Biomechanical Analysis of Reducing Sacroiliac Joint Shear Load
by Optimization of Pelvic Muscle and Ligament Forces
J. J. M. PEL,C.W.SPOOR,A.L.POOL-GOUDZWAARD,G.A.HOEK VAN DIJKE, and C. J. SNIJDERS
Department of Biomedical Physics and Technology, Erasmus MC, PO Box 2040, Rotterdam 3000 CA, The Netherlands
(Received 5 April 2007; accepted 18 September 2007; published online 18 January 2008)
Abstract—Effective stabilization of the sacroiliac joints (SIJ)
is essential, since spinal loading is transferred via the SIJ to
the coxal bones, and further to the legs. We performed a
biomechanical analysis of SIJ stability in terms of reduced
SIJ shear force in standing posture using a validated static
3-D simulation model. This model contained 100 muscle
elements, 8 ligaments, and 8 joints in trunk, pelvis, and upper
legs. Initially, the model was set up to minimize the
maximum muscle stress. In this situation, the trunk load
was mainly balanced between the coxal bones by vertical SIJ
shear force. An imposed reduction of the vertical SIJ shear
by 20% resulted in 70% increase of SIJ compression force
due to activation of hip ﬂexors and counteracting hip
extensors. Another 20% reduction of the vertical SIJ shear
force resulted in further increase of SIJ compression force by
400%, due to activation of the transversely oriented M.
transversus abdominis and pelvic ﬂoor muscles. The M.
transversus abdominis crosses the SIJ and clamps the sacrum
between the coxal bones. Moreover, the pelvic ﬂoor muscles
oppose lateral movement of the coxal bones, which stabilizes
the position of the sacrum between the coxal bones (the
pelvic arc). Our results suggest that training of the M.
transversus abdominis and the pelvic ﬂoor muscles could
help to relieve SI-joint related pelvic pain.
Keywords—Static forces, Sacroiliac joints, Pelvis, Pelvic ﬂoor
muscles, Human posture.
The human body uses an ingenious 3-D framework
of bones, joints, muscles, and ligaments for posture
and movement. In upright posture, the trunk load
passes the sacroiliac joints (SIJ). The orient ation of the
SIJ surfaces, however, is more or less in line with the
direction of loading, which induces high shear forces
between sacrum and coxal bones.
The SIJ have a
strong passive, viscoelastic ligamentous system for
providing stability. These ligaments are vu lnerable for
creep during constant trunk load and need to be pro-
tected against high SIJ shear forces.
From a biome-
chanical point of view, an active muscle corset that
increases the compression force between the coxal
bones and the sacrum could protect the ligamentous
system and support the transfer of trunk load to the
legs and vice versa. Interlocking of the SIJ may be
promoted by transversely oriented muscles, e.g., M.
transversus abdominis, M. piriformis, M. gluteus
maximus, M. obliquus externus abdominis, and M.
obliquus internus abdominis, which has been described
However, due to the complex lines
of action of (counteracting) muscles and ligaments in
the pelvic region, it is difﬁcult to demonstrate the
contribution of transversely oriented muscles to SIJ
stability, in vitro as well as in vivo.
In the past, a number of biomechanical models of
the lumbosacral region (spine and pelvis) have been
developed to study the aetiology of low back pain
(LBP) in relation to (over)loading of the lumbar
and the pelvis.
Most of these models
dealt with mechanical stability in terms of muscle
and compression forces between (lumbar) verte -
A different approach is to relate LBP
to overloading of SIJ and nearby ligaments, for
example the iliolumba r ligaments.
The load trans-
fer through the SIJ was studied using a static, 3-D
biomechanical simulation model based on the muscu-
loskeletal anatomy of the trunk, pelvis, and upper
This simulation model calculates forces in
muscles, ligaments, and joints that are needed to
counterbalance trunk weight and other external forces.
It was shown that this simulation model underesti-
mated antagonistic muscle activity, but a good agree-
ment was found for agonist muscle activity. The
number of passive structures in the model was small,
for example no joint capsules were incorporated.
Therefore, the model was only valid for postures in
which none of the joints were near an end position.
