[ieee 2013 20th iranian conference on biomedical engineering (icbme) - tehran, iran...

Effect of considering stability requirements on

antagonistic muscle activities using a musculoskeletal

model of the human lumbar spine

Majid Hajihoseinali

Department of Mechanical Engineering

Sharif University of Technology

Tehran, Iran

[email protected]

Hasan Nickpour



Tehran, Iran

[email protected]

Navid Arjmand



Tehran, Iran

[email protected]

Farzam Farahmand



Tehran, Iran

[email protected]

Abstract— The recruitment pattern of trunk muscles is determined

using a three-dimensional model of the spine with two joints and

six symmetric pairs of muscles in which both equilibrium and

stability requirements are satisfied. Model predictions are verified

using Anybody Modeling System (AMS) and Abaqus. The model

is used to test the hypothesis that antagonistic muscle activities are

necessary for the spinal stability. The model with stability

constraints predicts muscle activities greater than those predicted

without stability consideration. In agreement with experimental

data, the stability-based model predicts antagonistic muscle

activities. It is shown that spinal stability increases with trunk

flexion and further improves at heavier tasks.

Keywords-Lumbar spine; Musculoskeletal modeling; Muscle

recruitment; stability

I. INTRODUCTION

Low back disorders (LBDs) are very prevalent and costly. More than 85% of adults experience low back pains at least once in their lifetime [1]. Epidemiological studies indicate that 52% of work-related disabilities is due to LBDs [2].

Biomechanical factors such as repetitive bending and twisting, routine tasks and vibrations increase the risk of LBDs in workplaces [3]. Excessive mechanical loads acting on the human spine as well as mechanical instability of the spine during daily/occupational/physical activities, amongst others, are known as important causes of LBDs [4, 5]. While there exists no direct method to measure spinal loads in vivo [3] some indirect approaches including intradiscal pressure measurement [6] to estimate disc compressive force and electromyography (EMG) [7] to predict muscle forces are proposed. Due to invasiveness nature of these in vivo means for quantification of spinal loads

and stability, biomechanical models have emerged as invaluable tools for this purpose.

A wide range of biomechanical models are used to simulate spine conditions in different postures and movements, from simple models with one degree of freedom and two muscles [8,9] to much more detailed models including finite element models with tens of muscle fascicles, ligaments, tendons and nonlinear properties [10-14]. Recent complex models however face the redundancy problem as the number of unknown variables (forces in muscles that span a given joint) exceeds those of the equilibrium equations (or number of degrees of freedom). To achieve a unique solution usually an optimization method is employed in which an arbitrary cost function is optimized while equilibrium equations are resolved [11, 13, 15, and 16]. Optimization approaches, however, fail to predict antagonistic muscle co-activities while in vivo studies show that these muscles are active during various tasks [10, 17, and 18]. Granata and Wilson [17] showed that with increasing the height of hand load while preserving its horizontal distance from body, antagonistic muscle activities increases showing that these activities are related to spinal stability.

In this study, a musculoskeletal model of the human spine is used to predict both agonistic and antagonistic muscle forces by introducing stability requirements (besides equilibrium ones) into the optimization procedure. It is hypothesized that considering stability into the optimization procedure enables the model to predict antagonistic activities.

II. METHOD

A. Musculuskeletal spine model

The three dimensional model of the spine similar to that of Granata and Wilson (2001) [17] with two spherical joints having

Proceedings of 20th Iranian Conference on Biomedical Engineering (ICBME 2013), University of Tehran, Tehran, Iran,December 18-20, 2013

978-1-4799-3232-0/13/$31.00 ©2013 IEEE 260

total of six rotational degrees of freedom is developed (Fig. 1). The lower segment, attached to pelvis, represent lower lumbar spine (L3-S1) and the upper one represents the upper lumbar spine (L1-L3).

B. Muscle modeling

Six pairs of trunk muscles (three back Erector Spinae fascicles (ES1, ES2, and ES3), Rectus Abdominis (RA), External Oblique (EO) and Internal Oblique (IO)) are modeled. Origins, insertion and cross-sectional area of the muscles are derived from literature [19, 20] (Table I). Trunk mass is represented by a concentrated weight of 420 N applied at 5 cm anterior to the superior surface of upper segment [17].

