Seediscussions,stats,andauthorprofilesforthispublicationat:https://www.researchgate.net/publication/7937624
Anequationtocalculateindividualmuscle
contributionstojointstability
ArticleinJournalofBiomechanics·June2005
ImpactFactor:2.75·DOI:10.1016/j.jbiomech.2004.06.004·Source:PubMed
CITATIONS
READS
48
37
2authors:
JimRPotvin
StephenHMBrown
McMasterUniversity
UniversityofGuelph
89PUBLICATIONS1,440CITATIONS
62PUBLICATIONS972CITATIONS
SEEPROFILE
Allin-textreferencesunderlinedinbluearelinkedtopublicationsonResearchGate,
lettingyouaccessandreadthemimmediately.
SEEPROFILE
Availablefrom:JimRPotvin
Retrievedon:18May2016
ARTICLE IN PRESS
Journal of Biomechanics 38 (2005) 973–980
www.elsevier.com/locate/jbiomech
www.JBiomech
An equation to calculate individual muscle contributions
to joint stability
Jim R. Potvina,, Stephen H.M. Brownb
a
Department of Kinesiology, University of Windsor, 401 Sunset Ave, Windsor, Ont., Canada N9B 3P4
b
Department of Kinesiology, University of Waterloo, Waterloo, Ont., Canada
Accepted 2 June 2004
Abstract
The purpose of the current paper was to use the energy approach to develop a simplified equation for quantifying individual
muscle contributions to mechanical stability about all three axes of a particular joint. Specific examples are provided for muscles
acting about the lumbar spine’s L4/L5 joint. The stability equation requires input of: (1) origin and insertion coordinates, relative to
the joint of interest, (2) muscle force, and (3) muscle stiffness. The muscle force must be derived from a biomechanical analysis that
first results in static equilibrium about all axes being studied. The equation can also accommodate muscles with nodes that change
the line of action, with respect to a particular joint, as it passes from the origin to insertion. The results from this equation were
compared to those from a Moment approach using more than two million simulated muscles with three-dimensional orientations.
The differences between approaches were negligible in all cases. The primary advantage of the current method is that it is very easy
to implement into any 2D or 3D biomechanical model of any joint, or system of joints. Furthermore, this approach will be useful in
dissecting total joint stability into the individual contributions of each muscle for various systems, joints, postures and recruitment
patterns.
r 2004 Elsevier Ltd. All rights reserved.
Keywords: Joint stability; Spine mechanics; Muscle stiffness; Buckling behavior
1. Introduction
Mechanical joint stability has become a popular topic
in the biomechanics literature. To date, most researchers, including ourselves, have merely assumed stability
levels based on a qualitative interpretation of agonist/
antagonist muscle activation. Spine instability has been
proposed as a risk factor in the development of lowback pain and injury (Panjabi, 1992) and this system has
been the focus of most of the rare attempts to actually
quantify muscle contributions to stability. However, due
to the complexity of these calculations, few have made
them and the role of stability in spine function, and thus
its direct link to injury, is not yet fully understood.
Corresponding author. Tel.: +1-519-253-3000x2461; fax: +1-519973-7056.
E-mail address: [email protected] (J.R. Potvin).
0021-9290/$ - see front matter r 2004 Elsevier Ltd. All rights reserved.
doi:10.1016/j.jbiomech.2004.06.004
Bergmark (1989) was the first to fully define and
examine the mechanical stability of a muscular system
which can be considered to be stable when the potential
energy ðV Þ of the entire system is at a relative minimum.
Thus, a system must always be able to return to its
original state of equilibrium in response to perturbations
around this original state. Mathematically, two conditions must be met in order for stability to exist. First, the
system must be in mechanical equilibrium, meaning that
the first derivative of V (net moment) must be zero.
Next, the second derivative of V must be positive
definite (greater than zero) indicating that V is at some
minimum.
A muscle can contribute to potential energy during a
perturbation, and subsequent length change, by storing
or releasing elastic energy related to its stiffness and by
performing positive or negative work related to its pretension. Muscle stiffness plays a very significant role in
ARTICLE IN PRESS
974
J.R. Potvin, S.H.M. Brown / Journal of Biomechanics 38 (2005) 973–980
stabilizing a joint and/or system of joints. A muscle with
a higher stiffness stores more energy in relation to the
distance it is stretched during a perturbation, thus
creating a higher level of stability in the system.