Address correspondence to J. J. M. Pel, Department of
Biomedical Physics and Technology, Erasmus MC, PO Box 2040,
Rotterdam 3000 CA, The Netherlands. Electronic mail: j.pel@
Annals of Biomedical Engineering, Vol. 36, No. 3, March 2008 ( 2008) pp. 415–424
0090-6964/08/0300-0415/0 2008 Biomedical Engineering Society
The aim of the present study was to determine which
muscles have to become active in the 3-D pelvic sim-
ulation model when there is an imposed reduction of
the vertical SIJ shear force.
MATERIALS AND METHODS
The 3-D Simulation Model
The present study was performed using the vali-
dated, 3-D simulation model as described by Hoek van
Dijke et al.
The model is based on the musculoskel-
etal anatomy of the trunk, pelvis, and upper legs,
including muscle and ligament attachment sites, cross-
sectional areas of muscles and the direction of muscle,
ligament, and joint reaction forces. The geometry of
this model was based on structures extracted from
MRI slices and from previously publis hed data on
and upper leg
geometry. Figure 1a
illustrates, in frontal and median view, the bones on
which muscle and ligament forces act in the simulation
model. These are the lowest thoracic vertebra, ﬁve
lumbar verte brae, the sacrum, the left and right coxal
bones and the left and right femurs. The vertebrae are
treated as a single structure. The arrangement of the
bones depends on the static posture for which muscle
forces are calculated, for example standing with or
without trunk ﬂexion. A description of the model
equilibrium, its optimization scheme and validation of
some of the parameters is presented in the Appendix.
In the present study, we focus on the compression and
shear forces in the SIJ. These forces are represented as
perpendicular vectors. The normal vector of the SIJ
surface has an oblique direction (xyz = 0.365,
±0.924, 0.114). Compression force is deﬁned along
this normal vector and can only vary in magnitude.
One of the two components of the SIJ shear force was
deﬁned in the YZ-plane (xyz = 0, ±0.123, 0.992).
This force is denoted as the vertical SIJ shear force.
Directions of the SIJ compression force and the SIJ
vertical shear force in the YZ-plane are shown in
Fig. 1a, left panel. Figure 1b shows the vectors repre-
senting the most important muscle and ligament forces
in the pelvic region superimposed on the bones. Bone
shapes are for illustration purpose only; they are not
part of the simulation model. In total, the model
contains 100 vectors for muscle forces, 8 vectors for
ligament forces, and 22 vectors for joint forces; see
Table 1 for a list of all the structures.
Simulations and Dat a Analyses
A ﬁrst simulation, with the model in standi ng pos-
ture and a trunk weight of 500 N, showed that the
vertical shear SIJ force was 563 N on each side of the
sacrum. To ﬁnd the muscles that promote sacroiliac
joint stability, the maximum value for the vertical SIJ
shear force was decreased in steps of 30 N (~5% of
the initial vertical SIJ shear force). Theoretically,
lowering of the imposed vertical SIJ shear force to
0 N could induce a non-physiological equilibrium
between muscle, ligament, and joint forces. In addi-
tion, when the model was set in 30 ﬂexion, the force in
the iliolumbar, the sacrotuberal, and the posterior
sacroiliac ligaments was 250 N. This value was set as
a maximum physiological ligament force in the sim-
ulation model in the upright position to prevent
overloading of the pelvic ligaments that were imple-
mented in the model. The following criteria were
deﬁned to warrant a physiological solution for muscle
and ligament forces.
FIGURE 1. Panel (a) shows the bones on which the muscles,
ligaments and joint reaction forces act in the frontal plane
(left) and the median plane (right): the lowest thoracic verte-
bra, ﬁve lumbar vertebrae, the sacrum, the left and right coxal
bones, and the left and right femurs. The coordinate system is
deﬁned with the origin halfway between the rotation centers
of the hip joints. Axes: x posterior, y left, z vertical. Panel (b)
shows, superimposed on the bones, the vectors representing
the most important force components in the frontal plane (left)
and the median plane (right), see also Table 1. The labels in
this panel refer to a selection of the muscle structures listed in
PEL et al.416
1. Muscle tension must not exceed 240 kPa
2. Lowering of the maximum vertical SIJ shear force
must result in reduction of the total SIJ shear force
(combination of vertical and horizontal shear);
3. Ligament force must not exceed 250 N.
A muscle was included for further analysis when it
produced at least 15% of the maximum muscle stress
during the simulation. For all muscles, the maximum
muscle stress depended on the calculated minimum
muscle stress (see optimization criterion 1 in the
Appendix). Two muscle groups were analyzed sepa-
rately: (1) the muscles that increased at least 80% in
force after the ﬁrst simulation step and (2) the muscles
that increased at least 10 times in force after comple-
tion of the simulation series.
