, = I,,, + I,,, + + 1omnand l,, = I,,, + l,,, + + lpmnthen

The energy stored in linear springs (U,) can be expressed as follows:

(B3) where F, = instantaneous muscle force (N), K, = instantaneous muscle stiffness (N/m), I,,,,,, Ir,,,, = original (‘frozen’ in a given frame) and perturbed muscle lengths (4 and

W)

14

Ctin. Biomecb.

Vol. 11, No. 1, 1996

Since tbc length of a given muscle I, (dropping the muscle subscript ‘m’ at this pointJ is given by the vector sum of the length components in the X, Y and Z axes direction,

(B7) then

~=(l;,+l;,“+l;,)-l’z (

l,,~+l,,~+lp,~ / 1

w9

.I

L

where h is a rotation matrix, L is the vector of vertebral segment lengths taken between the adjacent joints, X, Y, 2 are coordinates of the muscle attachment points in the reference posture, OX, OY, 02 are coordinates of the rotation centre (a joint) of a given segment. Partial derivatives of the elements of rotation matrices were easily programmed on a computer by inserting the appropriate derivatives of the trigonometric functions. To obtain the elastic energy, which is stored in all of the torsional springs, we need to integrate the Equation (1) with respect to the relative joint angles and sum it over the 6 joints:

UTx=

i j=O

Mxjl(~j-t$j+l)=,$o~

[ebw(h--cPl+‘)

s

Xl

and

UT,=

a21 l= auiaaj

-(l;x+l;,+l;r)-3’2

I

(

i‘,lp*G+‘.i dz ai px--.E+’ (\lhj

a1 1,,-g+l,~+l,,~

aat

I

byj

F

s

[eb’j@+J+” 2.1

-bzAej-ej+ 111

(B9)

W3)

The first partial derivatives of UT will have two terms belonging adjacent intervertebral joints:

to the two

= axj[ebd@f - 43+ l)- 11 + K($lj - ej+ 1)

Wj

(B6). (B7) and (B8) into (B4) yields

-~,(j-l)[&~-

I)(~~,I-~~)-l]-K(~j_l--j)

al

au, ~=F,(l;,+1;y+l;,)1~2

lb,

f?? [&(*J-$l+l)

III+ K(4j-4j+ IXll/j-+j+ 1)

M,d(B,--8,+,)=,jo j=O

-au, Substituting

i j=o

s

-hyj($j-$j+

I

lpg~+lpz~ +(l;,+l;,+1;,)-“2 %i ‘j ) -a21,, a1 al,, px aaiaaj+xtf z

a21 PY+“‘P azPz b’ a21,, -I-‘PY iJolidcljal.j ihi "aaiaclj>

M,j~(~j-~j+l)=

i j=O

lpx,+l,~+lpz~ i

I

I

) ‘(B10) au, __=aZj[&(‘V4+1)-

aej

11+a,(j-l,[eb~-‘)(e~~~-e,)-

and

11 0314)

For the negative angles, coefficients ‘a’ and ‘b’ will appear with a minus sign and the appropriate constants will be inserted in the case of flexion. Now, there are six second partial derivatives of the UT possible for the general case:

azu, = -~,,j-l,~,,j- l)eb~-~‘)(+-+J) Gja+j- I

--= au,

a$jWj+ 1 It remains to evaluate partial derivatives of muscle length components I,,, I,,, I, in relation to all 18 rotation angles 01,. If the muscle originates on a skeletal segment ‘w’ and inserts onto the segment ‘a’ (Figure 3). then its length vector

u, w=O,...6,

w>u

0312)

a2u,

- a ,b &,W-4,+4

Xl x.l

aw,

K

a~ja+j-l=a~ja*j+,=-

An identical equation format results if the UT formulation of twist is differentiated twice. Flexion/extension has the same general format as (B15), except K = 0 in this case. The external work W performed by the load P is a dot product of the force and displacement vectors:

Cholewicki

and McGill:

W= + *A%= P,(h,, - h,,) + Py(hpy- h,,) + ~,(h,, - h,,)

where h, and h, are the perturbed Thus,

azw -F azh,, -5&+F,g&+Fz~ auiauj

Mechanical

stability

of the in vivo lumbar

spine

15

Since the load P is always applied to the ribcage,

and the original points of force application.

W7)

The derivatives of the rotation matrix [IL] are the same in Equation (B12). Because the global axes system is imbedded into the pelvis, the last term in Equation (B18) vanishes upon the differentiation. Once calculated, all partial derivatives were inserted into the Hessian matrix in Equation (2).