Bergmark (1989) and subsequent researchers (Crisco
and Panjabi, 1991; Cholewicki and McGill, 1996;
Gardner-Morse et al., 1995; Gardner-Morse and Stokes,
1998; Granata and Wilson, 2001) have investigated
stability of the lumbar spine by developing a stiffness
and load matrix incorporating V , with respect to each
generalized coordinate, and examining the eigenvalues
for each joint degree of freedom. For the system to be
stable each eigenvalue, and the global matrix determinant, must be positive definite. The smallest eigenvalue
is considered to be the critical stiffness. Loads that
compromise this critical stiffness would cause the system
to become unstable such that it would buckle. Recently,
Howarth et al. (2004) have shown that the global
determinant and smallest eigenvalue (weakest link)
provide similar trends for interpreting system stability.
While it is important to quantify spine stability, the
complexity in applying the required mathematical
analyses can be quite significant and limiting to many
researchers. To this end, recent researchers (Cholewicki
et al., 1999; Granata and Orishimo, 2001) have been
able to demonstrate a relationship between muscle
moment arm, length and stiffness that allows for a
relatively simple, yet accurate, assessment of stability for
the purposes of their studies. However, their models,
and thus their equations, were proposed to be limited to
one upright single-equivalent flexor and one upright
single-equivalent extensor muscle. Such a simplification
was not suggested to apply to the complex musculature
of human joints for the sake of a full-scale analysis of
stability.
The purpose of the current paper was to use the
energy approach to develop a simplified equation for
quantifying individual muscle contributions to stability
about the three axes of a particular joint. Subsequently,
a method will be proposed for estimating total stability
in a multi-muscle system. Specific examples will be
provided of muscles acting about the lumbar spine’s L4/
L5 joint.
(B2X, B2Y, B2Z)
(BX, BY, BZ)
l2
θz
l
Joint at (0,0,0)
(AX, AY, AZ)
Fig. 1. Initial ðBX ; BY ; BZ Þ and final ðB2X ; B2Y ; B2Z Þ coordinates of
the node, or insertion point where no node exists, for a small rotation
about the joint. The ðAx ; Ay ; Az Þ coordinates refer to the muscle origin.
All coordinates are relative to the joint at (0,0,0). For illustration
purposes, the rotation angle ðyÞ is exaggerated beyond the small angles
generally used for stability analyses. The ‘ and ‘2 indicate the initial
and final distance from the origin to the insertion/node. In this
example, the XY plane is represented and stability will be calculated
about the z-axis.
muscles are modeled to pass through nodes between the
origin and insertion. These nodes serve to change the
line of action of the muscle. In such cases, (Bx , By , Bz )
will represent the node, and not the final insertion point
(Fig. 1). All calculations here assume rotation about one
axis of one joint, with muscles that have a maximum of
one node. However, these equations can be used for
rotations about multiple joints, axes and nodes if the
total 3D length of the muscle is calculated.
For a given rotation about the z-axis:
2
3 2
32
3
B2X
BX
cos yZ sin yZ 0
6
7 6
76
7
cos yZ 0 5;
ð1Þ
4 B2Y 5 ¼ 4 BY 54 sin yZ
0
0
1
B2Z
BZ
B2X ¼ BX cos yZ BY sin yZ ;
ð2Þ
B2Y ¼ BX sin yZ þ BY cos yZ ;
ð3Þ
B2Z ¼ BZ ;
ð4Þ
2. Methods
The following equations will outline the development
of a simple equation for the calculation of individual
muscle contributions to joint stability. Calculations will
be made for stability about the z-axis, but similar
equations can be used for stability about the x- and yaxes.
A muscle can be observed before and after rotation
about the z-axis of a particular joint, assumed to be at
(0,0,0) m (Fig. 1). In the biomechanical literature, many
where AX ; AY ; AZ are the origin coordinates with
respect to the joint of interest at (0,0,0) m, BX ; BY ; BZ
the initial node or insertion (without nodes) coordinates
with respect to joint and B2X ; B2Y ; B2Z the rotated node
or insertion (without nodes) coordinates with respect to
joint.