Table 2 summarize s the muscle (de)activation pat-
tern when the maximum vertical SIJ shear force was
stepwise decreased. Initially, vertical SIJ shear force
was 563 N (on each side of the sacrum) at a trunk load
of 500 N. The angle between the normal direction of
the SIJ surface and the direction of the total SIJ
TABLE 1. List of muscles, ligaments, and joints and the number of vectors describing the forces used in the simulation model on
transferring trunk load from lumbar spine via the pelvis to the upper legs (unilateral), see also Fig. 1.
Name of structure Number of elements Remark
1 M. adductor brevis 2 Upper and lower muscle
2 M. adductor longus 1
3 M. adductor magnus 3 Upper, middle and lower muscle
4 M. biceps femoris 1
5 M. coccygeus 1 Pelvic ﬂoor muscle
6 M. iliococcygeus 1 Pelvic ﬂoor muscle
7 M. pubococcygeus 1 Pelvic ﬂoor muscle
8 M. gemellus inferior 1
9 M. gemellus superior 1
10 M. gluteus maximus 2 Femur—sacrum and ilium muscle
11 M. gluteus maximus facia 2 Ilium—femur and trunk muscle
12 M. gluteus medius 3 Upper, middle, and lower muscle
13 M. gluteus minimus 3 Upper, middle, and lower muscle
14 M. gracilis 1
15 M. iliacus 1
16 M. longissimus 1
17 M. iliocostalis 1
18 M. multiﬁdus 1
19 M. obliquus externus abdominis 2 Ventral and dorsal muscle
20 M. obliquus internus abdominis 2 Ventral and dorsal muscle
21 M. obturatorius externus 1
22 M. obturatorius internus 1
23 M. pectineus 1
24 M. piriformis 1
25 M. psoas 2 Upper and lower muscle
26 M. quadratus femoris 1
27 M. quadratus lumborum 5 Sacrum—rib12, L1, L2, L3 and L4 muscle
28 M. rectus abdominis 1
29 M. rectus femoris 1
30 M. sartorius 1
31 M. semimembranosus 1
32 M. semitendinosus 1
33 M. tensor fasciae latae 1
34 M. transversus abdominis 1
A Iliolumbar ligament 1 Transversal plane
B Posterior sacroiliac ligament 1 Transversal plane
C Sacrospinal ligament 1
D Sacrotuberous ligament 1
I L5-S1 joint 3 Shear (two directions) and compression
II SI joint 3 Shear (two directions) and compression
III Hip joint 3 Shear (two directions) and compression
IV Knee joint 3 Shear (two directions) and compression
V Pubic symphysis 1 Compression
Sacroiliac Shear and Compression Forces 417
reaction force was 81, indicating that mainly vertical
shear force acted through the SIJ, see Fig. 2a. Force
equilibrium was mainly achieved by activation of M.
abdominal oblique (internus and externus), M. iliacus,
M. psoas, M. rectus abdominis, M. rectus femoris, M.
tensor fasciae latae, and loading of the sacrotuberous
When the maximum vertical SIJ shear force was
decreased from 563 to 443 N in steps of 30 N, the SIJ
compression force increased by about 70%. Force
equilibrium was obtained, amongst others, by activa-
tion of some of the muscles with a hip ﬂexion com-
ponent (M. adductor longus, M. iliacus, M. pectineus,
and M. sartorius, M. rectus femoris) and some of the
counteracting hip extensors (MM. gluteus medius an d
minimus and M. piriformis). Most of these muscles
became (more) active after we lowered the maximum
vertical SIJ shear force by 30 N. This led to unloading
of the sacrotuberous ligaments and loading of the
sacrospinal ligaments. The angle between the normal
direction of the SIJ surface and the direction of the
total SIJ reaction force was reduced to 72, indicating
that a combination of reduced vertical SIJ shear and
increased SIJ compression could ba lance the trunk
load on the sacrum, see Fig. 2b.
