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Towards a model for force predictions in the human shoulder
J. Biondwnics
Vol. 25, No. 2, pp. 189-199.
1992.
OoZl-9290/92
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C: 1991 Pergamon Press
Printed in Great Britain
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TOWARDS A MODEL FOR FORCE PREDICTIONS IN THE HUMAN SHOULDER DAN KARLSSONand Bo PETERSON Center for Biomechanics, Chalmers University of Technology, S-412 96 GBteborg, Sweden
Abstrati-In
this paper the concept of a thme~imensional biomechanical model of the human shoulder is introduced. This model is used to analyze static. load sharing between the muscles, the bones and the ligaments. The model consists of all shoulder structures, which means that different positions and different load situations may be analyzed using the same model. Solutions can be found for the complete range of shoulder motion. However, this article focuses only on elevation in the scapular plane and on forces in structures attached to the humerus. The intention is to expand the model in future studies to also involve the forces acting on the other shoulder bones: the scapula and the clavicle. The musculoskeletal forces in the shoulder complex are predicted utilizing the opti~~tion technique with the sum of squared muscle stresses as an objective function. Numerical results predict that among the muscles crossing the ~enoh~eral joint parts of the deltoideus, the infraspinatus, the supraspinatus, the subscapularis, the pectoralis major, the coracobrachialis and the biceps are the muscles most activated during this sort of abduction. Muscle-force levels reached values of 150 N when the hand load was 1 kg. The results from the moderseem ta be qualitatively accurate, but it is concluded that in the future development of the model the direction of the contact force in the glenohumeral joint must be constrained.
number of shoulder components internal force (N) internal force vector (N) cross-sectional area for muscle i (mm”) power used in the objective function external force (N) external force vector (N) position vector of attachment point of force (m) x-component of external force vector Pi (N) x-component of the vector ri x Pi (N m) direction coefficient matrix for force and torque minimum value of the force F, (N) maximum value of the force F, (N) constant for maximum muscle stresses (N mm-2)
Shoulder pains of muscular many working
situations.
origin are common in High muscle forces seem to
be an important etiological factor (e.g. Herberts et al., 1984). The shoulder is a complex system from a biomechanical point of view; it consists of three bones and more than 20 muscles. Due to the fact that the shoulder is an indeterminate mechanical system, it is hard to predict individual internal forces. The number of unknown forces far exceeds the number of force and moment equilibrium equations available for the three shoulder bones: the clavicula, the scapula and the humerus. To date, no biomechanical shoulder model exists that is capable of predicting all of the musculoskeletal forces, and no model exhibits the flexibility needed for the analysis of different positions
Received i~~na~~rrn
27 December 1990.
and load situations. For ergonomic planning of work sites both in prevention and rehabilitation, such predictions could prove useful. Attempts have been made earlier to construct a biomechanical model of the shoulder (e.g. DeLuca and Forrest, 1973; Ringelberg, 1985; Dul, 1989). However, these models have often been restricted to single-motion patterns and have not included all the shoulder muscles, just the ones expected to be the most active components. We believe that this simplification is unacceptable when analyzing more complex load situations. The role of individual components in the shoulder has very often been studied using electromyography (EMG) (e.g. Herberts et al., 1980; Sigholm et al., 1984: JSirvholm et al., 1989; Kronberg et al., 1990). Jiirvholm et al. (1989) also studied the intramuscular pressure (IMP) in the supraspinatus muscle. Although these methods give some appreciation of muscle effort, the relationship between EMG and muscle force is not clear (Kadefors, 1978), especially when studying situations where the muscle length varies. It is also hard, if not impossible, to measure the EMG in all shoulder muscles or muscle parts at the same time. Nor is the relationship clear between the IMP and the muscle force. Another way of investigating the role of certain shoulder muscles is to block their nerves by local anesthesia. This was done by Colachis et al. (1969) and Howe1 et al. (1986) with the axillaris nerve. The latter group also studied the effects of blocking the suprascapularis nerve. Results show that when the arm is elevated, the torque-producing role is shared equally between the deltoideus and the supraspinatus muscles, Still, it is not possible to obtain information about all of the shoulder components by using EMG, IMP, or by studying the effects of local anesthesia (the serratus anterior muscle, for example, is anatomic~ly
189
190
D.
KARLSSONand
hard to reach). To obtain a complete picture of the role of the different components, it is necessary to use some kind of a biomechanical model. In general, in the human musculoskeletal system, there are more muscles crossing a joint than are strictly necessary, from a mechanical point of view, for performing most tasks. The muscles are activated simultaneously during most movements (synergistic muscle action). So far, the most efficient way to solve these indeterminate mechanical problems has been to formulate an objective function and use optimization techniques. This approach is based on the assumption that during learned activities the body selects a unique way of distributing the forces, and that this distribution is in some way governed by certain physiological criteria. The criterion often suggested is, for example, that synergistic muscles share the load proportional to their size or capacity, or that the total ‘effort’ to perform a specific task is minimized. It is often said that the capacity of a muscle is proportional to its cross-sectional area (A,), but it has also been suggested that the fiber-type composition must be taken into consideration (Dul et al., 1984). Objective functions in the optimization problem are often the sum of muscle stresses, or the sum of squared or cubed muscle stresses. In the mathematical formulation, finding a solution means the minimization of c (Fi/Ai)‘, where the power p is given the value 1,2 or 3 [a survey of this subject is given, for example, by Dul et al. (1984)]. To find a solution to our indeterminate problem, we use the objective function C(Fi/Ai)‘, but other ‘logical’ objective functions can, of course, easily be used as well. The authors have tried objective functions with powers of 1.5, 3 and 4, and these results do not seem to differ very much. However, this is beyond the scope of this paper. The purpose of this study is to test whether a three-dimensional shoulder model can be used to predict all the internal musculoskeletal forces acting on the humerus in different load situations, in
B. PETERSON
the case of a static external load in a static working position. Force predictions in this study are restricted to the shoulder structures attached to the humerus. Our biomechanical description of the shoulder is based on the works of Hiigfors et al. (1987,1988) and the relative capacities of the shoulder muscles (i.e. the normalized cross-sectional areas) which are examined in a study made by Karlsson and Jarvholm (1988), where they presented data on the cross-sectional areas of the shoulder muscles according to the method of Brand et al. (1981). The numerical results are compared with EMG studies (Jarvholm et al., 1989) and the results from Poppen and Walker (1978). These comparisons, however, show that there are large differences between some predictions from the model and the other studies mentioned above. It is concluded that the model shows qualitatively good results. However, in future developments the direction of the glenohumeral contact force must be taken into consideration and a constraint for this must be incorporated in the model. It might be interesting to validate the model with, for example, more EMG studies. It is also desirable to obtain a better understanding of the physiological criteria of muscle-force distributions and their relationship to objective functions. Further examination of the human shoulder rhythm is also interesting in order to get more reliable quantitative data.
THE BIOMECHANICAL MODEL
Biomechanical analysis of the shoulder
Hiigfors et al. (1987) described an idealized mechanical model of the shoulder. In this model the shoulder bones are considered as rigid bodies. The shoulder joints are modelled as ball joints. Muscles or muscle sections are described as stretched strings following the shortest path between attachment points,
Fig. 1. The shoulder muscles modelled as ‘strings’(Hiigfors et al., 1987).
A model for force predictions in the human shoulder
possibly constrained by other structures, i.e. lines or surfaces (Fig. 1). Internal forces actin on the rigid bodies are divided into muscle forces a f ting along the strings representing the muscles, joint contact forces and ligament forces. To get the complete geometry of the shoulder, we have to know the relationship between the position and direction in space of the humerus and the scapula, and the same between the humerus and the clavicle. This interplay between the motions of the constituent parts of the human shoulder is known as a ‘shoulder rhythm’. For example, Pronk (1989) described this subject, and quantified results have been presented by Inman et al. (19441,and by Poppen and Walker (1976). Here, we have designed this ‘shoulder rhythm’ by extrapolation of the
191
and CriXFi=-CriXPi.
(4)
The external loads Piconsidered here are all gravitational forces (it is an easy matter to extend these Pi to more general loads or even introduce force couples). Using a fixed reference system where the zaxis is vertically oriented, each force Pi takes the form: 0 0
Pi= I
piz
.