ARTICLE IN PRESS
J.R. Potvin, S.H.M. Brown / Journal of Biomechanics 38 (2005) 973–980
Given these coordinates, a muscle’s initial length,
rotated length and change in length can be calculated as
follows:
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
‘ ¼ ðBX AX Þ2 þ ðBY AY Þ2 þ ðBZ AZ Þ2 ;
ð5Þ
‘2 ¼
steps can be found in the appendix:
d2 UðmÞ
dy2Z
AX BX þ AY BY ðAX BY AY BX Þ2
¼F
‘
‘3
ðAX BY AY BX Þ2
þk
:
‘2
SðmÞZ ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ðB2X AX Þ2 þ ðB2Y AY Þ2 þ ðB2Z AZ Þ2
ð6Þ
substituting (2)–(4) into (6) yields:
‘2 ¼
By substituting (5) and (7) into (8), the change in
muscle length for a small rotation about the z-axis is
calculated as
where ‘ is the initial distance from ðAX ; AY ; AZ Þ to
ðBX ; BY ; BZ Þ, ‘2 the distance from ðAX ; AY ; AZ Þ to
ðB2X ; B2Y ; B2Z Þ, after a small rotation ðyZ Þ and D‘ the
change in muscle length, from the origin to the insertion/
node, for a small rotation ðyZ Þ.
The elastic potential energy stored in a given muscle is
calculated by
ð9Þ
where UðmÞ is the sum of the energy stored and work
done by, or on, the muscle, F the muscle force (N), and
k the muscle stiffness (N/m).
The stability contribution of a muscle, about the
z-axis, is calculated as the second derivative of stored
potential energy, with respect to a small rotation
about z:
d2 UðmÞ
;
dy2Z
ð10Þ
where SðmÞZ is the stability contribution of a muscle
about the z-axis of a joint.
The following steps were then performed to yield Eq.
(11): (a) substituting (8) into (9), (b) applying a Taylor
Series expansion (third-order approximation), (c) differentiating twice with respect to y, (d) substituting (5) and
simplifying. A more detailed representation of these
ð7Þ
are determined from the cross products of the radius
ðBX ; BY ; BZ Þ and the muscle force unit vector, from
ðBX ; BY ; BZ Þ to ðAX ; AY ; AZ Þ. This value represents the
D‘ ¼ ‘2 ‘
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
¼ ðBX cos yZ BY sin yZ AX Þ2 þ ðBX sin yZ BY cos yZ AY Þ2 þ ðBZ AZ Þ2
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ðBX AX Þ2 þ ðBY AY Þ2 þ ðBZ AZ Þ2 ;
SðmÞZ ¼
(11)
In addition, the functional moment arms of the muscle
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ðBX cos yZ BY sin yZ AX Þ2 þ ðBX sin yZ BY cos yZ AY Þ2 þ ðBZ AZ Þ2 :
UðmÞ ¼ F D‘ þ 12 k D‘2 ;
975
(8)
moment about z, created by a resultant muscle force of
1 N. For rotations about the z-axis, this simplifies to
rZ ¼
BX AY AX BY
;
‘
ð12Þ
where rZ is the functional moment arm of the muscle
about the z-axis (m).
Substituting (12) into (11) yields
AX BX þ AY BY r2Z
SðmÞZ ¼ F
ð13Þ
þ kr2Z :
‘
Bergmark (1989) calculated muscle stiffness as
k¼
qF
;
L
ð14Þ
where L is the total muscle length from origin to
insertion, q the proportionality constant relating muscle
force and length to stiffness.
Substituting (14) into (13) and simplifying yields
AX BX þ AY BY r2Z qr2Z
SðmÞZ ¼ F
þ
:
ð15Þ
‘
L
Nodes serve as pulleys to change the line of action of the
muscle (Fig. 2).
The total stability of the system requires the
examination of all sources of potential energy, including
the work done by external loads acting on the body
ARTICLE IN PRESS
J.R. Potvin, S.H.M. Brown / Journal of Biomechanics 38 (2005) 973–980
976
where W is the work done during the rotation (J), P is
the external load (kg), and h is the initial height of the
load (m).
