Journal of Biomechanics 45 (2012) 461–468

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Posture strategies generated by constrained optimization

Robert Pettersson a, Asa Bartonek b, Elena M. Gutierrez-Farewik a,b,n

a KTH Mechanics, Royal Institute of Technology, Osquars backe 18, SE-100 44 Stockholm, Swedenb Department of Women’s & Children’s Health, Karolinska Institute, Stockholm, Sweden

a r t i c l e i n f o

Article history:

Accepted 30 November 2011For people with motion disorders, posture can impact fatigue, discomfort or deformities in the long

term. Orthopedic treatments such as orthoses or orthopedic surgeries which change geometric

Keywords:

Posture

Multibody system

Static optimization

90/$ - see front matter & 2011 Elsevier Ltd. A

016/j.jbiomech.2011.11.053

esponding author at: KTH Mechanics, Roy

backe 18, SE-100 44 Stockholm, Sweden. Te

6 87969850.

ail address: lanie@mech.kth.se (E.M. Gutierre

: http://www.mech.kth.se (E.M. Gutierrez-Far

a b s t r a c t

properties can improve posture in these individuals. In this study, a model has been created to study

posture strategies in such situations. A 3D mechanical model consisting of eight rigid segments and

30 muscle groups is used in which varying moment arms along the ranges of motion and biarticular

muscles are considered. The method is based on static optimization, both to solve the load sharing in

the muscle system and to choose posture strategy. The optimization computes the specific posture with

minimal required effort (level of muscle activations), while fulfilling constraints containing subject

specific ranges of motion, muscle strength/weakness and external support if present. Anthropometry

and strength were scaled to each individual, based on reported pediatric anthropometry and strength

values, combined with each individual’s physical assessment. A control group of 10 able-bodied

subjects as well as three subjects with motion disorders were studied, and simulated posture was

compared with experimental data. The simulation showed reasonable to good agreement and ability to

predict the effect of motion disorders and of external support. An example of application in parameter

studies was also presented wherein ankle orthosis angles were varied. The model allows the user to

study muscle activity at the muscle group level, position of center of mass and moments at joints in

various situations.

& 2011 Elsevier Ltd. All rights reserved.

1. Introduction

An individual’s chosen standing posture could among able-bodied be generally the most comfortable posture, due to generalhealth, body build and flexibility (Rybski, 2004). In persons withmotion disorders the presence of lower limb contractures(restricted range of motion (ROM)), muscle weakness and spas-ticity can greatly affect standing posture (Bartonek et al., 2005).Contractures can be a consequence of spasticity during growth(Michael and Barnes, 2008), and weakness, of flaccid paresis(Beckung, 2002) or spasticity-reduction treatment, among others(Bjornson et al., 2007). For persons with deviating posture, it isdifficult to predict the effect of prescribed orthopedic treatment(Nicolopoulos et al., 2000). Furthermore many parameters can bechanged to affect posture: orthosis design (Owen, 2004), heel lifts(Eslami et al., 2006; Rodriguez and Aruin, 2002), muscle strength

ll rights reserved.

al Institute of Technology,

l.: þ46 87907719;

z-Farewik).

ewik).

(Kuo and Zajac, 1993), weight/obesity (Gilleard and Smith, 2007),and even anthropometry (Rybski, 2004). In these cases mechan-ical theory may help clinicians make more informed decisions.

In an ambitious and detailed musculoskeletal model, Amankwahet al. (2006) studied how passive joint properties affect postures,wherein reduced muscle effort and joint stiffness restrict theworkspace of possible postures. They showed that stiffness inplantarflexion and hip flexion can greatly affect required muscleeffort for static standing.

Even if several postures are possible, it is not always clearhow a subject may respond to a specific parameter change,e.g. orthotic treatment, altered strength, surgical release of con-tracture or weight change. Few orthotic treatment plans usescientific testing to evaluate their performance; it is thereforeimportant to understand the amount of control obtained throughorthotic treatment (Nicolopoulos et al., 2000). A mechanicalmodel with parameter studies on subject-specific data could bothhelp treatment decisions and give further insight of posturestrategies. The objectives of this study were to create such amodel to run subject-specific simulations allowing the user tospecify lower extremity properties found at routine clinicalassessment in order to have a feasible tool in a clinical context,

R. Pettersson et al. / Journal of Biomechanics 45 (2012) 461–468462

and to explore the model’s design and potential in predicting thestanding posture in three subjects with motion disorders as wellas in a group of able-bodied controls.