I
Equations (3) and (4) can now be used to formulate the six equilibrium equations written together in matrix form: 0
=
_dPi*
-CMix -CMiy
(6)
0 results given by Hiigfors et al. (1988), providing quantitative data for the rhythm over a limited range of movement for the arm (see Appendix). The positions of the bone-fixed coordinate systems are given by three Euler angles for each bone (see Appendix for definition). The external forces acting on the arm are the weight of the upper arm, the weight of the lower arm, the weight of the hand-held load. The shoulder structures capable of carrying an internal load give us 46 forces which are listed in Table 1. In this study, though, we are considering only the forces acting on the humerus and the arm, which are the first 23 forces of Table 1. If we now look at this humerus-arm system we may use vector expressions and let Pistand for external forces, Fi for internal forces and ri for the position vector of the attachment points of the forces. This gives the following force and moment equilibrium equations for the humerus {including the rest of the arm):
~Fi+~PI=O
(1)
and
These equations must be satisfied for each static position, but since there are only six equations [equations (1) and (2)] with 23 unknown internal forces, no unique solution exists. Optimization technique The equilibrium equations (1) and (2) can be written in the form:
xFi=--xPi
or
CalCFI=C-PI,
(7)
where al1 the elements in the matrix [a] are known. [These quantities can be computed in a straightforward manner using the modelling of muscles, ligaments and contact forces from the work of Hiigfors et al. (1989)]. This equation system has an infinite number of solutions. To find a unique solution among these one must add some criterion, and in our case we use the sum of squared muscle stresses (as mentioned earlier). The optimization problem can now be summarized as follows: minimize
C(Fi/Aij2
(8)
subject to
(1) CalEFl=E--PI
(9)
FiminG Fi G Fimax3
(10)
(II)
The first constraint is the equilibrium equations given by (I) and the second is the lower and upper limit for each internal force (II). The vector [Ff in equation (9) contains the listed internal forces to be predicted For the muscles we have the lower limits Fimin=O and the upper limits Finax= kAi, where k is a constant which depends on the maximum tension in the muscles. For the contact forces we did not apply constraint (II), as we assumed that the pressure between the skeletal bones could, in reality, become much higher than the muscle tensions. The ligament force involved in this study had its lower limit equal to zero and had no upper limit. The value of the constant k has been suggested to be in the range 0.4-1.0
192
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KARLSSON
and B. PETERSON
Table 1. Internal forces in structures attached to the humerus Number
Normahzed crosssectional area*
Shoulder component
(Hagfors et al., 1987) Com~nenrs ~ttac~d to the ~u~rus Latissimus dorsi 1 (superior part) 1 Latissimus dorsi 2 (inferior part) 2 Pectoralis major (sternum part) 3 Pectoralis major (clavicle part) 4 Deltoideus anterior 5 Deltoideus medialis 6 Deltoideus posterior 7 Coracobrachialis 8 Infraspinatus 1 (superior part) 9 Infraspinatus 2 (inferior part) 10 Subscapuiaris 1 (superior part) 11 Subscapularis 2 (middle part) 12 Subscapularis 3 (inferior part) 13 Supraspinatus 14 Teres major 15 Teres minor 16 Biceps brachii (long head) 17 Biceps brachii (short head) 18 Triceps (part attached to the scapula) XE components of the contact force? 21 Yz- 1 22 23
2.61 3.15 4.50 2.75 4.29 5.63 5.21 1.91 3.87 4.05 3.13 2.82 3.87 3.83 4.48 1.89 2.14 1.89 -8.87
Force in the coracohumeral ligament
The rest of the shoulder components (not investigated in this paper}
24 25 26 ;;: ;;: 31 32 33 34 35 36 :; 39 40 41 42 43 44 45 46
Trapezius 4 Trapezius 3 Trapezius 1 Trapezius 2 Levator scapulae Rhomboideus major Rhomboideus minor Pectoralis minor Serratus anterior 1 Serratus anterior 2 Serratus anterior 3 Omohyoideus Sternocleidomastoideus Sternohyoideus Subclavius XY- force components between the scapula and the calvicle Z- 1 51:
force components between the clavicle and the sternum 1 kperior contact force between the scapula and the thorax Inferior contact force between the scapula and the thorax
2.87 4.01 2.76 2.34 1.91 2.31 1.14 2.08 3.02 2.55 4.15 -
-
*Karlsson and Jfrvholm (1988). The areas are normalized so that the values listed above are the percentages of the total sum of the cross-sectional areas of the shoulder muscles. TContact force in the glenohumeral joint between the scapula and the humerus.
MN rns2 (Cro~ins~ield and Brand, 1981). Wood et stated that an average maximum stress in the muscles is of the order lOOpsi (equivalent to 0.7 MN mV2). In our study we used k=0.7 MN rnm2, Such a stress value is never reached in any muscle in our study. The objective function [equation (S)] is used only to minimize the muscle forces so the forces (Fi) from the contact and the ligament forces are not included in this objective function. al. (1989)
To solve the op~~tion problem, we designed a computer program which is schematically described in Fig. 2. The computation of the muscle force directions and the search for the solution determined by the objective function were done by computer program subroutines from the IMSL- and NAG-subroutine libraries. The program is written so that constraints more complicated than those given by equation (lo), even nonlinear ones, can be handled. Non-
193
A model for force predictions in the human shoulder \ Input data: - bandload - arm and body position G specified antropometric data)
and muscle trajectories*
Computation of muscle force directions
Mechanical equilibrium equations
I
l**Karlsson & Jiirvholm W88)
Fig. 2. How the computer works to find a solution to the force distribution problem.
linear constraints can, for example, be used for contact forces where a direction-dependent force limit is desired). The plane of quasi-static motion is described in Fig. 3. Calculations were made for static positions with intervals of 15” and the elevation angle (0 in Fig. 3) varied between O-120”. The hand load was set to 1 kg. Anthropometric data for masses and mass centers of the arm segments were taken from the literature (Pheasant, 1986). These data correspond to a young male person. FORCE PREDICTIONS
z
abduction pl:
& coronal plane
FOR ELEVATION
In every position, numerical solutions of the force levels were found for the force directions of the components involved. The predicted muscle forces did not reach their upper force limits in any of the computations. In abduction, almost all muscle activity was confined to the following muscles acting around the glenohumeral joint: the deltoideus anterior, the deltoideus medialis, the infraspinatus, the supraspinatus, the clavicular part of the pectoralis major, the coracobrachialis and the biceps muscles. Forces in these components are shown in Fig. 4, a result which is in good agreement with other investigations (for example, Inman et al., 1944). In these muscles, the results showed force levels up to 150 N for 1 kg hand load [the deltoideus medialis Fig. 4(a)]. The rest of the muscles crossing the glenohumeral joint were predicted to have zero or very low forces. The contact force in the glenohumeral joint reached a value of 650 N in a 60” abduction position [Fig. 4(d)]. The direction of the contact force never diverged more
Fig. 3. The plane of elevation used in this study.
than 27” from the normal to the cavitus glenoidalis (except for 0” abduction). In Fig. 5 horizontal and dorsal views of the contact force in the joint are shown for elevation angles of 15, 45, 75 and 105”. The dashed line in each view is a reference line fixed in the scapula and it goes through the medial angle of the scapula and the center of the glenohumeral joint (a line which has been used by other investigators, for example, Kronberg, 1989). Note that one direction of this line is not the same as the normal direction in the middle of the glenoid
D. KARLSSON and B. PETERSON
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angle
(Cl 150
E B
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E Q E
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M0
1 0
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-
arbscap. 1 wbscap. 2 subscap.3
100
angle Fig. 4. Predictions of force levels in the shoulder components in abduction. Dorsal view
Top view
(a) Elevation angle: 15’
//
Angle difference*: 25?
1 i (
(b) Elevation angle: 45’ Angle difference? 26.9O
(cl
& M
1’ /’ I
/
/’
/’
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Elevation angle: 75’ Angle difference*: 10.1’ J
(d) Elevation angle: 105’ Angle difference*:
L&0/
*._
/ / / .-_&
* The difference between the direction of the joint force and the normal direction in the middle of the glenoid cavity.
Fig. 5. The direction of the predicted glenohumeral joint force in four abduction angles.