Further stepwise reduction of the vertical SIJ shear
force resulted in a sharp rise of the maximum muscle
stress. The simulation series ended with exceeding the
maximum physiological muscle stress when the vertical
SIJ shear force was decreased to about 60% of its
initial value. Surprisingly, activation of some of the hip
ﬂexors and extensors had decreased or even disap-
peared. This was not the case for the MM. gluteus
medius and minimus. In this simulation, force equi-
librium was obtained by activation of the transversel y
oriented M. transversus abdominis (ventral to the SIJ)
TABLE 2. Summary of the structures that stabilize the sacroiliac joints in terms of lowered shear.
Reduction of sacroiliac shear Initial Preset value (N)
SIJ (vertical shear) 563 533 503 473 443 413 383 353 323
Structures Force (N)
M. adductor longus 918181818 9
M. coccygeus 1111241020
M. iliococcygeus 1111241020
M. pubococcygeus 126
M. gluteus medius (lower) 711131415274430
M. gluteus medius (middle) 5 8 11 10 10 10 29 41
M. gluteus medius (upper) 42 90
M. gluteus minimus (lower) 333347
M. gluteus minimus (middle) 510111111 51012
M. gluteus minimus (upper) 817181817 5111625
M. iliacus 47 47 50 54 58 68 88 85 102
M. obliquus externus abdominis 21 18 17 14 12
M. obliquus internus abdominis 15 20 20 23 27 34 29
M. obturatorius externus 15 4 4 4 5 8
M. pectineus 818181818 6
M. piriformis 18 26 28 25 22
M. psoas (lower) 64 50 46 27
M. rectus abdominis 27 27 29 31 34 37 51 76 83
M. rectus femoris 34 34 36 39 42 49 64 62 50
M. sartorius 16 17 18 19 16 22 14
M. tensor fasciae latae 31 31 33 35 38 44 58 85 79
M. transversus abdominis 3 5 5 6 721325382
Iliolumbar ligament 53 250 250
Posterior sacroiliac ligament 26 49 62 73 147 250 250 250
Sacrospinal ligament 38 150 159 147 145 132 106 74
Sacrotuberous ligament 206 151 21
SIJ (compression) 92 121 130 142 154 229 473 607 633
SIJ (horizontal shear) -132 -141 -160 -154 -142 -154 -208 -226 -233
Total SIJ shear 579 551 528 497 465 441 436 419 398
Angle of SIJ reaction force () 817876747263433532
Maximum muscle tension (kPa) 37 37 39 42 45 53 69 125 247
Included are those muscles that produced at least 15% of the maximum muscle stress after each simulation.
The muscles printed italic increased at least 80% in force after the ﬁrst simulation step. The muscles printed bold increased at least 10 times
in force after completion of the simulation series.
PEL et al.418
and the pelvic ﬂoor muscl es, i.e., the M. coccygeus, the
M. iliococcygeus, and the M. pubococcygeus (caudal
to the SIJ). This resulted in further reduction of the
angle between the normal direction of the SIJ surface
and the direction of the total SIJ reaction force to 35,
see Fig. 2c. This indicates that the SIJ compression
force, which increased by about 400% and the reduced
vertical SIJ shear force, now clamped the sacrum
between the coxal bones, see Fig. 2c. The MM. gluteus
medius and minimus contributed to some extent to this
increased compression due to a distinct force compo-
nent in the transverse direction. To maintain force
equilibrium, increased SIJ compression led to loading
of the iliolumbar and posterior sacroiliac ligaments to
the preset maximum value of 250 N.