2. Method

2.1. Model

The model is a 3D system with eight rigid links and 20 degrees of freedom

(dof) (Fig. 1). Anthropometric data, i.e. segment inertia proportions, was derived

from Jensen (1986, 1989) scaled for age, body mass, height and segment lengths.

The pelvis segment length was defined as the distance between the midpoint of

hip joint centers and an approximated center of rotation point 50 mm anterior the

1st sacral vertebrae marker. The ground reaction force (GRF) was limited to act

under each foot segment, 1/10 foot length from the heel and 2/10 from the toe

(Hof et al., 2005) and without lateral width, i.e. the base of support (BoS) is the

resulting area of a parallelogram outlined by the feet. The inclusion of solid ankle–

foot orthoses (AFO) was performed using a stiff torsional spring with a specified

q7

q8

q9

q1

q2

q3

q6 q5

q4

q13

q14

q15

q16

q10

q18

q17

q20

q19

q12

q11

Fig. 1. The mechanical model with eight rigid links representing the feet, shanks,

thighs, pelvis and a head-arms-trunk segment, i.e. the joints/segments most likely

to be addressed clinically to improve standing. The single arrow head corresponds

to translational dof while the double arrowhead rotational dof according to the

right hand rule. Each dof are as follows: q1–q3—translational dof of the pelvis

segment, q4–q6—frontal, sagittal and oblique tilt respectively of the pelvis

segment, q7–q9 are the right hip ab/adduction, flexion/extension and rotation

respectively, q10 is the right knee flexion/extension, q11 and q12 are the right foot

plantar/dorsiflexion and inversion/eversion respectively, q13, q14 and q15 are the

left hip ab/adduction, flexion/extension and rotation respectively, q16 is the left

knee flexion/extension, q17 and q18 are the left foot plantar/dorsiflexion and

inversion/eversion respectively, q19 and q20 are the head-arms-trunk segments

posterior/anterior tilt and lateral tilt respectively.

neutral angle as observed experimentally. Joints’ limited ROMs were modelled

with exponential torsional springs MendofROM ¼�eð100ðcontracture angle�ðjoint angleÞÞÞ ,

which were practically inactive unless the end of the ROM was reached (sign

dependent on direction of the contracture).

Equations of motion were derived using Lagrange theory and were reduced

such that the model became static. The static postures were made stable by

constraining the vertical projection of the center of mass (COM) to lie within the

BoS. The model allows asymmetries in strength, anthropometry, ROM and

orthoses if present. The model may then freely choose any relative foot placement,

but with both feet in contact with the ground (i.e. heel rise was not allowed).

2.2. Muscle system

Fifteen muscle groups in each leg and two control moments representing

frontal and sagittal plane abdominal moments were modelled (Table 1). Muscles

were collected into groups representing joint extensors and flexors but divided

such that biarticular muscles were separated. A factor fðqÞ for each muscle group

about each joint was derived from SIMM (Delp et al., 1990) and describes each

muscle group’s maximum moment as a function of joint angle, based on each

muscle’s length and moment arm along the spanned joint’s ROM. The factor

was also normalized such that the sumP

ifi,jointðqjointÞ for fi : muscles at joint,

q : angle within ROMg has a maximum value of 1. Thus for each joint, fi,joint

represents the relative contribution of the i-th muscle to a specific joint config-

uration. This relationship between moment and joint angle was assumed to be the

same for all subjects while segment lengths were scaled to each individual’s

stature, age and gender (Morse et al., 2008; O’Brien et al., 2009).