TABLE I. ANATOMY OF CONSIDERED MUSCLES INTO THE MODEL

Muscle PCSA (cm2)

Lower Insertion (cm) Upper Insertion (cm)

x y z segment x y z segment

ES1 13 4.5 -6 -4.2 Pelvis 4.5 -6 6.3 Lower

ES2 13 4.5 -6 6.3 Lower 4.5 -6 6.3 Upper

ES3 13 4.5 -6 -4.2 Pelvis 4.5 -6 6.3 Upper

RA 9 4 8 -3.1 Pelvis 4.5 13 10.5 Upper

EO 10.6 6 10 -3.1 Pelvis 13 6 10.5 Upper

IO 10.3 6.5 -6 0 Pelvis 10 2 10.5 Upper

Figure 1. Schematic of the modeled mscles and segments in Sagittal plane.

C. Governing equations

1) Equilibrium Muscle forces should be determined so that they balance

external forces and moments according to the mechanical equilibrium requirements:

∑ 𝑟𝑖𝑚⃗⃗ ⃗⃗ ⃗ × 𝐹𝑗

𝑚12𝑚=1 = 𝑀𝑒𝑥𝑡𝑖𝑗

𝑖 = 1,2 𝑗 = 1,2,3 (1)

In which 𝑖 and 𝑗 show joint and coordinate axis number respectively, 𝑟𝑖

𝑚 is the moment arm of 𝑚th muscle around 𝑖th

joint and 𝑀𝑒𝑥𝑡𝑖𝑗 is the total external moment around 𝑖th joint in

𝑗 direction. 𝐹𝑗𝑚 is 𝑗th componet of unknown force of muscle 𝑚.

2) Stability From a mechanical standpoint, a system is stable if its second

derivatives of total potential energy with respect to generalized coordinates (degrees of freedom) is positive. For an elastic structure, the potential energy 𝐸 is the sum of the internal strain energy 𝑈𝐿 minus the work 𝑈𝑊 of the external applied forces:

𝐸 = 𝑈𝐿 − 𝑈𝑊 (2)

Therefore, the second derivation can be expressed as follow:

∂2𝐸

𝜕𝜃𝑖𝑗 𝜕𝜃𝑘𝑛

= ∂2𝑈𝐿


− ∂2𝑈𝑊


(3)

in which 𝜃𝑖𝑗

is the rotation of 𝑖 th joint around 𝑗 th axis. The

Hessian matrix is formed by the second derivations of potential energy. If this matrix is positive definite, the system is stable [18]. The Hessian matrix of the model is given in (4):

𝐻(𝐸) =

[

𝜕2𝐸

𝜕2𝜃11

𝜕2𝐸

𝜕𝜃11 𝜕𝜃12

𝜕2𝐸

𝜕𝜃12 𝜕𝜃11

𝜕2𝐸

𝜕2𝜃12

… …

𝜕2𝐸

𝜕𝜃11 𝜕𝜃23

𝜕2𝐸

𝜕𝜃12 𝜕𝜃23

.

.

.

.

.

.

..

.

.

.

.

𝜕2𝐸

𝜕𝜃23 𝜕𝜃11

𝜕2𝐸

𝜕𝜃23 𝜕𝜃12…

𝜕2𝐸

𝜕2𝜃23 ]

(4)

Total strain energy in muscles is:

𝑈𝐿 = ∑ 𝐹𝑚(𝑙𝑝𝑚 − 𝑙0𝑚) +1

2𝑘𝑚(𝑙𝑝𝑚 − 𝑙0𝑚)

212𝑚=1 (5)

in which 𝐹𝑚 is the internal force of 𝑚th muscle. 𝑙𝑝𝑚 and 𝑙0𝑚 are

perturbed length and initial length of 𝑚th muscle, respectively. Muscle stiffness 𝑘𝑚 depends on length and force of the muscle, as follow:

𝑘 = 𝑞𝐹

𝐿 (6)

Coefficient 𝑞 is considered equal to 10 [19]. Now, the second derivative of strain energy can be expressed as follow:

𝜕2𝑈𝐿

𝜕𝜃𝑖𝑗 𝜕𝜃

𝑘𝑛

= ∑ 𝑘𝑚

𝜕𝑙𝑝𝑚

𝜕𝜃𝑖𝑗

.𝜕𝑙𝑝𝑚

𝜕𝜃𝑘𝑛

12

𝑚=1+ [𝐹𝑚

+ 𝑘𝑚(𝑙𝑝𝑚 − 𝑙0𝑚)]𝜕2𝑙𝑝𝑚

𝜕𝜃𝑖𝑗 𝜕𝜃

𝑘𝑛

(7)