In the present study, the simulation model predicted
muscle and ligament forces in the pelvic region when
there was an imposed reduction of the vertical SIJ
shear force. Initially, the forces acting through the SIJ
were mainly vertical shear forces, see Fig. 2a. These
forces were not only caused by trunk load, but also by
muscles that acted in the longitudinal direction of the
spine, for example the M. psoas and M. rectus abdo-
minis. As a result of the forward bending moment, the
sacrotuberous ligament was loaded. This large liga-
ment protects the SIJ against excessive ﬂexion of the
sacrum relative to the coxal bones. The controlled
reduction of the vertical SIJ shear force with 30 N
forced some muscles that act as hip ﬂexors and hip
extensors to become acti ve. Due to their transverse
orientation, especially the MM. g luteus medius and
minimus and M. piriformis contributed to the
increased compression force between the coxal bones
and the sacrum. However, these muscles did not con-
tribute enough to self-bracing of the SIJ, because the
total force through the SIJ still mainly acted in vertical
direction. When the vertical SIJ shear was further
reduced to about 60% of its initial value, the simula-
tion model predicted that self-bracing mainly resulted
from the transverse muscles ventrally (M. transversus
abdominis) and caudally (pelvic ﬂoor) to the SIJ. In
this situation, some of the hip ﬂexors and extensors
reduced in activity, for example the M. piriformis.
Although the M. piriformis has a transverse orienta-
tion and crosses the SIJ, its contribution was mini-
mized by the simulation program because this muscle
also induces vertical SIJ shear force. The pelvic ﬂoor
muscles, the M. coccygeus and M. pubo-, and ilio-
coccygeus, contribute to the stabilization with respect
to the sacrum. It has been suggested that this stabil-
ization by force closure has an analogy with a classical
FIGURE 2. Directions of the force in the frontal plane exerted
by the right ilium through the SIJ on the sacrum as a reaction to
trunk load, F
. Panel (a): initial loading condition without
limitation of the vertical shear component (563 N, see under
‘‘initial’’ in Table 2). This condition led to loading of the sacro-
tuberal ligaments, F
(solid thick line). Panel (b):
loading condition with the vertical shear component preset at a
120 N lower level than the initial value (see under 443 N in
Table 2). This condition led to loading of the sacrospinal liga-
(solid thick line). Panel (c): loading condi-
tion with the vertical shear component preset at a 240 N lower
level than the initial value (see under 323 N in Table 2). In this
situation, SIJ compression force increased by ~400%, mainly by
M. transversus abdominis, F
, and the pelvic
, muscle forces. The location of these muscles
is schematically drawn by the thick solid lines, including the
M. pubococcygeus, the M. iliococcygeus and the M. coccygeus
(as drawn from the mid to the lateral position). It also led to
loading of the iliolumbar ligaments to the maximum allowed
of 250 N, (solid thick line). 3-D images copy-
right of Primal Pictures Ltd. http://www.primalpictures.com.
Sacroiliac Shear and Compression Forces 419
When sideways displacement of both ends
of the arc is opposed, mechanical equilibrium of the
stones is achieved by compression forces and not by
shear forces. In the pelvis, the pelvic ﬂoor muscles may
help the coxal bones to support the sacrum by com-
pression forces, while shear forces between sacrum and
coxal bones are minimized, see Fig. 3. Note that the SI
compression force is deﬁned as the force acting per-
pendicular to the SIJ surface. Therefore, decreasing or
increasing this force will not alter the shear forces. The
articular surfaces of the SIJ are irregular which results
in bony interdigitation in the SI joint space. SIJ shear
force calculated in the simulation model thus reﬂects
the combination of real joint friction and friction
due to this intermingling of bones. The real joint fric-
tion forces may be extremely small consider ing the
extremely low coefﬁcients of friction between the
articular surfaces. The majority of the shear force is
effectuated as normal contact pressures due to the
bony interdigitation in the SI joint space. It was not
possible to calculate the percentage of shear in terms of
joint friction force. This requires a more detailed
description of the SIJ surfaces.
The simulation model predicts that simultaneous
contraction of the M. transversus abdominis and pelvic
ﬂoor muscles, i.e., the M. coccygeus, the M. iliococ-
cygeus, and the M. pubococcygeus, contribute to
lowering of the vertical SIJ shear forces, increasing of
the SIJ compression and hence increasing of the SIJ
stability. We emphasize that this simulation model was
set up to estimate the forces acting in the pelvic region
under static conditions and that the outcome of the
simulations must be interpreted with caution.
theless, in a previous study co-contraction was shown
of pelvic muscles and M. transversus abdominis.
result and the prediction of our simulation model
suggest that a protective mechanism against high SIJ
shear forces may exist in humans. This mechanism has
been investigated in vivo an d in vitro. The contribution
of the M. trans versus abdominis to SIJ stability was
shown in an in vivo study in patients with LBP.
vitro study in embalmed human pelvises showed that
simulated pelvic ﬂoor tension increased the stiffness of
the pelvic ring in female pelvises.