Total strength of each muscle group was based on regression equations from

Eek et al. (2006) determined by age, weight and gender. Plantarflexor strength data

was not reported in children over 9, but was, on observation of Eek’s data,

approximately twice that of dorsiflexor strength in younger children. This assump-

tion was therefore applied here. Among the subjects in our study, weaknesses were

present, approximated during a clinical assessment with Manual Muscle Testing

(MMT) on a 0–5 scale (Hislop and Montgomery, 1995). Since MMT is a rough scale,

we have attempted to semiquantify strength; grades 0 (no strength), 3 (able to

resist gravity) and 5 (normal strength) have objective definitions, and the corre-

sponding strength for these grades were computed for each subject. Grade 4 has

been shown not significantly distinguishable from grade 5 (Aitkens et al., 1989) and

was therefore assumed as full strength. Grade 2 was assumed to be half of grade 3’s

strength. An activation variable, ai, for each muscle was then introduced and the

total available moment about a joint can be written as (shown for the knee)

Mtot ¼Mmax,flexionða2f2,kneeðqkneeÞþa4f4,kneeðqkneeÞþa14f14,kneeðqkneeÞÞ

þMmax,extensionða6f6,kneeðqkneeÞþa7f7,kneeðqkneeÞÞ ð1Þ

The passive ROM at a joint did not take into account the angle at adjacent joints.

Table 1Muscle groups and their functions.

Function Muscles No.

Dorsiflexion tib_ant, per_t, ext_d, ext_h 1

Plantarflexion med_gas, lat_g 2

soleus, tib_p, flex_h, flex_d, per_b, per_l 3

Knee flexion med_gas, lat_g 2

semimem, semit 4

bifemsh 14

Knee extension rect_fem 6

vas_med, vas_i, vas_l, pat_l 7

Hip flexion rect_fem 6

add_long, add_b, pect, grac 8

glut_med1, glut_min1, sar, tfl 9

psoas, iliacus 15

Hip extension semimem, semit, bifem 4

glut_max2, glut_max3 5

add_mag1, add_mag2, add_mag3 10

glut_med3, glut_max1, glut_min3 11

Hip abduction glut_med1, glut_min1, sar, tfl 9

glut_med3, glut_max1, glut_min3 11

glut_med2, glut_min2, peri 12

Hip adduction semimem, semit, bifem 4

add_long, add_b, pect, grac 8

add_mag1, add_mag2, add_mag3 10

psoas, iliacus 15

Hip internal rotation glut_med1, glut_min1, sar, tfl 9

Hip external rotation quad_fem, gem 13

psoas, iliacus 15

R. Pettersson et al. / Journal of Biomechanics 45 (2012) 461–468 463

2.3. Optimization/problem formulation

The general constrained optimization problem can be written as

minx

f ðxÞ subject to

cðxÞr0

ceqðxÞ ¼ 0

lbrxrub ð2Þ

where x is a vector containing all variables, f is the scalar function to be

minimized, c and ceq are the inequality and equality constraints, respectively,

and lb and ub are the variables’ lower and upper bounds. The biomechanical

analogies are as follows: variables are angles at each dof, activations in each

muscle group and external forces such as orthosis moment and GRFs; cost

function defines the strategy by which the individual chooses posture; constraints

on variables define ROMs and strength limits; and finally, the nonlinear

constraints contain the equilibrium equations, the equations to keep the COM’s

position within the BoS and the friction limit (Appendix A).

The problem becomes a complex constrained nonlinear optimization problem

(78 variables and approximately 214 constraints) and requires a nonlinear solver.

The function fmincon from Matlab (MathWorks) was used.

While the constraints deal with mechanical properties and conditions, the cost

function determines which of the feasible solutions to choose. The cost function

used was the sum of squared activations of each muscle group, and the posture

which minimizes this cost is chosen, corresponding to the least overall required

muscle activation.

2.4. Motion capture and clinical assessment

Motion analysis was performed using an eight-camera motion analysis system

(Vicon) with two force plates (Kistler) and a conventional, whole-body marker set

(Vicon, Plug-In-Gait). Quiet standing was recorded during 25 s, of which at least

5 s were chosen to represent the posture. In connection with motion analysis, the

subjects underwent a physical examination during which a physiotherapist

evaluated strength with MMT, measured passive joint ROM with a goniometer

and assessed presence of spasticity.