Applying a Taylor Series expansion, and calculating
the second derivative with respect to yZ , produces:
(CX, CY, CZ)
(BX, BY, BZ)
lBC
rBC
2
l
(BX, BY, BZ)
lAB
(0,0,0)
L = lAB + lBC
r
rAB
V¼
(AX, AY, AZ)
(AX, AY, AZ)
N
X
Um W;
where V is the potential energy of the entire system and
m is a particular muscle.
The second derivative of V represents the total
stability S Z about the z-axis, and can be approximated
with
"
#
N
d2 V X
d2 U
d2 W
SZ ¼ 2 ¼
:
ð19Þ
dyZ m¼1 dy2Z m
dy2Z
Substituting (15) and (17) into (19) yields
SZ ¼
P
¼
d2 V
dy2Z
N
X
m¼1
h
Y
r3
r1
r4
Joint
X
m2
m3
m1
Sz =
Z
m4
d2 V
dθ z
2
N
=Σ
m=1
[
2
F (AX BX + AY BY – rz )
l
+
qFrz
L
2
]
– Ph
m
Fig. 3. Example of four muscles and a load ðPÞ and the calculation of
stability about the z-axis. The q is set as a constant. The stability
contributions of each muscle are summated and counteract the
destabilizing potential energy of the load ðPhÞ. For each muscle, the
stabilizing potential is a function of the origin, insertion and nodal
coordinates, length, moment arm and force.
(Fig. 3). For the sake of simplicity, the contributions of
passive structures will be ignored, although they will
contribute to energy storage. During a small rotation,
the work done by the external load at a given height is
W ¼ Phð1 cos yÞ;
ð18Þ
m¼1
Fig. 2. Illustration of the parameters used for the calculation of the
stability contribution of a muscle about one axis, in cases where there
is no node (left), and where there is a node associated with the joint
(right). With a node, the ‘ and L will differ, such that the final stability
equation cannot be further simplified. Note the larger moment arm
when the node is present. The insertion (left) and node (right) have
coordinates ðBX ; BY ; BZ Þ that will move with rotations about the lower
joint (0,0,0). The moment arms ðrÞ are for illustration purposes only
and are not exactly the functional moment arms derived from a 3D
analysis. Moment arms rAB and rBC reflect moment potential about the
z axis, relative to joints 1 and 2 respectively.
r2
ð17Þ
Therefore, for a system with multiple muscles, the
potential energy of the entire system can be calculated as
1
(0,0,0)
(0,0,0)
d2 W
¼ Ph:
dy2Z
ð16Þ
Fm
AX BX þ AY BY r2Z qr2Z
þ
‘
L
Ph: (20)
m
It is important to note here that stability cannot be
calculated until there is static equilibrium. Thus, the
individual muscle forces must first be determined such
that the net moment about each axis is zero. Once this
first step has been achieved, the net mechanical stability
about the z-axis can be calculated with Eq. (20).
To provide examples, the stability contributions of
two trunk muscles were evaluated in the upright posture
about the flexion/extension, lateral bend and axial twist
axes. The origin, insertion and nodal coordinates for
these muscles were taken from Cholewicki and McGill
(1996) and are provided in Table 1. The above equations
can also be converted for use with the lateral bend ðxÞ
and axial twist ðyÞ axes. For calculations about the
lateral bend axis, substitute x for z, y for x and z for y.
For the axial twist axis, substitute y for z, z for x and x
for y. The proposed equations can be used for each
lumbar joint as long as the A and B coordinates are
calculated with respect to the joint of interest.
The right rectus abdominis provided an example of a
muscle with no node, while the right L1 pars lumborum
was modeled to have one node at the L4 level. The q
value was assumed to be 10 and forces were set to 1 N.
The ‘‘Geometric Stability’’ was calculated to be the
muscle’s stability related solely to its orientation. This
value could then be multiplied by any muscle force to
obtain the actual stability contribution of the muscle.