It is worthwhile to
further investigate the contribution of both muscle
groups simultaneously, not only during stiffness mea-
surements of the SIJ but also during lumbo-pelvic
stability tests based on increased intra-abdominal
pressure (IAP). It was shown that the pelvic ﬂoor
muscles, in combination with abdominal muscles and
the diaphragm, may control and/or sustain IAP to
increase lumbar spine stability as well.
In the present study, the ligament forces were not
allowed to exceed 250 N. The distribution between
muscle and ligament forces depended on the maximum
muscle stress as formulated in the ﬁrst optimization
scheme as presented in the Appendix. Increasing the
maximum ligament forces might result in a lower
maximum muscle stress, which could lead to a diﬀerent
muscle activation pattern to stabilize the SIJ. A small
sensitivity test, however, showed that when the liga-
ment forces were allowed to exceed the 250 N up to
500 N and in a next step up to 750 N, the model cal-
culated a similar muscle activation pattern. The out-
come of the present study also depended on the choice
of optimization criteria and the magnitude of the cross-
sectional areas of the muscles. The inﬂuence of diﬀer-
ent criteria was previously investigated for muscle
forces in the leg.
Indeed, various choices led to dif-
ferent calculated forces, but the obtained solutions
were qualitatively similar, as was the case in our model.
When we developed the model, other optimization
criteria were also tested, for example minimization of
the sum of muscle forces. However, minimization of
the sum of squared muscle stresses yielded the
most plausible solutions. The model cannot account
for anatomical variations or detailed variation in
muscle attachment sites. Obviously, direct comparison
between the model predictions and the outcome of
in vivo force measurements in the SIJ are not available,
FIGURE 3. Analogy of pelvic bones supporting the trunk
with a classical stone arc. The M. transversus abdominis and
the pelvic ﬂoor muscles caudal to the SIJ mainly oppose lat-
eral movement of the coxal bones. Spinal loading is trans-
ferred mainly by compression forces through the SIJ to the
coxal bones and further down to the legs. 3-D images copy-
right of Primal Pictures Ltd. http://www.primalpictures.com.
PEL et al.420
so there is no data to conﬁrm the outcome of the
present study. Nevertheless, EMG recordings of
(superﬁcial) abdominal and back muscles in various
postures showed higher M. abdominal oblique inter-
nus activity when standing upright than resting on one
leg and tilting the pelvic backwards.
This muscle is
considered as one of the self-bracing muscles of the
SIJ. It was hyp othesized that when standing on one
leg, the shear load on the contralateral SIJ is dimin-
ished. Posterior tilt of the pelvis with less lumbar lor-
dosis may than lead to less M. psoas major muscle load
on the spine meaning less shear load on the SIJ. These
ﬁndings indirectly support our ﬁndings that trans-
versely oriented muscles reduce SIJ shear forces. We
emphasize that the present model served as a tool to
investigate the general relations between muscle an d
ligament forces in the pelvic region. The present
simulations results may lead to the development of a
new SIJ stabilizing training-program to reduce pain
induced by high SIJ shear forces. The effectiveness of
such a program , however, can only be tested with an
The simulation model predicted unloading of the
sacrotuberous and loading of the iliolumbar and pos-
terior sacroiliac ligaments when the vertical SIJ shear
was forced to reduce. This loading of the dorsal liga-
ments resulted from the absence of transversely ori-
ented muscles at the dorsal side of the SIJ to
counterbalance activation of the M. transverse abdo-
minis at the ventral side of the SIJ. Loading of the
iliolumbar ligament has been related to LBP.
shown that in sitting position, the stepwise backwa rd
movement of an erect trunk (from upright position
into a slouch ) resulted in forward ﬂexion of the spine
combined with backward tilt of the sacrum relative to
It was shown, that this mo vement into a
sudden or sustained slouch might cause loading of the
well-innervated iliolumbar ligaments near failure
The co-contraction that exists between the deep
abdominal M. transversus abdominis and the deep
back extensor M. multiﬁdus presumably retains lum-
In the future, we intend to extend
the model with co-contraction between the M. trans-
versus abdominis, the M. multiﬁdus, and the pelvic
ﬂoor muscles to study prevention of (over)loading of
pelvic ligaments at different static postures.