2.5. Application

The presented method was applied to 10 able-bodied controls and three

subjects with different motion disorders, and in a model-based parameter study.

Ethical approval was obtained. Participation was voluntary and in consent with

the children’s parents.

2.5.1. Control study

To test the simulation’s viability in predicting posture and postural changes,

standing data from a control group of 10 able-bodied children (5F/5M, mean(SD)

age 10.5(1.7)y, height 145(11) cm) who stood both with ordinary athletic shoes,

then with additional 30-mm heel wedges, was analyzed. Paired t-tests were used

to compare experimental vs. simulated pelvis angles, and a repeated measures

ANOVA was used to compare experimental vs. simulated hip flexion/extension,

hip ab/adduction, knee flexion/extension and ankle dorsi/plantarflexion angles.

Statistical comparison was then performed between the simulated vs. experi-

mental postural change due to heel elevation as a difference between the two

conditions.

2.5.2. Case 1

A male subject, 4.7 years old (height 111.5 cm, weight 19.7 kg), with arthro-

gryposis was studied. Passive ROM was in hip extension 01/01 (left/right), knee

extension 201/�101 and ankle dorsiflexion 201/�151, where 01 indicates a neutral

ankle or fully extended hip/knee and negative angles indicate that an angle less

than neutral was measured. The subject wore shoes and a solid AFO on the right

foot (151 shank anterior tilt). In this example the feet were constrained in the

model to have the heels aligned, as in the experiments. Standing with heel lifts of

10 mm, 20 mm, 30 mm was also studied. Strength in the left leg was normal (5)

(hip adductors (4)). Right knee extensor and hip adductor strength were nearly

normal (4) and hip abductors, plantarflexors and dorsiflexors had weakness (3).

2.5.3. Case 2

A female subject, 13.0 years old (height 157.0 cm, weight 41.4 kg), with

myelomeningocele was studied. Passive ROM in hip extension was 01/01 (left/

right), knee extension �101/�101 and ankle dorsiflexion �101/�101. Although

the dorsiflexion ROM was limited to �101, the shank orientation, standing with

heel–floor contact, could reach 321/221 (left/right) of anterior lean, due to subtalar

eversion, knee flexion and foot deformation. Therefore the shank orientation, gives

a better description of the dorsiflexion limit and was used as the constraint.

Strength in hip flexors, hip adductors and knee extensors was normal, hip

extension and knee flexion close to normal (4/4) but weakness was present in

hip abductors (2/2) dorsiflexors (1/1) and plantarflexors (1/1). Measurement and

simulation were performed both barefoot and with bilateral AFOs and shoes.

Two additional hypothetical situations were simulated for this subject in

barefoot stance: one with unlimited GRF placement under the foot, and one with a

penalty for resting against the end of the ankle ROM to represent discomfort.

2.5.4. Case 3

A female subject, 7.4 years old (height 119 cm, weight 21.9 kg), with

arthrogryposis was studied. Passive ROM was in hip extension 01/01 (left/right),

knee extension 151/151 and ankle dorsiflexion 01/01. The right leg was 1.5 cm

longer than the left. Solid AFOs and shoes were used in motion analysis, and the

measured shank angle (asymmetric) was used as input for the model’s dorsiflex-

ion limit. The subject had normal strength. Standing with a 30 mm heel lift was

also studied.

2.5.5. Parameter dependence, a simulation study

The neutral ankle angles of left and right AFOs were varied for a hypothetical

subject with normal strength in all joints except the ankles (anthropometry from

case 2 was used). AFO angles varied independently from 51 to 251 of dorsiflexion.

An asymmetry was studied with a limited knee extension of �201/�101 (left/

right). Otherwise normal ROM and strength were assigned.

3. Results

3.1. Control study

In the able-bodied group, the simulation predicted signifi-cantly more anterior pelvic tilt, slightly more pelvic lateral tilt,more hip flexion, slightly more hip adduction, and slightly lessknee extension than the experimental results. However, thesimulation predictions of the postural changes due to heelelevation were not significantly different from the experimentalvalues in any angle (Table 2).