ARTICLE IN PRESS
J.R. Potvin, S.H.M. Brown / Journal of Biomechanics 38 (2005) 973–980
977
Table 1
The coordinates used for the rectus abdominis and L1 pars lumborum as examples for the calculation of stability
Global Coordinates
L4/L5
Rectus Abdominis
L1 Pars Lumborum
Coordinates wrt L4/L5
X
Y
Z
X
Y
Z
Origin
Insertion
Pelvis
Rib
0.106
0.184
0.190
0.211
0.050
0.350
0.000
0.030
0.070
0.000
0.078
0.084
0.000
0.161
0.139
0.000
0.030
0.070
Origin
Node
Insertion
Pelvis
L4
L1
0.024
0.025
0.044
0.178
0.223
0.328
0.060
0.050
0.026
0.082
0.081
0.062
0.033
0.012
0.117
0.060
0.050
0.026
These coordinates were taken from Cholewicki and McGill (1996) and all units are in meters. For Eq. (15), all coordinates must be taken with respect
to the joint (L4/L5) and those values are presented on the right.
Table 2
Muscle parameters used to calculate Geometric Stability of the rectus abdominis and L1 pars lumborum about the three anatomical axes of the L4/
L5 joint
Length from origin to node/insertion (m)
Total length (m)
X -axis (lateral)
Moment Arm (m)
AY BY ðm2 Þ
AZ BZ ðm2 Þ
Geometric stability (N m)
‘
L
r
Y -axis (axial)
Moment Arm (m)
AZ BZ ðm2 Þ
AX BX ðm2 Þ
Geometric stability (N m)
r
Moment Arm (m)
AX BX ðm2 Þ
AY BY ðm2 Þ
Geometric stability (N m)
r
Z-axis (flex/ext)
S
S
S
Rectus Abdominis
L1 Pars Lumborum
0.3030
0.3030
0.0510
0.0224
0.0021
0.010
0.0461
0.1555
0.0514
0.0004
0.0030
0.169
0.0097
0.0021
0.0066
0.031
0.0165
0.0030
0.0066
0.221
0.0805
0.0066
0.0224
0.140
0.0793
0.0066
0.0004
0.403
The q value was set at 10 for all calculations.
For the sake of simplicity in these examples, it was
assumed that rotations only occurred at L4/L5.
To test the validity of the muscle stability equation
(15), a large number of muscles were simulated and
rotated by a very small angle. Insertions were simulated
with all combinations of x (lateral axis), y (vertical) and
z (flex/ext) coordinates ranging from 0 to 10, in
increments of 1 ðn ¼ 1331Þ. Origins were simulated with
all combinations of x and z coordinates from 0 to 10 and
y coordinates from 10 to 10 ðn ¼ 2541Þ for a total of
3,382,071 origin/insertion combinations (i.e. muscles).
From this, simulated muscles were removed if they
either created a negative moment and/or had an origin
that was superior to the insertion, leaving a total of
2,189,253 muscles (65% of the total). Each muscle was
given a q value of 10, a force of 100 N and the insertion
was rotated 1 108 radians. Eqs. (7) and (12) were
used to calculate the length ðlÞ and moment arm ðrÞ,
respectively. Eq. (15) was then used to estimate the
stability contribution of each muscle about the z-axis.
Stokes and Gardner-Morse (2003) noted that stability
can also be calculated as the change in moment divided
by the change in angle, during a very small rotational
perturbation. For each simulated muscle, the stability
was also calculated using this Moment method. The
change in moment, resulting from a 1 108 radian
rotation, was calculated using the product of the
original force and moment arm and subtracting it from
the product of the new force (new length times stiffness)
and new moment arm after rotation. A difference value
(error) was calculated for each muscle, by subtracting
the stabilities calculated with the energy method from
those obtained with the moment method.
3. Results
The stability values for the rectus abdominis and L1
pars lumborum were calculated for each axis (Table 2).