Eﬀective stabilization of the SIJ is essential in
transferring spinal load via the SIJ to the coxal bones
and the legs. A biomechanical analysis of the upright
standing posture showed that activation of trans-
versely oriented abdominal M. transversus abdominis
and pelvic ﬂoor, i.e., M. coccygeus and M. pubo- and
iliococcygeus muscles would be an eﬀective strategy to
reduce vertical SIJ shear force and thus to increase SIJ
stability. The force equilibrium in this situation
induced loading of the iliolumbar and posterior
sacroiliac ligaments. The M. transversus abdominis
crosses the SIJ and clamp s the sacrum between the
coxal bones. Moreover, the pelvic ﬂoor muscles oppose
lateral movement of the coxal bones, which stabilizes
the position of the sacrum (the pelvic arc).
This Appendix summarizes the model equilibrium,
its optimization scheme and model validation and
reliability. Part of the text is a short summary of pre-
viously published data.
A detailed description of the
model will be provided on request.
Biomechanical Model Equilibrium
Figure 1a illustrates, in frontal and median view, the
bones on which muscle and ligament forces act in the
simulation model. The origin of its coordinate system
is located between the rotation centers of the hip joints.
The ligament and joint forces are the result of coun-
teracting the external loads and the muscles forces. All
forces act along straight lines, and in principle, each
muscle (see also Table 1) or ligament has one line of
action, see Fig. 1b. The simulation model quantiﬁes
the muscle, ligament, and joint forces to warrant force
equilibrium of the complete model. In the present
study, the trunk load is the only external force applie d.
It is also possible to include an external reaction force
as a backing support. The trunk load transfers from
the lumbar spine, via the pelvis to the lower legs on the
basis of the prescribed posture. The static equilibrium
equations for the trunk, the coxal bones, and the
sacrum are based on a free-body diagram of each part.
All forces and moments exerted on each part, including
the weight, the external loads, the muscle, and ligament
forces, and the joint reaction forces, have to be in
equilibrium. Reaction forces in joints balance the other
forces. The trunk has a reaction force in the L5-S1
joint, the sacrum has additional reaction forces in the
SIJ, and the two coxal bones have reaction forces in
the SIJ and the hip joints. In the present study we focus
on the vertical SIJ shear forces. These are the net sum
of trunk load, muscle, and ligament forces, and reac-
tion forces acting in vertical direction on the vertebrae,
the sacrum, and the coxal bones. Reaction forces are
calculated in this order, that is starting with the trunk
and ending with the ground reaction force (which is
Sacroiliac Shear and Compression Forces 421
not given as output of the model). Leaving out the
ground reaction forces on the feet does not mean that
the model ﬂoats: these forces are just not needed in the
equilibrium equations of the trunk, the pelvis, and the
Force Vector Optimization Scheme
The pelvis was represented as three separate links
(left and right coxal bones and sacrum). The left and
right femurs and lowest thoracic vertebra were
included as links as well. Muscles and ligaments were
modeled as vector forces, and muscle, ligament, and
joint reaction forces between the separate links bal-
anced the loading of the pelvis. Several sources were
combined to complete the anatomical data set. A set of
164 transversal MRI-scans of a healthy young male
adult enabled the quantiﬁcation of the 3-D coordinates
of muscle and ligament attachments sites and the
magnitude of cross-sectional areas to calculate muscle
stresses. A computer program was written to describe
the positions of all attachment sites in diﬀerent pos-
tures. For each load situation, the equilibrium of the
links had to be calculated. Since the model is statically
indeterminate, additional criteria were formulated to
come to unique solution of muscle, ligament, and joint
reaction forces . The program GAMS
was used to
determine the optimum solution for the forces. It cal-
culated the magnitudes of 130 force vectors based on a
preset load (e.g., the trunk load) using ﬁrst a linear and
second a quadratic optim ization criterion. Since the
present study focu sed on static postures, the ﬁrst linear
criterion was as follows:
The maximum muscl e stress was mini mized. The
upper limit for muscle stress valid for all muscles
was minimized, which was considered as maximizing
the endurance time. This criterion only minimized
muscle stresses, not the ligament forces. The liga-
ment and joint reaction forces were just part of the
optimum solution for the muscle forces.