3.2. Case 1 (Fig. 2)

The optimization suggested a posture with 48% body weightsupported by a hyperextended left knee (Fig. 2a). The actualsubject did hyperextend his left knee (Fig. 2b), but rested most(79%) of his body weight on it.

Heel lifts increased the required effort in the simulation(increasing value of computed cost function: f ¼ ½0:0363,0:0587,0:0966,0:1811�, for 0 mm, 10 mm, 20 mm, 30 mm respectively).The individual muscle groups’ computed activations (Table 3)were qualitatively similar for the four simulated postures. Themain differences in muscle activation due to heel elevation wereincreased activation of the left plantarflexors (Nos. 2, 3 in Table 1)and hip flexors (Nos. 6, 9, 15), right knee extensors (Nos. 6, 7) anda hip extensor muscle group (No. 11).

3.3. Case 2

The simulation’s suggested posture was similar to the subject’sactual posture (Fig. 3) in barefoot standing in the lower body butpredicted an upright trunk instead of the anterior trunk lean observedexperimentally. This was also reflected in the COM positions relativeto the ankles (Table 4 and Fig. 3) which are closer in the measuredposture. With AFOs the COM is positioned further away from theankle in the measured postures than in the simulated. The optimalposture with AFOs required 5% of the effort of the barefoot posture.

In the simulation with unconstrained GRF positions under thefeet (Fig. 4a), the computed posture required less effort than withthe constraint (Fig. 3a) but does not consider the instability of thisposture due to ankle muscle weakness. In the simulation withpenalty for resting against the end of ROM (Fig. 4b), the posturehad dorsiflexion and flexed knees, but unloaded the plantarflexors

SIMULATED

SIMULATED with HEEL LIFT MEASURED with HEEL LIFT

MEASURED

Fig. 2. The resulting posture in Case 1 with and without heel lift: (a) simulation

without heel lift, (b) measured without heel lift, (c) simulation with heel lift, (d)

measured with heel lift. In this figure, as in Figs. 3–5, there is a sagittal, frontal and

oblique view of each posture. The ground reaction force at each foot are plotted for

the simulated postures, where direction and origin together with relative size can

be seen. COM positions are also shown.

Table 2Statistical values of the model’s viability in terms of predicting chosen postures and postural changes due to heel elevation.

Segment orientation Posture without heel wedge Postural change with heel wedge

Mean (SD)

simulation

Mean (SD)

experimental

p-Valuea Mean (SD)

simulation

Mean (SD)

experimental

p-Valuea

Pelvis tilt, sagittal (1)b 26 (7) 12 (5) 0.001 0 (4) 1 (1) 0.839

Pelvis obliquity, frontal (1)c�3 (5) 0 (2) 0.033 �1 (5) 0 (1) 0.565

Hip flexion (1) Right 24 (7) 3 (5) o0:001 3 (6) 2 (1) 0.662

Left 24 (7) 4 (4) 1 (4) 1 (1)

Hip ad-/abduction (1) Right 3 (2) �2 (2) 0.034 �1 (2) �1 (1) 0.792

Left 2 (3) 1 (3) 0 (2) 0 (1)

Knee flexion (1)d Right 0 (1) �5 (3) 0.004 4 (9) 1 (2) 0.354

Left 0 (1) �3 (4) 0 (0.1) 0 (1)

Ankle dorsi-/plantarflexion (1) Right 2 (0.3) 2 (3) 0.818 �11 (5) �10 (2) 0.103

Left 2 (1) 2 (3) �13 (0.7) �11 (1)

a Only one p-value is shown for bilateral data, calculated from a repeated measures ANOVA with between-subject factor of side (left or right) and analysis type

(simulated or experimental).b Positive indicates anterior tilt.c Positive indicates right tilt.d Negative indicates hyperextension.

R. Pettersson et al. / Journal of Biomechanics 45 (2012) 461–468464

by moving the center of pressure close to the ankles throughposterior trunk tilt.