The calculation of the mechanical stability contribution
ARTICLE IN PRESS
978
J.R. Potvin, S.H.M. Brown / Journal of Biomechanics 38 (2005) 973–980
of the L1 pars lumborum about the z-axis (flexion/
extension) will be presented here. Using a q value of 10,
force of 1.0 N and the xy coordinates, the values from
Table 2 were input into Eq. (15) to obtain
ð0:0066Þ þ ð0:0004Þ ð0:0793Þ2
SðL1ÞZ ¼ F
0:0461
2
ð10Þð0:0793Þ
þ
¼ 0:403F :
(21)
0:1555
With a geometric stability of 0.403 m, the stability for a
muscle force of, for example, 500 N would be 201.5 N m.
Without the node, the L1 pars lumborum flexor moment
arm decreases 5% from 0.0793 to 0.0750 m, but this
results in a 17% decrease in geometric stability to
0.335 m.
Of the 2,189,253 muscles simulated, the average
stability value was calculated to be 25.1 N m. Amongst
all these comparisons between the energy and moment
methods, the largest error was only 0.00013 N m and this
was only 0.0003% of the mean stability.
4. Discussion
This paper presents a simple equation that calculates
the magnitude of individual muscle contributions to
mechanical stability about any axis of a particular joint.
The stability equation requires the input of: (1) origin
and insertion coordinates, relative to the joint of
interest, (2) muscle force and, (3) muscle stiffness. The
muscle force must be derived from a biomechanical
analysis that first results in static equilibrium about all
three axes. The equation can also accommodate muscles
with nodes that change the muscle’s line of action, with
respect to a particular joint, as it passes from the origin
to insertion. The primary advantage of the current
method is that it is very easy to implement into any 2D
or 3D biomechanical model of any joint, or system of
joints. This paper uses trunk muscles for illustrating the
equation’s utility. To date, almost all attempts to
directly quantify muscle contributions to joint rotational
stability, have done so with applications to the spine.
The author’s are aware of no such calculations with
other joints. However, it is proposed that this method
can be used for any joint where the location of the axis
of rotation as well as muscle origin, insertion and nodal
coordinates are available.
Eq. (15) is sensitive to a number of factors that affect
muscle contributions to stability. Muscle force generated prior to the perturbation (pre-tension) will directly
affect the work done by the muscle during a small
rotational perturbation (first half of Eq. (15)). This pretension will also determine muscle stiffness (k ¼ qF =L in
Bergmark, 1989), and affect the stability contributions
of strain energy stored in the muscle during the
perturbation. In addition, the origin and insertion
coordinates will determine the muscle length (Eq. (5))
and moment arm of the muscle acting about the axis of
interest (Eq. (12)). When the muscle passes through a
node that deviates the line of action from that of the line
joining the origin and insertion, this is accounted for by
replacing the insertion coordinates with that of the node
so that the origin–node length ð‘Þ and origin–insertion
length passing through all nodes ðLÞ can be calculated
separately. In fact, Eq. (15) could be re-written into a
much longer equation, using only muscle force, Bergmark’s q and the coordinates for the origin and insertion
(and node, where applicable).
Cholewicki et al. (1999) and Granata and Orishimo
(2001) have presented simplified models of muscle
contributions to stability, using only stiffness and
moment arm ðS ¼ kr2 Þ. However, Eq. (13) indicates
that stability is more complex than this, as it is also
dependent on the origin and insertion locations. For a
linear spring, stiffness and pre-tension are independent.
While the stiffness always makes a positive contribution
to stability, pre-tension can either increase or decrease
stability, depending on the origin and insertion/node
locations (i.e. AX BX and AY BY polarities). Unlike linear
springs, however, muscle stiffness is dependent on
muscle force/activation so that it is not as easy to
differentiate the relative contributions of muscle stiffness and pre-tension to joint stability. For most muscles,
increasing force has a larger effect on increasing the
stiffness contribution to stability than any potential
destabilizing work done by the initial force. In fact, of
the 2,189,253 muscles simulated, 93.5% had positive
geometric stabilities such that relatively few muscles had
orientations that made the joint less stable as their pretension was increased.
With the proposed equation, the relative contribution
of muscle stiffness depends on the selection of a q value.