A model for force predictions in the human shoulder cavity. The angular difference between the latter direction and the direction of the predicted contact force is noted in Fig. 5 for each elevation. DISCUSSION From the results in the previous section it is evident that solutions to the indeterminate shoulder problem can be found using an optimization technique. The reliability of the predicted results is, of course, dependent on the approximations and assumptions that are made in order to obtain a solution. The essential approximations made in our model are the following: (a) The glenohumeral joint is considered a ball joint. This seems to be a good approximation for the relatively small external loads used here. (b) Our shoulder rhythm is a ‘handmade’ extension to the whole motion range of the shoulder, based on a relatively thorough, but not complete, investigation (Hiigfors et al., 1988). It is unlikely that minor defects in the rhythm would produce large errors in the forces predicted. However, an investigation of the differences in predictions, between different and slightly different shoulder rhythms, could prove valuable. Furthermore, extreme angles cannot be handled at present (i.e. the highly elevated arm, which may cause contact between the humerus and the acromion). (c) The muscles are treated as strings, which seems to be a good idealization when used for long and thin muscles. However, the wider a muscle, the more strings it has to be divided into. It is increasingly important to base this sort of idealization on a physical/physiological foundation as the number of divisions increase. Most investigators have separated the deltoideus muscle into an anterior, a middle and a posterior part, which seems to be a ‘natural’ subdivision. However to split up the trapezius muscle into four independent parts, as we have done in our study, is not a common practice, and the reliability of this subdivison must be considered. For example, we do not know if each of the four sections can be considered totally independent of the force conditions in the other parts. In fact, we have found no references which describe how muscle parts can be activated independently. (d) The muscle force distribution is such that the total sum of muscle tensions is ‘as low as possible’. Since this sum does not include ligament tensions, no attempt is made to take into consideration the force level in the coracohumeral ligament. Moreover, the ligament is considered totally passive; it does not have a muscle’s active ability to shorten its length. Here our model might be slightly improved by adding the influence of the ligament length and its viscoelastic properties in the calculations. (e) The capsular ligament between the humerus and the scapula is not represented in our model. Some of its action must then be considered incorporated in the contact force. However, this makes it more com-
195
plicated to interpret the actual contact force between the two bones involved. (f) The humerus and the lower arm are together considered as a rigid body. No equilibrium equations have been formulated for the elbow joint. This introduces errors because the biceps brachii muscles, as well as the triceps, are crossing this joint. In reality, they are involved when the lower arm is moved together with other muscles not included in our mode1 (for example, the brachioradialis muscle). To get a complete and reliable shoulder model the muscles crossing the elbow joint should be analyzed and incorporated into the model. This is also our intention, as a next step, in the future development of the model. (g) In the mode1 we study only the equilibrium equations for the humerus-arm system, together with the objective function, including forces for this system. This can possibly lead to solutions which are not feasible for the larger system including the clavicle, the scapula, the humerus and the lower arm. This drawback is, of course, shared by all simplified models not including the equilibrium equations for all bones. The influence of this simplification cannot be clarified until the larger system is studied. (h) No constraint on the direction of the glenohumeral contact force is incorporated into the model. Investigators have reported that an angle difference for the so-called glenoid tilt (Saha, 1971; Brewer et al., 1986) between healthy shoulders and those with a tendency for luxation could be only 1%20” (Kronberg, 1989). A shift in the direction of the glenohumeral force angle of almost 30”, which is shown in Fig. 5 for our predicted results, is not satisfactory. In the future development of the mode1 this result shows the need for a constraint on this angular direction. Our solution should be seen in the light of the above approximations as optimal, when using our objective function. It is not clear whether this optimality corresponds to a physiological criterion used by man. Most of the predictions of muscle forces in abduction seem to be ‘reasonable’, i.e. the muscles predicted to be activated are those named in the literature as being the main abductors in elevation. JHrvholm et al. (1989) reported EMG studies on the supraspinatus, the infraspinatus and the anterior deltoid muscle for the same sort of elevation as used in this paper. In Fig. 6 comparisons are shown between the predicted results of our model and these measurements. Although it is hard to see any significant relationship between these measurements and our predicted results, the EMG results show more ‘smooth curves whereas the curves for the predicted forces show more ‘go up and down’. Our predicted forces deviate considerably from the EMG measurements for abduction angles exceeding 90”. It is noted that in this range, extreme muscle shortening makes the EMG results unreliable. However, a predicted zero result from our model when there seems to be an EMG
D. KARLSSONand B. PETERSON
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‘(a)
-
Norm.
-
force [N]
EMG
-150
1.5 -
w 70 120 angle
l(b)
60
EMG
-L
Norm.
--t-
force [N]
_
force [N]
1;0
angle
Norm. EMG
150
60 angle
Fig. 6. Comparisons
for the infraspinatus, the supraspinatus and the anterior deltoid muscle between EMG (Jlrvholm et al, 1989) and the predicted muscle force.
activity in the same muscle may indicate some drawback in our model. For example, this seems to be the
case for the deltoideus medialis muscle in the elevation angle 45”. The zero activity in the subscapularis muscle for small elevation angles, and the same for the infraspinatus and supraspinatus muscles for large angles, shows the same difference with EMG. On the other hand, such muscles could be used for stabilizing the shoulder. This means that together they might not produce any larger torque around the glenohumeral joint even if their muscle forces show high values. Figure 4(d), showing the contact forces during elevation in the plane of the scapula, can be compared with results from other studies. Poppen and Walker
(1978) found the glenohumeral joint force (i.e. the resultant contact force) to be close to body weight in abduction. However, their results are based on EMG and are not fully explained in their paper. In our computer model we use anthropometric data corresponding to a young man with a mass of 75 kg. In Fig. 4(d) we obtain a contact force around 600 N in 60-90” elevation, or about 0.8 times the body weight, which is very close to the result obtained by Poppen and Walker (1978). The magnitude of our predicted contact force seems to be confirmed by these results. If, instead, we focus on the direction of this force which was shown in Fig. 5, we do not have such a satisfactory result. In the top views it can be seen
A model for force predictions in the human shoulder
how the contact force differs in direction compared to the reference line. One can also see a shift in direction beween the 45 and 75” elevation angles. Comparing this with Fig. 4(a) and (c)shows that for low angles the supraspinatus and infraspinatus show high force levels at the same elevation angles where their socalled antagonist, the subscapularis, shows no or not so large activity. The opposite is found for high elevation angles. Such observations lead to the conclusion that the direction of the glenohumeral contact force must be considered in the model. Hopefully, this will give more simultaneous activity in all muscles in the rotator cuff around the joint. Although some results from our model seem to be reasonable in comparison with earlier studies, most of the muscle force results are difficult to analyze and validate. The comparisons with EMG results are not thoroughly satisfying, as seen in Fig. 6. Other studies are restricted to certain positions or motion patterns (very often elevation in the scapular plane). In the future, we intend to incorporate also the other shoulder bones and muscles in our computer model. From these investigations we would, for example, get predictions for the forces in the trapezius muscle. This muscle is often discussed in ergonomics and comparisons could prove useful. Predictions of muscle forces from our model could also be validated by designing EMG experiments on muscles for which there are currently no activity data. This, however, would benefit from a better understanding of the refationship between the EMG signal and the force output of a muscle, in particular, when the muscle length changes. CONCLUSIONS l It is demonstrated in the present study that although the shoulder has a complex geometry and many components, a shoulder model can be devised which gives solutions for the forces in different load situations even when all muscles are taken into account. An optimization criterion based on minimizing the sum of squared muscle tensions is found to be applicable. l Some of these results seem ‘reasonable’ while others do not. However, there are several doubts when comparing with, for example, EMG results. a The direction of the contact force in the ~enohumera~ joint must be taken into account when designing a shoulder model.
Acknowledgements-We
highly appreciate the support and inspiration of Christian Hiigfors during our research. This study was supported by the Swedish Work Environment Fund.
REFERENCES Brand, P. W., Beach, R. B. and Thompson, D. E. (1981) Relative tension and potential excursion ofthe muscles in the forearm and hand. J. Hand Surg. 6,209-219.
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Brewer, B. J., Wubben, R. C. and Carrera, G. F. (1986) Excessive retroversion of the glenoid cavity. J. Bone Jt Surg. 68A, 124-131.