Since the derivatives are evaluated at the unperturbed point, that

is (𝑙𝑝𝑚 − 𝑙0𝑚) = 0, the above equation reduces to:

∂2𝑈𝐿


= ∑ 𝑘𝑚𝜕𝑙𝑝𝑚

𝜕𝜃𝑖𝑗

.𝜕𝑙𝑝𝑚

𝜕𝜃𝑘𝑛

12𝑚=1 + 𝐹𝑚

𝜕2𝑙𝑝𝑚


(8)

The external work is the dot product of the force 𝑃 and the displacement ∆𝑊.

𝑈𝑊 = ∑ �⃗� 𝑘. (�⃗⃗⃗� 𝑝𝑘 − �⃗⃗⃗�

0𝑘)𝑘 (9)


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in which �⃗⃗⃗� 0𝑘 and �⃗⃗⃗�

𝑝𝑘 are the original and perturbed points of

force application. Therefore, the second derivatives of external forces are:

𝜕2𝑈𝑊


= ∑ ∑ 𝑃𝑘𝑗

𝑊𝑝𝑘𝑗


3𝑗=1𝑘 (10)

in which 𝑃𝑘𝑗 and 𝑊𝑝𝑘𝑗

are the 𝑗th component of �⃗� 𝑘 and �⃗⃗⃗� 𝑝𝑘 in

reference coordinates, respectively. Determinant of the Hessian matrix can be considered as an indicator of stability (SI).

D. Optimization

Although mechanical equilibrium provides three equations of moments at each joint and therefore six equations in total, unknown muscle forces are 12. To resolve the redundancy problem, an optimization method whose cost function is sum of cubed muscle stresses is used. Minimizing this cost function while solving equilibrium equations provides an appropriate mean to predict muscle recruitments and yields a unique solution [15]. Furthermore, the Hessian matrix has six eigenvalues representing stability in joints in the direction of corresponding eigenvector. All eigenvalues should be positive in order for the system to be stable. Therefore, the optimization problem is as follows:

minimize∑ (𝐹𝑚

𝐴𝑚)

312𝑚=1 (11)

𝑆𝑢𝑏𝑗𝑒𝑐𝑡 𝑡𝑜:

{

∑ 𝑟𝑖𝑚⃗⃗⃗⃗ ⃗ × 𝐹𝑗

𝑚12𝑀=1 = 𝑀𝑒𝑥𝑡𝑖𝑗

𝑖 = 1,2 𝑎𝑛𝑑 𝑗 = 1,2,3

𝑒𝑖𝑔(𝐻𝑒𝑠𝑠𝑖𝑎𝑛) > 00 ≤ 𝐹𝑚 ≤ 𝜎𝐴𝑚 𝑚 = 1,2, … ,12

in which, 𝑚 is the muscle number. The objective is to find muscle forces 𝐹𝑚. First constraint grantees the equilibrium of the system; the second one satisfies stability; and the third one indicates that muscle forces cannot be compressive and cannot exceed a maximum limit. Maximum muscle forces are considered to be proportional to cross-sectional area and

maximum stress 𝜎 which is considered to be 60𝑁

𝑐𝑚2 for all

muscles [21]. The optimization problem is solved using MATLAB (The MathWorks, Natick, MA, USA) Optimization

ToolboxTM (fmincon command).

E. Verification

In this study, one task in standing posture and two tasks in flexed postures are studied. The tasks are simulated in two modes: with considering the stability criteria and without it. Anybody Modeling System (AMS) and Abaqus are used to evaluate and verify the model predictions in these modes, respectively.

In order to verify the equilibrium equations, the model was reconstructed in AMS (Fig. 2). The cost function of ‘sum of cubed muscle activities’ was used to estimate muscle forces. This software does not consider stability constraints while estimating muscle forces. Moreover, to verify the stability equations, the model is built in the finite element Abaqus software while considering muscles as uniaxial spring elements with stiffness proportional to muscle force according to (6).

To verify the stability equations of the model using Abaqus, combination of static and perturbation analyses are used. Static external load of 420 N is first applied under the calculated muscle (spring) forces and subsequently the model is perturbed using a unit forces of 1 N exerted at the point of application of the external load. A large deflection in perturbation analysis indicates that the external load is equal to the critical load of the system. The iterations for finding critical load is done utilizing Matlab and Python programming.