Large q values will increase the magnitude of the kr2
component of Eq. (13), and diminish the relative effect
of the AX BX þ AY BY r2 component. Estimates of q in
the literature have been in the large range from
approximately 0.5–50, with a mean of approximately
10 (Crisco and Panjabi, 1991; Cholewicki and McGill,
1995). Recent work has estimated the critical q (the
minimum q at which the spine is stable) to be in the
range of approximately 0.3–7.3 (Gardner-Morse et al.,
1995; Gardner-Morse and Stokes, 1998; Stokes and
Gardner-Morse, 2003). It appears that future work is
needed to determine the most appropriate q for
biomechanical analyses. Cholewicki and McGill (1995)
describe a method for estimating muscle stiffness without the need to estimate q. They modified the crossbridge bond distribution moment (DM) model of Ma
and Zahalak (1991) which was based on the sliding
filament theory of Huxley (1957). Eq. (13) allows for the
use of either method to estimate muscle stiffness.
ARTICLE IN PRESS
J.R. Potvin, S.H.M. Brown / Journal of Biomechanics 38 (2005) 973–980
One benefit of the proposed equation, is that it allows
for a muscle’s contribution to stability to be broken into
two components: (1) the capacity to generate force
(generally related to muscle cross sectional area, length
and velocity) and, (2) the orientation of the muscle
(referred to here as geometric stability). This orientation
can be defined completely by the coordinates of the
origin, insertion and, where applicable, a node. For the
two trunk muscle examples presented in Tables 1 and 2,
it is apparent that geometric stability is sensitive to both
the length and moment arm of the muscle. For example,
the rectus abdominis and L1 pars lumborum have
similar moment arms about the x-axis (about 0.051 m)
but given that the L1 muscle has a node and shorter
muscle length, its geometric stability of 0.169 is much
larger than the 0.010 of the rectus abdominis. Based on
Eq. (15), it appears that the ideal stabilizer would have a
short length and a long moment arm, with moment arm
being the more important of the two.
Cholewicki and McGill (1996) have presented the
most complete, multi-joint method to examine multiple
muscle contributions to spine stability. Specifically,
matrices of the stiffness contributions of the different
system components have been examined, with respect to
the V of each generalized coordinate, to obtain the
eigenvalues and the overall stability determinant of the
system. However, the complexity of their method has
presented other researchers with a substantial obstacle
to quantifying joint stability, so that many provide only
qualitative interpretations based on muscle activation
patterns. While less complex, the present method does
allow for an accurate quantitative assessment of the
joint stability contributions of multiple muscle systems.
It must be stressed that the stability equations presented
here are limited to a single joint and a single degree of
freedom. However, the equation could be used for all
joints and axes in a complex system like the spine, by
sequentially calculating the origin, insertion and node
coordinates relative to the joint of interest. It should be
noted that it is the muscles acting in opposition to the
perturbation that affect stability, however muscle forces
will often be required on both sides of the joint to
achieve static equilibrium. In addition, any complete
analysis of joint stability must include the contributions
of passive tissue stiffness. The proposed equations
provide only for each muscle’s direct effect on stability.
Recent work by Stokes and Gardner-Morse (2003)
would indicate that muscle force will also indirectly
contribute to joint stiffness by increasing axial compression force. Any use of the proposed equations would
have to account for this factor, relative to the
characteristics of the specific joint being studied.
The primary advantage of this method is that it
provides a simple formula to quantify muscle contributions to joint stability under various static conditions.
Furthermore, this approach will be useful in dissecting
979
total joint stability into the individual contributions of
each muscle for various systems, joints, postures and
recruitment patterns.