Colachis, S. C., Strohm, B. R. and Brechner, V. L. (1969) Effects of axillary nerve block on muscle force in the upper extremity. Arch. Phys. Med. Rehab. 50,645654. Crowninshield, R. D. and Brand, R. A. (1981) A phydologically based criterion of the muscle force prediction in locomotion. J. &mechanics 14, 793-801. DeLuca, C. J. and Forrest, W. J. (1973) Force analysis of individual muscles acting simultaneously on the shoulder joint during isometric abduction. J. Biomechanics 6, 385-393. Dul, 1. (1989) A biom~hanicai model to quantify shoulder load at the work place. C&n. B~o~chunjcs 3, 124-128. Dul, J., Townsend, M. A., Shiavi, R. and Johnson, G. E. (1984) Muscular synergism-II. A fatigue criterion for load sharing between synergestic muscles. J. Biomechanics 17, 663-673. Dul, J., Johnson, G. E., Shiavi, R. and Townsend, M. A. (1984) Muscular synergism-I. A criteria for load sharing between synergestic m&es. J. Bio~ckun~cs 17,675-684. Herberts. P.. Kadefors. R. and Broman. H. 09801 Arm DOSitioning in’ manual tasks-an electro&yo&apdic study of localized muscle fatigue. Ergonomics 23, 655-665. Herberts, P., Kadefors, R., HBgfors, C. and Sigholm, G. (1984) Shoulder pain and heavy manual labour. Clin. Ortkop. 191, 166178. Howell, S. M., Imobersteg, A. M., Seger, D. H. and Marone, P. J. (1986) Clarification of the role of the supraspinatus muscle in shoulder function. J. Bone ft Surg. @$A, 398-404. Hiigfors, C., Sigholm, G. and Herberts, P. (1987) Biomechanical model of the human shoulder--I. Elements. J. Biomechanics 20, 157-166. Hijgfors, C., Peterson, B., Sigholm, G. and Hetberts, P. (1988) Biom~hanic~ model of the human shoulder-II. The shoulder rhythm. Preprint 1988~1,Centre for Biomechanics CTH. Inman, V. T., Saunders, M. and Abbot, L. C. (1944) Observations on the function of the shoulder joint. J. Bone Jt Surg. 26, l-30. Jirvholm, U., Palmer&, G., Herberts, P., Hdgfors, C. and Kadefors, R. (1989) Intramuscular pressure and electromyo~aphy in the supraspinatus muscle at shoulder abduction. Clin. Ortkon. Rel. Res. 244. 45-52. Jiirvholm, U., Palmerud,‘G,, Karlsson, b., Herberts, P. and Kadefors, R. (1989) Intramuscular pressure and EMG in four shoulder muscles. J. orthop. Rex (submitted). Kadefors, R. (1978) Application of electromyography in ergonomics: new vistas. Stand. J. Rehab. Med. 10, 127-133. Karlsson, D. and JLrvholm, U. (1988) Force-producing ability in all shoulder muscles, as determined by cross-sectional areas. Preprint 1988:2, Centre for Biomechanics CTH. Kronberg, M. (1989) Shoulder joint stability-aspects on muscle function and skeletal anatomy. Ph.D. thesis, Department of Orthopaedic Surgery, Karolinska Hospital, Stockholm, Sweden. Poppen, N. K. and Walker, P. S. (1976) Norma1 and abnormal motion of the shoulder. J. B&e Jt Surg. 58A, 195-201. Popoen, N. K. and Walker. P. S. (19781 Forces at the &nohumeral joint in abduction,’ C& Orthop. 135, 165-170. Pheasant, S. (1986) Bodyspace: Anthropometry, Ergonomics and Design, pp. 121-134. Taylor & Francis, London. Petrofsky, J. C. and Lind, A. R. (1979) Isometric endurance in fast and slow muscle in cat. Am. J. Pkysiol. 236, C185-Cl91. Pronk, G. M. (1989) A cinematic model of the shoulder girdle: a r&urn&. J. Med. Engng Tecknol. 13, 119-123. Ringelberg, J. A. (1985) EMG and force production of some
D. KARLSSONand B. PETERSON
198
human shoulder muscles during isometric abduction. J. Biomechanics 18, 939-947.
Saha, A. K. (1971) Dynamic stability of the glenohumeral joint. Acta orthop. scand. 42, 491-505. Sigholm, G., Herberts, P., AImstrom, C. and Kadefors, R. (1984) Electromyographic analysis of shoulder muscle load. J. orthop. Res. 1, 379-386. Wood, J. E., Meek, S. G. and Jacobsen, S. C. (1989) Quantitation of human shoulder anatomy for prosthetic arm
control-I.
Surface
modelling.
J.
Biomechanics
APPENDIX
Five coordinate systems are necessary (and sufticient) for our description of the shoulder. One laboratory-fixed, one body- (thorax-) fixed system and one for each bone (Fig. 7). These systems are chosen as follows (Hcigfors et al., 1987).
CLAVICULA
SCAPULA
Fig. 7. A graphic presentation of the coordinate systems used for the description of the shoulder.
Table 2. Handmade extension of the shoulder rhythm by Hiigfors et al. (1988); Euler angles for the shoulder bones in elevation of the arm in the scapular plane used in this paper. When the arm is elevated from &120”, the following Euler angles are given for the bones according to the bone coordinate systems suggested by Hijgfors et al. (1987). Humerus ah=45 /%= -9O+o
(when 0 is the elevation angle shown in Fig. 3) as= -4S+a,(l-O/360)+(135-a&l) sin[0.50(1+ab/90)] (for ‘neutral’ rotation of the upper arm) a, = - 50 + 30 cos [0.75(/& + 90)]
/?,=24{1-cos[O.75(~,+90)]} y,=l5{1-cos[o.75(f?~+9o)]}+3
22,
273-292.
(0.5+a,/90)+9
Scapula cr,=200+20c0s[0.75(~,+90)} /?,= -140+94cos[0.75(/?,+90)
(l-y,/270)] y.= 82 + 8 cos {(ab + 10) sin [0.75(& +90)]} The Euler angles refer to a fixed orthonormal laboratory system with the z-axis vertical and the y-axis directed forward in a sagittal plane. The indices h, c and s (on a, /3 and 7) stand for humerus, clavicula and scapula, respectively.
A model for force predictions in the human shoulder
199
The sternum system has its origin R in the center of the stemoclavicular joint. It is such that the l-axis goes through the middle of the articular surfaces, i.e. it is normal to the sagittal plane and directed away from it. The entire system is or~honormal and the l-2 planecontains the midpoint of the first thoracic vertebrae. The 2-axis and the 3-axis are directed forwards and upwards, respectively. The sternum system thus defined is right-oriented for the right shoulder and leftoriented for the left shoulder. The claoicula system has the same origin R as the sternum system. Its l-axis goes through the center of the acromial surface. The 2-axis is orthonormal to the l-axis and parallel to the upper planar surface on the lateral end of the clavicula. The 2-axis is directed forwards and the 3-axis upwards. The system is orthonormal and right-oriented for the right shoulder. The scapula system has its origin in the point N, which is the acromioclavicular joint between the clavicula and scapula. The l-axis of the system is directed through the inferior angle. The l-2 plane contains the superior angle in its first quadrant. The system is orthonormal and right-oriented for the right shoulder. The last system is the humerus system. The gliding surface of the humeral head has an essentially spherical shape. The center of this sphere is used as the origin Rh of the humerus system. The l-axis is directed along the humerus through the end of the ridge between the coronoid fossa and the radial fossa. The 2-axis lies in the plane of the l-axis and the angular mobility direction of the ulna having positive 2coordinates. This system is orthonormal and right-oriented for the right shoulder. The positions for the bones are described with a set of Euler angles described in Fig. 8. The directions when all Euler angles are set to zero are such that the y-axes are pointing horizontally in a sagittal plane and the z-axes are pointing upwards in a vertical direction. The x-, y- and z-axis used here correspond to the l-, 2- and 3-axis used for the bone-fixed coordinate systems. The shoulder rhythm used in this paper describing the different bone positions is listed in Table 2.
Fig. 8. Euler angles used in this paper.
Vol. 25, No. 2, pp. 189-199.
1992.