Figure 2. AnyBody model: left: frontal view; right: lateral view

III. RESULTS

Predicted force in the muscles by the model (with and without stability constraints) and by the AMS software for three tasks including relaxed upright standing and two flexed postures are compared (Table II). The determinant of Hessian matrix is represented as an index of stability; the greater the determinant is, the more stable the spine in that posture is.

TABLE II. FORCE IN MUSCLESA FOR THREE DIFFERENT TASKS

Muscle

Standing 20o Flexion 30o Flexion

EQ b AMS SB c EQ b AMS SB c EQ b AMS SB c

ES1 40 36 143 142 153 169 181 173 214

ES2 62 59 170 118 92 176 134 145 151

ES3 102 121 72 164 169 143 297 288 243

RA 0 0 0 0 0 0 0 0 0

EO 0 0 74 0 0 0 0 0 0

IO 35 32 78 64 76 93 128 104 144

SId

(J/rad2) -7.5e+3 - 5.0e+3 -1.8e+3 - 4.4e+4 -5.7e+2 - 3.8e+4

a. All forces are in N

b. Considering only equilibrium

c. Considering both equilibrium and stability

d. Stability Index

Figures 4, 5 and 6 represent results of Abaqus simulation for the three tasks with a unit perturbation force. Vertical and horizontal axes represent the external load (trunk weight) and its displacement under the unit perturbed force, respectively. Clearly, the displacement is very large under the trunk weight (420 N) indicating that muscle forces resulted from the main


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model provide marginal stability to the spine. Therefore, the model performed well in calculating muscle forces so that the system is minimally stable. The Abaqus analyses therefore confirms that the stability equations considered in the musculoskeletal model is accurate (model verification).

Figure 3. Displacement of point of application of external force (trunk

weight) as a result of unit perturbation in standing posture


weight) as a result of unit perturbation in 20o flexion.


weight) as a result of unit perturbation in 30o flexion.

IV. DISCUSSION

In this study, a musculoskeletal model of the spine was developed to evaluate the effect of considering stability requirements on predicting antagonistic muscle activities. Although this model provides only six degrees of freedom, it can model local (between pelvis and lumbar spine) and global (between pelvis and thorax) muscles. Moreover, the two spherical joints allow controlling lumbar lordosis and trunk flexion as well as offering asymmetrical postures and local and global buckling.

Comparison of the critical load with results of Abaqus and of muscle forces with those of AMS software verify the model’s predictions. The differences between muscle forces of the EQ and AMS models are likely due to the failure of the AMS in finding the global minimum of the optimization cost function. In most models, optimization is used to achieve muscle recruitment patterns using only equilibrium criteria. However, the results does not grantee a stable system. Considering stability requirement increases the muscle forces so much as the system is marginally stable. In all simulated tasks, muscle forces resulted from only equilibrium consideration were not able to provide spinal stability (negative SI as given in Table II).

Considering stability not only increases muscle activities to stiffen the spine but also changes the muscle recruitment pattern in a way that it activates the antagonist muscles. As it is shown in Table II, by adding stability constraint, external oblique becomes active although it has no role in counterbalancing external loads as an antagonistic muscles. Experimental results also show that this muscle is active in standing posture [17]. Therefore, it can be inferred that adding stability constraint enables the model to predict antagonistic muscle activities in accordance with in vivo data.

Muscle forces increase with increasing flexion angle (Table II) because center of mass of the body moves away from the origin and spinal moment increases. Consequently, spinal stability increases which is shown by stability index (SI) in Table II. Former studies also showed that in heavier tasks, the


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spine is more stable [18]. Therefore, in heavy tasks, the risk of low back injuries is mainly due to greater forces on the discs. On the contrary, in light tasks, although the forces are small, there is a noticeable risk of injuries because of spinal instability [22].

The present model, similar to its counterparts, has some simplifications. Although both equilibrium and stability equations were verified using Abaqus and AMS, validation needs experimental tests of EMG and intradiscal pressure. In this study, parameter 𝑞, which has a direct influence on spinal loads and stability, is considered to be 10 based on earlier studies [19]. However, its values ranges from 5 to 100 in different studies [24] and there is no physiological basis to determine this parameter. Moreover, determinant of the Hessian matrix is considered as the index of overall stability but other indicators such as smallest eigenvalue or average eigenvalue are also suggested [25].

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