Appendix A
Eq. (8) was substituted into (9) and a Taylor Series
expansion (third-order approximation) was applied to
yield:
ðBY AY ÞBX ðBX AX ÞBY
UðmÞ ¼F
y
1=2
½ðBX AX Þ2 þ ðBY AY Þ2 þ ðBZ AZ Þ2 F ðBX AX ÞBX þ B2Y ðBY AY ÞBY þ B2X 2
y
þ
2 ½ðBX AX Þ2 þ ðBY AY Þ2 þ ðBZ AZ Þ2 1=2
F
½ðBY AY ÞBX ðBX AX ÞBY 2
y2
2 ½ðBX AX Þ2 þ ðBY AY Þ2 þ ðBZ AZ Þ2 3=2
þ
k
½ðBY AY ÞBX ðBX AX ÞBY 2
y2
2 ðBX AX Þ2 þ ðBY AY Þ2 þ ðBZ AZ Þ2
þ 0ðy3 Þ;
(A.1)
substituting (5) into (A.1) and simplifying yields
AX BY AY BX
UðmÞ ¼ F
y
‘
F AX BX þ AY BY
ðAX BY AY BX Þ2 2
y
þ
2
‘
‘3
k ðAX BY AY BX Þ2 2
þ
(A.2)
y :
2
‘2
The first derivative of UðmÞ, with respect to a small
rotation about the z-axis, is also the moment equilibrium equation. This must add to zero (static equilibrium) in a stability analysis of all muscles in a system:
dUðmÞ
AX BY AY BX
¼F
dyZ
‘
AX BX AY BY
ðAX BY AY BX Þ2
y
þF
‘
‘3
ðAX BY AY BX Þ2
þk
y:
(A.3)
‘2
Stability is calculated as the second derivative of UðmÞ,
with respect to a small rotation about the z-axis. This is
also the first derivative of moment with respect to
rotation angle:
d2 UðmÞ
AX BX þ AY BY
ðAX BY AY BX Þ2
¼
F
‘
‘3
dy2Z
ðAX BY AY BX Þ2
þk
:
(A.4)
‘2
ARTICLE IN PRESS
980
J.R. Potvin, S.H.M. Brown / Journal of Biomechanics 38 (2005) 973–980
References
Bergmark, A., 1989. Stability of the lumbar spine: a study in mechanical
engineering. Acta Orthopaedica Scandinavica 230 (Suppl.), 1–54.
Cholewicki, J., McGill, S.M., 1995. Relationship between muscle force
and stiffness in the whole mammalian muscle: a simulation study.
Journal of Biomechanical Engineering 117, 339–342.
Cholewicki, J., McGill, S.M., 1996. Mechanical stability of the in vivo
lumbar spine: implications for injury and chronic low back pain.
Clinical Biomechanics 11, 1–15.
Cholewicki, J., Juluru, K., McGill, S.M., 1999. Intra-abdominal
pressure mechanism for stabilizing the lumbar spine. Journal of
Biomechanics 32, 13–17.
Crisco, J.J., Panjabi, M.M., 1991. The intersegmental and multisegmental muscles of the lumbar spine: A biomechanical model
comparing lateral stabilizing potential. Spine 16, 793–799.
Gardner-Morse, M.G., Stokes, I.A.F., 1998. The effects of abdomincal
muscle coactivation on lumbar spine stability. Spine 23, 86–91.
Gardner-Morse, M.G., Stokes, I.A.F., Laible, J.P., 1995. Role of
muscles in lumbar spine stability in maximum extension efforts.
Journal of Orthopaedic Research 13, 802–808.
Granata, K.P., Orishimo, K.F., 2001. Response of trunk muscle
coactivation in protecting against changes in spinal stability.
Journal of Biomechanics 34, 1117–1123.
Granata, K.P., Wilson, S.E., 2001. Trunk posture and spinal stability.
Clinical Biomechanics 16, 650–659.
Howarth, S.J., Allison, A.E., Grenier, S.G., Cholewicki, J., McGill,
S.M., 2004. On the implications of interpreting the stability index: a
spine example. Journal of Biomechanics, 37, 1147–1154.
Huxley, A.F., 1957. Muscle structure and theories of contraction. In:
Butler, J.A.V., Katz, B. (Eds.), Progress in Biophysics and
Biophysical Chemistry. Pergamon Press, New York, The Macmillan Company, pp. 6–318.
Ma, S.P., Zahalak, G.I., 1991. A Distribution-Moment Model of
energetics in skeletal muscle. Journal of Biomechanics 24, 21–35.
Panjabi, M.M., 1992. The stabilizing system of the spine. Part I.
Function, dysfunction, adaptation and enhancement. Journal of
Spinal Disorders 5, 383–389.
Stokes, I.A.F., Gardner-Morse, M., 2003. Spinal stiffness
increases with axial load: another stabilizing consequence of
muscle action. Journal of Electromyography and Kinesiology 13,
397–402.