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A GLOBAL OPTIMIZATION METHOD FOR PREDICTION OF MUSCLE FORCES OF HUMAN
MUSCULOSKELETAL SYSTEM
+ Li G; + ** Pierce JE;
+ Bioengineering Laboratory, Department of Orthopaedic Surgery, Massachusetts General Hospital/Harvard Medical School, Boston, MA
[email protected]
INTRODUCTION
Quantitative data of in-vivo muscle forces, joint reaction forces and
moments of human musculoskeletal joints have been pursued for
decades using inverse dynamic optimization methods (1,2,3). However,
the traditional formulation of the optimization method violated the joint
moment equilibrium condition. The predicted muscle and joint reaction
forces are coordinate system-dependent (4).
This paper presented an improved optimization method, in which the
instantaneous rotation center of the joint is used as an optimization
variable and accurate moment equilibrium equation is formulated with
respect to the joint rotation center. The new method was used to
calculate muscle forces of a human elbow joint.
METHODS
A 3D anatomic model has been developed previously using a cadaveric
human forearm specimen (female, 62 years old) (4). The model included
bony geometry, insertion and origin areas of major muscles, and the
action lines of the muscles (Fig. 1). The geometric center of the trochlea
was chosen as the elbow joint center (1,5), and was used as the origin of
the coordinate system. Six muscles were simulated: the biceps brachii
(BIC), brachialis (BRA), brachioradialis (BRD), pronator teres (PRT),
anconeus (ANC), and triceps brachii (TRI) (1). The model was used to
simulate the equilibrium state of the elbow at 90° of flexion in the
neutral forearm position under the muscle and external loads (Fig. 1).
v
If the actual rotation center of the joint is at the position rrot with respect
to the origin of the coordinate system, then the moment equilibrium
v
v
fiM , rrot
under constraints:
6

v
v
v
 ( a)
fiM ei + F Jo i n t = F i n t

i= 1

6
v int
v

M v v
f i (( ri − rr o t) × e i ) = M r o t
 ( b)

i=1

M
 ( c ) 0 ≤ fi ≤ σ ⋅ Ai


∑
∑
(2)
The average mass of the lower arm is 1.36 kg. External loads include
axial moments (internal-external moment) changing between ±5 Nm
along the x-axis and a 50 N vertical compressive force applied at the
distal end of the radius (Fig. 1). The optimization procedure predicted
the muscle forces and the joint reaction forces. The results of the global
optimization method were compared to those of traditional optimization
method. The converged objective function values were presented as a
ratio of the value predicted using traditional method. The location of the
instantaneous rotation center was also presented.
RESULTS
The global optimization method converged to smaller objective function
values (cubic muscle stress) compared to the traditional optimization
method (Fig. 2a). For example, when an internal moment of 5 Nm was
applied, the cubic muscle stress was only 20% of that predicted by the
global optimization method.
The predicted rotation center moved from medial to lateral around the
geometric center of the elbow joint with the internal-external moment
(Fig. 2b). As the internal moment increased (positive values), the
rotation center moved almost linearly to the lateral side of the geometric
center.
Z
Y
X
Medial
Lateral
Brachialis
Brachialis
Triceps
Ratio of objective function value
v
v
i= 1
is i-th muscle force and M ri on tt is the intersegamental moment
calculated with respect to the actual rotation center. This treatment
makes the moment equilibrium equation independent to the origin of the
v
coordinate sy stem. Since the joint rotation center rrot is unknown a
priori, it should be calculated as an optimization variable through the
optimization procedure. Therefore, a new optimization procedure can be
formulated to calculate muscle forces,
Minv
J ,
(1)
fi
M
v
∑ f iM ((ri − rr o t) × ei ) = M int
r o t , where
Pronator
Teres
Pronator Teres Brachioradialis
Biceps
Biceps
Weight
of Arm
Axial
Joint
Reaction Force
compressive
Joint
andReaction
Moment Anconeus
Anconeus
load
Force and
Internal/external
moment
moment
Fig. 1. An anatomic forearm model with muscle force orientations and application point.
12
1.0
a
0.8
0.6
0.4
0.2
0.0
-5 -4 -3 -2 -1 0 1
2
3
4 5
Joint rotation center location
Medial
Lateral
6
equation can be accurately written as
The predicted muscle reaction forces were also different compared to
those calculated from the traditional optimization method (Table 1).
Under the ±5 Nm internal-external moment, both BIC and BRA were
predicted to contract by the new optimization method while the
traditional optimization method only predicted recruitment of BIC
muscle under the external moment, and only BRA muscle under the
internal moment.
DISCUSSION
In traditional optimization method, eliminating the joint reaction
moment term in the moment equilibrium violates the joint equilibrium
condition and is equivalent to symplify the joint as a frictionless
rotational joint fixed at the origin of the coordinate system. Therefore,
the optimization procedure cannot converge to the actual minimal value
of the objective function. Consequently, choosing different origins will
lead to different solutions of muscle forces (4).
This study introduced an improved optimization method to predict
the muscle and joint reaction forces, where the actual joint rotation
center was treated as a variable in the optimization procedure. The
moment equilibrium constraint is accurately established with respect to
the actual joint rotation center. A unique minimal value for the objective
function could be obtained. The predicted muscle and joint reaction
forces are independent to the joint coordinate system. This method can
be widely applicable to predict muscle and joint reaction forces in all
other joints such as the knee, hip, and ankle joints.
b
8
4
0
-4
-8
-12
-5 -4 -3 -2 -1 0 1
2 3
4
5
Internal-external moment (Nm)
Internal-external moment (Nm)
Fig. 2 (a) Ratio of the converged objective functions of the global to traditional
optimizations; (b) Predicted rotation center location changed with external loads;
Table 1. Results of of traditional and global minimizations when the forearm
is under a combined load
Moment
(Nm)
-5 Nm
Optimization
method
TRI
Muscle Forces (N)
BIC BRA BRD ANC
PRT
Traditional
43
354
0
0
48
148
Global
84
241
246
0
0
153
Traditional
0
0
700
50
0
42
Global
0
204
313
37
30
0
5 Nm
REFERENCES
1. An, J Biomech 1981. 2. Pedotti, Math Biosciences 1978. 3.
Crowninshield, J Biomech 1981; 4. Pierce et al, J Biomech, 2004; 5.
Morrey, J Bone Joint Surg, 1976. ** Department of Mechanical
Engineering, Massachusetts Institute of Technology, Cambridge, MA
51st Annual Meeting of the Orthopaedic Research Society
Poster No: 0578