OoZl-9290/92
Sml
+ .#
C: 1991 Pergamon Press
Printed in Great Britain
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TOWARDS A MODEL FOR FORCE PREDICTIONS IN THE HUMAN SHOULDER DAN KARLSSONand Bo PETERSON Center for Biomechanics, Chalmers University of Technology, S-412 96 GBteborg, Sweden
Abstrati-In
this paper the concept of a thme~imensional biomechanical model of the human shoulder is introduced. This model is used to analyze static. load sharing between the muscles, the bones and the ligaments. The model consists of all shoulder structures, which means that different positions and different load situations may be analyzed using the same model. Solutions can be found for the complete range of shoulder motion. However, this article focuses only on elevation in the scapular plane and on forces in structures attached to the humerus. The intention is to expand the model in future studies to also involve the forces acting on the other shoulder bones: the scapula and the clavicle. The musculoskeletal forces in the shoulder complex are predicted utilizing the opti~~tion technique with the sum of squared muscle stresses as an objective function. Numerical results predict that among the muscles crossing the ~enoh~eral joint parts of the deltoideus, the infraspinatus, the supraspinatus, the subscapularis, the pectoralis major, the coracobrachialis and the biceps are the muscles most activated during this sort of abduction. Muscle-force levels reached values of 150 N when the hand load was 1 kg. The results from the moderseem ta be qualitatively accurate, but it is concluded that in the future development of the model the direction of the contact force in the glenohumeral joint must be constrained.
number of shoulder components internal force (N) internal force vector (N) cross-sectional area for muscle i (mm”) power used in the objective function external force (N) external force vector (N) position vector of attachment point of force (m) x-component of external force vector Pi (N) x-component of the vector ri x Pi (N m) direction coefficient matrix for force and torque minimum value of the force F, (N) maximum value of the force F, (N) constant for maximum muscle stresses (N mm-2)
Shoulder pains of muscular many working
situations.
origin are common in High muscle forces seem to
be an important etiological factor (e.g. Herberts et al., 1984). The shoulder is a complex system from a biomechanical point of view; it consists of three bones and more than 20 muscles. Due to the fact that the shoulder is an indeterminate mechanical system, it is hard to predict individual internal forces. The number of unknown forces far exceeds the number of force and moment equilibrium equations available for the three shoulder bones: the clavicula, the scapula and the humerus. To date, no biomechanical shoulder model exists that is capable of predicting all of the musculoskeletal forces, and no model exhibits the flexibility needed for the analysis of different positions
Received i~~na~~rrn
27 December 1990.
and load situations. For ergonomic planning of work sites both in prevention and rehabilitation, such predictions could prove useful. Attempts have been made earlier to construct a biomechanical model of the shoulder (e.g. DeLuca and Forrest, 1973; Ringelberg, 1985; Dul, 1989). However, these models have often been restricted to single-motion patterns and have not included all the shoulder muscles, just the ones expected to be the most active components. We believe that this simplification is unacceptable when analyzing more complex load situations. The role of individual components in the shoulder has very often been studied using electromyography (EMG) (e.g. Herberts et al., 1980; Sigholm et al., 1984: JSirvholm et al., 1989; Kronberg et al., 1990). Jiirvholm et al. (1989) also studied the intramuscular pressure (IMP) in the supraspinatus muscle. Although these methods give some appreciation of muscle effort, the relationship between EMG and muscle force is not clear (Kadefors, 1978), especially when studying situations where the muscle length varies. It is also hard, if not impossible, to measure the EMG in all shoulder muscles or muscle parts at the same time. Nor is the relationship clear between the IMP and the muscle force. Another way of investigating the role of certain shoulder muscles is to block their nerves by local anesthesia. This was done by Colachis et al. (1969) and Howe1 et al. (1986) with the axillaris nerve. The latter group also studied the effects of blocking the suprascapularis nerve. Results show that when the arm is elevated, the torque-producing role is shared equally between the deltoideus and the supraspinatus muscles, Still, it is not possible to obtain information about all of the shoulder components by using EMG, IMP, or by studying the effects of local anesthesia (the serratus anterior muscle, for example, is anatomic~ly
189
190
D.
KARLSSONand
hard to reach). To obtain a complete picture of the role of the different components, it is necessary to use some kind of a biomechanical model. In general, in the human musculoskeletal system, there are more muscles crossing a joint than are strictly necessary, from a mechanical point of view, for performing most tasks. The muscles are activated simultaneously during most movements (synergistic muscle action). So far, the most efficient way to solve these indeterminate mechanical problems has been to formulate an objective function and use optimization techniques. This approach is based on the assumption that during learned activities the body selects a unique way of distributing the forces, and that this distribution is in some way governed by certain physiological criteria. The criterion often suggested is, for example, that synergistic muscles share the load proportional to their size or capacity, or that the total ‘effort’ to perform a specific task is minimized. It is often said that the capacity of a muscle is proportional to its cross-sectional area (A,), but it has also been suggested that the fiber-type composition must be taken into consideration (Dul et al., 1984). Objective functions in the optimization problem are often the sum of muscle stresses, or the sum of squared or cubed muscle stresses. In the mathematical formulation, finding a solution means the minimization of c (Fi/Ai)‘, where the power p is given the value 1,2 or 3 [a survey of this subject is given, for example, by Dul et al. (1984)]. To find a solution to our indeterminate problem, we use the objective function C(Fi/Ai)‘, but other ‘logical’ objective functions can, of course, easily be used as well. The authors have tried objective functions with powers of 1.5, 3 and 4, and these results do not seem to differ very much. However, this is beyond the scope of this paper. The purpose of this study is to test whether a three-dimensional shoulder model can be used to predict all the internal musculoskeletal forces acting on the humerus in different load situations, in
B. PETERSON
the case of a static external load in a static working position. Force predictions in this study are restricted to the shoulder structures attached to the humerus. Our biomechanical description of the shoulder is based on the works of Hiigfors et al. (1987,1988) and the relative capacities of the shoulder muscles (i.e. the normalized cross-sectional areas) which are examined in a study made by Karlsson and Jarvholm (1988), where they presented data on the cross-sectional areas of the shoulder muscles according to the method of Brand et al. (1981). The numerical results are compared with EMG studies (Jarvholm et al., 1989) and the results from Poppen and Walker (1978). These comparisons, however, show that there are large differences between some predictions from the model and the other studies mentioned above. It is concluded that the model shows qualitatively good results. However, in future developments the direction of the glenohumeral contact force must be taken into consideration and a constraint for this must be incorporated in the model. It might be interesting to validate the model with, for example, more EMG studies. It is also desirable to obtain a better understanding of the physiological criteria of muscle-force distributions and their relationship to objective functions. Further examination of the human shoulder rhythm is also interesting in order to get more reliable quantitative data.
THE BIOMECHANICAL MODEL
Biomechanical analysis of the shoulder
Hiigfors et al. (1987) described an idealized mechanical model of the shoulder. In this model the shoulder bones are considered as rigid bodies. The shoulder joints are modelled as ball joints. Muscles or muscle sections are described as stretched strings following the shortest path between attachment points,
Fig. 1. The shoulder muscles modelled as ‘strings’(Hiigfors et al., 1987).
A model for force predictions in the human shoulder
possibly constrained by other structures, i.e. lines or surfaces (Fig. 1). Internal forces actin on the rigid bodies are divided into muscle forces a f ting along the strings representing the muscles, joint contact forces and ligament forces. To get the complete geometry of the shoulder, we have to know the relationship between the position and direction in space of the humerus and the scapula, and the same between the humerus and the clavicle. This interplay between the motions of the constituent parts of the human shoulder is known as a ‘shoulder rhythm’. For example, Pronk (1989) described this subject, and quantified results have been presented by Inman et al. (19441,and by Poppen and Walker (1976). Here, we have designed this ‘shoulder rhythm’ by extrapolation of the
191
and CriXFi=-CriXPi.
(4)
The external loads Piconsidered here are all gravitational forces (it is an easy matter to extend these Pi to more general loads or even introduce force couples). Using a fixed reference system where the zaxis is vertically oriented, each force Pi takes the form: 0 0
Pi= I
piz
.