This criterion did not necessarily lead to a unique
solution. For example: in case a trunk muscle, e.g., the
M. erector spinae determined the lowest possible value
for the maximum muscle stress, still an inﬁnite number
of solutions may exist for balancing muscle forces
around the hip. Thus, a supplementary, quadratic
criterion was used:
The sum of squared muscle stresses was minimized,
with the additional constraint that no muscle stress
was allowed to exceed the value that followed from
the ﬁrst criterion. This led to distribution of the load
over synergic muscles within the boundary of the ﬁrst
criterion. However, strictly upholding this additional
constraint led to unrealistic activity of muscles when a
small decrease of the maximum muscle stress was
achieved by recruiting many muscl es. In addition, this
second criterion could enhance the increase of liga-
ment forces, since the constraint was only applied to
muscle stresses. To prevent that ligament forces
became too high, the maximum muscle stress was
allowed to exceed the calculated lowest maximum
muscle stress from the ﬁrst criterion by 10%. This
increase of the maximum stress was allow ed to reduce
the recruitment of inefﬁcient muscles, but it also
tempered excessive loading of ligament forces.
Figure 4 shows a ﬂow chart of optimizing muscle
and ligament forces and stresses in terms of position,
loading, forces, and stresses.
Model Validation and Reliability
The presented biomechanical model enabled the
analysis of load transfer from the upper body to
the pelvis and legs. The number of passive structures in
FIGURE 4. A ﬂow chart of the optimization scheme used to
calculate the optimum set of muscle, ligament and joint
reaction forces on the basis of geometry, positions, external
forces, and force limitations.
PEL et al.422
the model is small, for instance no joint capsules were
incorporated into the model; it was only valid for
postures that were not close to the ends of the range of
motion of the joints. All forces acted along straight
lines, and in principle, each muscle or ligament had one
line of action, see Fig. 1b. However, muscles with
broad attachment sites were either deﬁned by a number
of lines of action in parallel, e.g., the M. gluteus
maximus and the M. obliquus externus abdominis or
in series, e.g., the M. psoas major, see Ref. 15 for a
detailed description. For example: the M. gluteus
maximus had attac hments to the sacrum (partially via
the sacrotuberous ligament), the coxal bone and the
thoracolumbar fascia. Since forces in these three parts
had fundamentally different effects on the load in the
SIJ, this muscle was modeled as three separate force
vectors. The three lines of action that represented the
M. gluteus maximus, were given a relative cross-
sectional area according to the mass proportion of
these muscle parts.
The reliab ility of the model was tested by compari ng
calculated muscle stresses with measured EMG activ-
ities described in the literature. The experiments in the
papers that were selected had to be accurately de-
scribed and EMG activity of various muscles in at least
one load situation or EMG activity of a single muscle
in a variety of load situations had to be reported. This
resulted in model simulations of (1) forward hipﬂexion
with the trunk held straight,
(2) an isometric axial
trunk torque exercise,
and (3) various asymmetric
sagittal and frontal plane loading conditions.
average, a good correlation between muscle EMG
activity and calculated muscle stresses was found,
ranging from 0.869 to 0.984. For example, the nor-
malized EMG values (using a scaling factor 1% of the
maximum voluntary contraction of 0.0037 N/mm
and calculated muscle stresses at different hip ﬂexion
angles showed good proportionality (correlation coef-
ﬁcient of 0.904). This was not the case for the activity
of the M. quadratus lumborum (slightly higher EMG
value) and the M. erector spinae (slightly lower EMG
value). Overall, a good agreement was found for ago-
nist muscle activity, but antagonistic muscle activity
was underest imated. The aim of the present model was
the development of a tool to investigate the general
relations between muscle and liga ment forces in the
pelvic region, rather than to focus on the contribution
of individual muscles. To enable the use of EMG data
as input in the model, a facility was made to deﬁne
upper and lower limits for each muscle force vector. It
also provides the possibility to simulate co-contraction
of antagonistic muscles in the pelvic region. A similar
facility was made for ligament and joint reaction for-
ces. In the present study, the ligament forces were not
allowed to exceed a maximum of 250 N.
This research was supported by the Anna Founda-
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