I
Equations (3) and (4) can now be used to formulate the six equilibrium equations written together in matrix form: 0
=
_dPi*
-CMix -CMiy
(6)
0 results given by Hiigfors et al. (1988), providing quantitative data for the rhythm over a limited range of movement for the arm (see Appendix). The positions of the bone-fixed coordinate systems are given by three Euler angles for each bone (see Appendix for definition). The external forces acting on the arm are the weight of the upper arm, the weight of the lower arm, the weight of the hand-held load. The shoulder structures capable of carrying an internal load give us 46 forces which are listed in Table 1. In this study, though, we are considering only the forces acting on the humerus and the arm, which are the first 23 forces of Table 1. If we now look at this humerus-arm system we may use vector expressions and let Pistand for external forces, Fi for internal forces and ri for the position vector of the attachment points of the forces. This gives the following force and moment equilibrium equations for the humerus {including the rest of the arm):
~Fi+~PI=O
(1)
and
These equations must be satisfied for each static position, but since there are only six equations [equations (1) and (2)] with 23 unknown internal forces, no unique solution exists. Optimization technique The equilibrium equations (1) and (2) can be written in the form:
xFi=--xPi
or
CalCFI=C-PI,
(7)
where al1 the elements in the matrix [a] are known. [These quantities can be computed in a straightforward manner using the modelling of muscles, ligaments and contact forces from the work of Hiigfors et al. (1989)]. This equation system has an infinite number of solutions. To find a unique solution among these one must add some criterion, and in our case we use the sum of squared muscle stresses (as mentioned earlier). The optimization problem can now be summarized as follows: minimize
C(Fi/Aij2
(8)
subject to
(1) CalEFl=E--PI
(9)
FiminG Fi G Fimax3
(10)
(II)
The first constraint is the equilibrium equations given by (I) and the second is the lower and upper limit for each internal force (II). The vector [Ff in equation (9) contains the listed internal forces to be predicted For the muscles we have the lower limits Fimin=O and the upper limits Finax= kAi, where k is a constant which depends on the maximum tension in the muscles. For the contact forces we did not apply constraint (II), as we assumed that the pressure between the skeletal bones could, in reality, become much higher than the muscle tensions. The ligament force involved in this study had its lower limit equal to zero and had no upper limit. The value of the constant k has been suggested to be in the range 0.4-1.0
192
D.
KARLSSON
and B. PETERSON
Table 1. Internal forces in structures attached to the humerus Number
Normahzed crosssectional area*
Shoulder component
(Hagfors et al., 1987) Com~nenrs ~ttac~d to the ~u~rus Latissimus dorsi 1 (superior part) 1 Latissimus dorsi 2 (inferior part) 2 Pectoralis major (sternum part) 3 Pectoralis major (clavicle part) 4 Deltoideus anterior 5 Deltoideus medialis 6 Deltoideus posterior 7 Coracobrachialis 8 Infraspinatus 1 (superior part) 9 Infraspinatus 2 (inferior part) 10 Subscapuiaris 1 (superior part) 11 Subscapularis 2 (middle part) 12 Subscapularis 3 (inferior part) 13 Supraspinatus 14 Teres major 15 Teres minor 16 Biceps brachii (long head) 17 Biceps brachii (short head) 18 Triceps (part attached to the scapula) XE components of the contact force? 21 Yz- 1 22 23
2.61 3.15 4.50 2.75 4.29 5.63 5.21 1.91 3.87 4.05 3.13 2.82 3.87 3.83 4.48 1.89 2.14 1.89 -8.87
Force in the coracohumeral ligament
The rest of the shoulder components (not investigated in this paper}
24 25 26 ;;: ;;: 31 32 33 34 35 36 :; 39 40 41 42 43 44 45 46
Trapezius 4 Trapezius 3 Trapezius 1 Trapezius 2 Levator scapulae Rhomboideus major Rhomboideus minor Pectoralis minor Serratus anterior 1 Serratus anterior 2 Serratus anterior 3 Omohyoideus Sternocleidomastoideus Sternohyoideus Subclavius XY- force components between the scapula and the calvicle Z- 1 51:
force components between the clavicle and the sternum 1 kperior contact force between the scapula and the thorax Inferior contact force between the scapula and the thorax
2.87 4.01 2.76 2.34 1.91 2.31 1.14 2.08 3.02 2.55 4.15 -
-
*Karlsson and Jfrvholm (1988). The areas are normalized so that the values listed above are the percentages of the total sum of the cross-sectional areas of the shoulder muscles. TContact force in the glenohumeral joint between the scapula and the humerus.
MN rns2 (Cro~ins~ield and Brand, 1981). Wood et stated that an average maximum stress in the muscles is of the order lOOpsi (equivalent to 0.7 MN mV2). In our study we used k=0.7 MN rnm2, Such a stress value is never reached in any muscle in our study. The objective function [equation (S)] is used only to minimize the muscle forces so the forces (Fi) from the contact and the ligament forces are not included in this objective function. al. (1989)
To solve the op~~tion problem, we designed a computer program which is schematically described in Fig. 2. The computation of the muscle force directions and the search for the solution determined by the objective function were done by computer program subroutines from the IMSL- and NAG-subroutine libraries. The program is written so that constraints more complicated than those given by equation (lo), even nonlinear ones, can be handled. Non-
193
A model for force predictions in the human shoulder \ Input data: - bandload - arm and body position G specified antropometric data)
and muscle trajectories*
Computation of muscle force directions
Mechanical equilibrium equations
I
l**Karlsson & Jiirvholm W88)
Fig. 2. How the computer works to find a solution to the force distribution problem.
linear constraints can, for example, be used for contact forces where a direction-dependent force limit is desired). The plane of quasi-static motion is described in Fig. 3. Calculations were made for static positions with intervals of 15” and the elevation angle (0 in Fig. 3) varied between O-120”. The hand load was set to 1 kg. Anthropometric data for masses and mass centers of the arm segments were taken from the literature (Pheasant, 1986). These data correspond to a young male person. FORCE PREDICTIONS
z
abduction pl:
& coronal plane
FOR ELEVATION
In every position, numerical solutions of the force levels were found for the force directions of the components involved. The predicted muscle forces did not reach their upper force limits in any of the computations. In abduction, almost all muscle activity was confined to the following muscles acting around the glenohumeral joint: the deltoideus anterior, the deltoideus medialis, the infraspinatus, the supraspinatus, the clavicular part of the pectoralis major, the coracobrachialis and the biceps muscles. Forces in these components are shown in Fig. 4, a result which is in good agreement with other investigations (for example, Inman et al., 1944). In these muscles, the results showed force levels up to 150 N for 1 kg hand load [the deltoideus medialis Fig. 4(a)]. The rest of the muscles crossing the glenohumeral joint were predicted to have zero or very low forces. The contact force in the glenohumeral joint reached a value of 650 N in a 60” abduction position [Fig. 4(d)]. The direction of the contact force never diverged more
Fig. 3. The plane of elevation used in this study.
than 27” from the normal to the cavitus glenoidalis (except for 0” abduction). In Fig. 5 horizontal and dorsal views of the contact force in the joint are shown for elevation angles of 15, 45, 75 and 105”. The dashed line in each view is a reference line fixed in the scapula and it goes through the medial angle of the scapula and the center of the glenohumeral joint (a line which has been used by other investigators, for example, Kronberg, 1989). Note that one direction of this line is not the same as the normal direction in the middle of the glenoid
D. KARLSSON and B. PETERSON
194
4 t 2 -t : E
100
Jo
ant.
--e--
deltoid
-
deno!dmed
-
rnwc4,ach.
-
inrraspin 1
-
bicepssholl
--t
infraspin2
-t-
bicepsbrig
-
suprarpin
‘-t
pect.maj.-da
0 0
50
100
angle
(Cl 150
E B
100
E Q E
:
M0
1 0
50
-
arbscap. 1 wbscap. 2 subscap.3
100
angle Fig. 4. Predictions of force levels in the shoulder components in abduction. Dorsal view
Top view
(a) Elevation angle: 15’
//
Angle difference*: 25?
1 i (
(b) Elevation angle: 45’ Angle difference? 26.9O
(cl
& M
1’ /’ I
/
/’
/’
/ // .-.-
Elevation angle: 75’ Angle difference*: 10.1’ J
(d) Elevation angle: 105’ Angle difference*:
L&0/
*._
/ / / .-_&
* The difference between the direction of the joint force and the normal direction in the middle of the glenoid cavity.
Fig. 5. The direction of the predicted glenohumeral joint force in four abduction angles.
A model for force predictions in the human shoulder cavity. The angular difference between the latter direction and the direction of the predicted contact force is noted in Fig. 5 for each elevation. DISCUSSION From the results in the previous section it is evident that solutions to the indeterminate shoulder problem can be found using an optimization technique. The reliability of the predicted results is, of course, dependent on the approximations and assumptions that are made in order to obtain a solution. The essential approximations made in our model are the following: (a) The glenohumeral joint is considered a ball joint. This seems to be a good approximation for the relatively small external loads used here. (b) Our shoulder rhythm is a ‘handmade’ extension to the whole motion range of the shoulder, based on a relatively thorough, but not complete, investigation (Hiigfors et al., 1988). It is unlikely that minor defects in the rhythm would produce large errors in the forces predicted. However, an investigation of the differences in predictions, between different and slightly different shoulder rhythms, could prove valuable. Furthermore, extreme angles cannot be handled at present (i.e. the highly elevated arm, which may cause contact between the humerus and the acromion). (c) The muscles are treated as strings, which seems to be a good idealization when used for long and thin muscles. However, the wider a muscle, the more strings it has to be divided into. It is increasingly important to base this sort of idealization on a physical/physiological foundation as the number of divisions increase. Most investigators have separated the deltoideus muscle into an anterior, a middle and a posterior part, which seems to be a ‘natural’ subdivision. However to split up the trapezius muscle into four independent parts, as we have done in our study, is not a common practice, and the reliability of this subdivison must be considered. For example, we do not know if each of the four sections can be considered totally independent of the force conditions in the other parts. In fact, we have found no references which describe how muscle parts can be activated independently. (d) The muscle force distribution is such that the total sum of muscle tensions is ‘as low as possible’. Since this sum does not include ligament tensions, no attempt is made to take into consideration the force level in the coracohumeral ligament. Moreover, the ligament is considered totally passive; it does not have a muscle’s active ability to shorten its length. Here our model might be slightly improved by adding the influence of the ligament length and its viscoelastic properties in the calculations. (e) The capsular ligament between the humerus and the scapula is not represented in our model. Some of its action must then be considered incorporated in the contact force. However, this makes it more com-
195
plicated to interpret the actual contact force between the two bones involved. (f) The humerus and the lower arm are together considered as a rigid body. No equilibrium equations have been formulated for the elbow joint. This introduces errors because the biceps brachii muscles, as well as the triceps, are crossing this joint. In reality, they are involved when the lower arm is moved together with other muscles not included in our mode1 (for example, the brachioradialis muscle). To get a complete and reliable shoulder model the muscles crossing the elbow joint should be analyzed and incorporated into the model. This is also our intention, as a next step, in the future development of the model. (g) In the mode1 we study only the equilibrium equations for the humerus-arm system, together with the objective function, including forces for this system. This can possibly lead to solutions which are not feasible for the larger system including the clavicle, the scapula, the humerus and the lower arm. This drawback is, of course, shared by all simplified models not including the equilibrium equations for all bones. The influence of this simplification cannot be clarified until the larger system is studied. (h) No constraint on the direction of the glenohumeral contact force is incorporated into the model. Investigators have reported that an angle difference for the so-called glenoid tilt (Saha, 1971; Brewer et al., 1986) between healthy shoulders and those with a tendency for luxation could be only 1%20” (Kronberg, 1989). A shift in the direction of the glenohumeral force angle of almost 30”, which is shown in Fig. 5 for our predicted results, is not satisfactory. In the future development of the mode1 this result shows the need for a constraint on this angular direction. Our solution should be seen in the light of the above approximations as optimal, when using our objective function. It is not clear whether this optimality corresponds to a physiological criterion used by man. Most of the predictions of muscle forces in abduction seem to be ‘reasonable’, i.e. the muscles predicted to be activated are those named in the literature as being the main abductors in elevation. JHrvholm et al. (1989) reported EMG studies on the supraspinatus, the infraspinatus and the anterior deltoid muscle for the same sort of elevation as used in this paper. In Fig. 6 comparisons are shown between the predicted results of our model and these measurements. Although it is hard to see any significant relationship between these measurements and our predicted results, the EMG results show more ‘smooth curves whereas the curves for the predicted forces show more ‘go up and down’. Our predicted forces deviate considerably from the EMG measurements for abduction angles exceeding 90”. It is noted that in this range, extreme muscle shortening makes the EMG results unreliable. However, a predicted zero result from our model when there seems to be an EMG
D. KARLSSONand B. PETERSON
196
‘(a)
-
Norm.
-
force [N]
EMG
-150
1.5 -
w 70 120 angle
l(b)
60
EMG
-L
Norm.
--t-
force [N]
_
force [N]
1;0
angle
Norm. EMG
150
60 angle
Fig. 6. Comparisons
for the infraspinatus, the supraspinatus and the anterior deltoid muscle between EMG (Jlrvholm et al, 1989) and the predicted muscle force.
activity in the same muscle may indicate some drawback in our model. For example, this seems to be the
case for the deltoideus medialis muscle in the elevation angle 45”. The zero activity in the subscapularis muscle for small elevation angles, and the same for the infraspinatus and supraspinatus muscles for large angles, shows the same difference with EMG. On the other hand, such muscles could be used for stabilizing the shoulder. This means that together they might not produce any larger torque around the glenohumeral joint even if their muscle forces show high values. Figure 4(d), showing the contact forces during elevation in the plane of the scapula, can be compared with results from other studies. Poppen and Walker
(1978) found the glenohumeral joint force (i.e. the resultant contact force) to be close to body weight in abduction. However, their results are based on EMG and are not fully explained in their paper. In our computer model we use anthropometric data corresponding to a young man with a mass of 75 kg. In Fig. 4(d) we obtain a contact force around 600 N in 60-90” elevation, or about 0.8 times the body weight, which is very close to the result obtained by Poppen and Walker (1978). The magnitude of our predicted contact force seems to be confirmed by these results. If, instead, we focus on the direction of this force which was shown in Fig. 5, we do not have such a satisfactory result. In the top views it can be seen
A model for force predictions in the human shoulder
how the contact force differs in direction compared to the reference line. One can also see a shift in direction beween the 45 and 75” elevation angles. Comparing this with Fig. 4(a) and (c)shows that for low angles the supraspinatus and infraspinatus show high force levels at the same elevation angles where their socalled antagonist, the subscapularis, shows no or not so large activity. The opposite is found for high elevation angles. Such observations lead to the conclusion that the direction of the glenohumeral contact force must be considered in the model. Hopefully, this will give more simultaneous activity in all muscles in the rotator cuff around the joint. Although some results from our model seem to be reasonable in comparison with earlier studies, most of the muscle force results are difficult to analyze and validate. The comparisons with EMG results are not thoroughly satisfying, as seen in Fig. 6. Other studies are restricted to certain positions or motion patterns (very often elevation in the scapular plane). In the future, we intend to incorporate also the other shoulder bones and muscles in our computer model. From these investigations we would, for example, get predictions for the forces in the trapezius muscle. This muscle is often discussed in ergonomics and comparisons could prove useful. Predictions of muscle forces from our model could also be validated by designing EMG experiments on muscles for which there are currently no activity data. This, however, would benefit from a better understanding of the refationship between the EMG signal and the force output of a muscle, in particular, when the muscle length changes. CONCLUSIONS l It is demonstrated in the present study that although the shoulder has a complex geometry and many components, a shoulder model can be devised which gives solutions for the forces in different load situations even when all muscles are taken into account. An optimization criterion based on minimizing the sum of squared muscle tensions is found to be applicable. l Some of these results seem ‘reasonable’ while others do not. However, there are several doubts when comparing with, for example, EMG results. a The direction of the contact force in the ~enohumera~ joint must be taken into account when designing a shoulder model.
Acknowledgements-We
highly appreciate the support and inspiration of Christian Hiigfors during our research. This study was supported by the Swedish Work Environment Fund.
REFERENCES Brand, P. W., Beach, R. B. and Thompson, D. E. (1981) Relative tension and potential excursion ofthe muscles in the forearm and hand. J. Hand Surg. 6,209-219.
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Brewer, B. J., Wubben, R. C. and Carrera, G. F. (1986) Excessive retroversion of the glenoid cavity. J. Bone Jt Surg. 68A, 124-131.
Colachis, S. C., Strohm, B. R. and Brechner, V. L. (1969) Effects of axillary nerve block on muscle force in the upper extremity. Arch. Phys. Med. Rehab. 50,645654. Crowninshield, R. D. and Brand, R. A. (1981) A phydologically based criterion of the muscle force prediction in locomotion. J. &mechanics 14, 793-801. DeLuca, C. J. and Forrest, W. J. (1973) Force analysis of individual muscles acting simultaneously on the shoulder joint during isometric abduction. J. Biomechanics 6, 385-393. Dul, 1. (1989) A biom~hanicai model to quantify shoulder load at the work place. C&n. B~o~chunjcs 3, 124-128. Dul, J., Townsend, M. A., Shiavi, R. and Johnson, G. E. (1984) Muscular synergism-II. A fatigue criterion for load sharing between synergestic muscles. J. Biomechanics 17, 663-673. Dul, J., Johnson, G. E., Shiavi, R. and Townsend, M. A. (1984) Muscular synergism-I. A criteria for load sharing between synergestic m&es. J. Bio~ckun~cs 17,675-684. Herberts. P.. Kadefors. R. and Broman. H. 09801 Arm DOSitioning in’ manual tasks-an electro&yo&apdic study of localized muscle fatigue. Ergonomics 23, 655-665. Herberts, P., Kadefors, R., HBgfors, C. and Sigholm, G. (1984) Shoulder pain and heavy manual labour. Clin. Ortkop. 191, 166178. Howell, S. M., Imobersteg, A. M., Seger, D. H. and Marone, P. J. (1986) Clarification of the role of the supraspinatus muscle in shoulder function. J. Bone ft Surg. @$A, 398-404. Hiigfors, C., Sigholm, G. and Herberts, P. (1987) Biomechanical model of the human shoulder--I. Elements. J. Biomechanics 20, 157-166. Hijgfors, C., Peterson, B., Sigholm, G. and Hetberts, P. (1988) Biom~hanic~ model of the human shoulder-II. The shoulder rhythm. Preprint 1988~1,Centre for Biomechanics CTH. Inman, V. T., Saunders, M. and Abbot, L. C. (1944) Observations on the function of the shoulder joint. J. Bone Jt Surg. 26, l-30. Jirvholm, U., Palmer&, G., Herberts, P., Hdgfors, C. and Kadefors, R. (1989) Intramuscular pressure and electromyo~aphy in the supraspinatus muscle at shoulder abduction. Clin. Ortkon. Rel. Res. 244. 45-52. Jiirvholm, U., Palmerud,‘G,, Karlsson, b., Herberts, P. and Kadefors, R. (1989) Intramuscular pressure and EMG in four shoulder muscles. J. orthop. Rex (submitted). Kadefors, R. (1978) Application of electromyography in ergonomics: new vistas. Stand. J. Rehab. Med. 10, 127-133. Karlsson, D. and JLrvholm, U. (1988) Force-producing ability in all shoulder muscles, as determined by cross-sectional areas. Preprint 1988:2, Centre for Biomechanics CTH. Kronberg, M. (1989) Shoulder joint stability-aspects on muscle function and skeletal anatomy. Ph.D. thesis, Department of Orthopaedic Surgery, Karolinska Hospital, Stockholm, Sweden. Poppen, N. K. and Walker, P. S. (1976) Norma1 and abnormal motion of the shoulder. J. B&e Jt Surg. 58A, 195-201. Popoen, N. K. and Walker. P. S. (19781 Forces at the &nohumeral joint in abduction,’ C& Orthop. 135, 165-170. Pheasant, S. (1986) Bodyspace: Anthropometry, Ergonomics and Design, pp. 121-134. Taylor & Francis, London. Petrofsky, J. C. and Lind, A. R. (1979) Isometric endurance in fast and slow muscle in cat. Am. J. Pkysiol. 236, C185-Cl91. Pronk, G. M. (1989) A cinematic model of the shoulder girdle: a r&urn&. J. Med. Engng Tecknol. 13, 119-123. Ringelberg, J. A. (1985) EMG and force production of some
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human shoulder muscles during isometric abduction. J. Biomechanics 18, 939-947.
Saha, A. K. (1971) Dynamic stability of the glenohumeral joint. Acta orthop. scand. 42, 491-505. Sigholm, G., Herberts, P., AImstrom, C. and Kadefors, R. (1984) Electromyographic analysis of shoulder muscle load. J. orthop. Res. 1, 379-386. Wood, J. E., Meek, S. G. and Jacobsen, S. C. (1989) Quantitation of human shoulder anatomy for prosthetic arm
control-I.
Surface
modelling.
J.
Biomechanics
APPENDIX
Five coordinate systems are necessary (and sufticient) for our description of the shoulder. One laboratory-fixed, one body- (thorax-) fixed system and one for each bone (Fig. 7). These systems are chosen as follows (Hcigfors et al., 1987).
CLAVICULA
SCAPULA
Fig. 7. A graphic presentation of the coordinate systems used for the description of the shoulder.
Table 2. Handmade extension of the shoulder rhythm by Hiigfors et al. (1988); Euler angles for the shoulder bones in elevation of the arm in the scapular plane used in this paper. When the arm is elevated from &120”, the following Euler angles are given for the bones according to the bone coordinate systems suggested by Hijgfors et al. (1987). Humerus ah=45 /%= -9O+o
(when 0 is the elevation angle shown in Fig. 3) as= -4S+a,(l-O/360)+(135-a&l) sin[0.50(1+ab/90)] (for ‘neutral’ rotation of the upper arm) a, = - 50 + 30 cos [0.75(/& + 90)]
/?,=24{1-cos[O.75(~,+90)]} y,=l5{1-cos[o.75(f?~+9o)]}+3
22,
273-292.
(0.5+a,/90)+9
Scapula cr,=200+20c0s[0.75(~,+90)} /?,= -140+94cos[0.75(/?,+90)
(l-y,/270)] y.= 82 + 8 cos {(ab + 10) sin [0.75(& +90)]} The Euler angles refer to a fixed orthonormal laboratory system with the z-axis vertical and the y-axis directed forward in a sagittal plane. The indices h, c and s (on a, /3 and 7) stand for humerus, clavicula and scapula, respectively.
A model for force predictions in the human shoulder
199
The sternum system has its origin R in the center of the stemoclavicular joint. It is such that the l-axis goes through the middle of the articular surfaces, i.e. it is normal to the sagittal plane and directed away from it. The entire system is or~honormal and the l-2 planecontains the midpoint of the first thoracic vertebrae. The 2-axis and the 3-axis are directed forwards and upwards, respectively. The sternum system thus defined is right-oriented for the right shoulder and leftoriented for the left shoulder. The claoicula system has the same origin R as the sternum system. Its l-axis goes through the center of the acromial surface. The 2-axis is orthonormal to the l-axis and parallel to the upper planar surface on the lateral end of the clavicula. The 2-axis is directed forwards and the 3-axis upwards. The system is orthonormal and right-oriented for the right shoulder. The scapula system has its origin in the point N, which is the acromioclavicular joint between the clavicula and scapula. The l-axis of the system is directed through the inferior angle. The l-2 plane contains the superior angle in its first quadrant. The system is orthonormal and right-oriented for the right shoulder. The last system is the humerus system. The gliding surface of the humeral head has an essentially spherical shape. The center of this sphere is used as the origin Rh of the humerus system. The l-axis is directed along the humerus through the end of the ridge between the coronoid fossa and the radial fossa. The 2-axis lies in the plane of the l-axis and the angular mobility direction of the ulna having positive 2coordinates. This system is orthonormal and right-oriented for the right shoulder. The positions for the bones are described with a set of Euler angles described in Fig. 8. The directions when all Euler angles are set to zero are such that the y-axes are pointing horizontally in a sagittal plane and the z-axes are pointing upwards in a vertical direction. The x-, y- and z-axis used here correspond to the l-, 2- and 3-axis used for the bone-fixed coordinate systems. The shoulder rhythm used in this paper describing the different bone positions is listed in Table 2.
Fig. 8. Euler angles used in this paper.