3.4. Case 3

The simulation suggested an upright posture of the pelvis andhead-arms-trunk (HAT) segments, while the subject insteadextended her knees and hips and compensated with slightlyposterior trunk tilt (Fig. 5). The COM’s position was also moreanteriorly located in the experiment. There was good agreementbetween the simulated and measured postures with 30 mm heel

elevation. The pelvic tilt in the frontal plane was present in bothsimulation and experiments. The simulated heel lift of 30 mm,however, required 11 times the effort level as without.

3.5. Parameter dependence, a simulation study

The computed cost function was lowest for AFO angles of 17–211and 13–171 (left/right) (Fig. 6a), which provide symmetric posturesdespite asymmetric knee contractures. The higher cost function witha left AFO angle of 51 is due to the model’s compensation of largeanterior trunk flexion, resulting in larger hip and lower backmoments. If it is clinically relevant instead to decrease left kneeextension moment (Fig. 6b), the optimal orthosis angle combinationwas the same. However, a left orthosis angle of 51 also results in lowleft knee moment.

In the knee extensors, specifically rectus femoris, the activa-tion level in the left leg was less for postures with highest costfunctions, i.e. postures with forward trunk lean. With forwardflexion of the upper body, it was not optimal to activate a musclewhich partly flexes the hip (Fig. 6c).

4. Discussion

The optimization model was able to study posture strategiesdependent on subject properties and orthopedic aids with reason-able accuracy and to speculate on optimal orthotic specificationsfor a particular case. We also, however, demonstrate the need tobetter mechanically describe the subject in order to draw moreprecise conclusions about the choice of posture and eventually aidin treatment planning.

The optimization model was able to successfully predict thepostural changes due to heel elevation observed in a group ofable-bodied persons. The differences observed between the simu-lated and observed angles are predominantly attributable to thedefinition adopted to define pelvic–HAT segment connection,evident in the large differences in anterior pelvic tilt and hipflexion (a relative angle), and decreasing differences in the moredistal joints. The present model assumes that the HAT’s weight istransferred only through the joint between the HAT and pelvis, andits location is therefore important for the determination of theneutral angle of the pelvis. Furthermore, the model does notconsider stretch resistance within the ROM at any joint, even thoughpelvic orientation is dependent on the trunk and femur orientation

Table 3Subject 1’s computed muscle groups activity level (range 0–1) for standing without and with heel lifts.

Muscles Without heel lift 10 mm heel lift 20 mm heel lift 30 mm heel lift

R L R L R L R L

tib_ant, per_t, ext_d, ext_h 0 0 0 0 0 0 0 0

med_gas, lat_g 0 0.0700 0 0.0800 0 0.0924 0 0.1741

soleus, tib_p, flex_h, flex_d, per_b, per_l 0.0013 0.0648 0.0018 0.0736 0.0011 0.0957 0.0041 0.1537

bifemsh 0 0 0 0 0 0 0 0

semimem, semit 0 0 0 0 0 0 0 0

rect_fem 0.0517 0.0512 0.0646 0.0738 0.0482 0.1004 0.0782 0.0777

vas_med, vas_i, vas_l, pat_l 0.0879 0.0002 0.1165 0.0002 0.0750 0.0165 0.2524 0

glut_max2, glut_max3 0 0 0 0 0 0 0 0

add_mag1, add_mag2, add_mag3 0.0114 0 0.0094 0 0 0 0.0146 0

glut_med3, glut_max1, glut_min3 0.0277 0 0.0467 0 0.0671 0.0248 0.1523 0

add_long, add_b, pect, grac 0 0 0 0 0 0 0 0

glut_med1, glut_min1, sar, tfl 0 0.0585 0 0.0920 0.0480 0.1515 0.0036 0.0977

glut_med2, glut_min2, peri 0.0074 0.0193 0.0180 0.0337 0.0600 0.0783 0.0414 0.0486

quad_fem, gem 0 0 0 0 0 0 0 0

psoas, iliacus 0 0.0495 0 0.0717 0 0.0821 0 0.0810

MEASURED BAREFOOTSIMULATED BAREFOOT

SIMULATED with AFO MEASURED with AFO

Fig. 3. The resulting posture in Case 2: (a) barefoot simulation, (b) barefoot

measured, (c) AFO simulation, (d) AFO measured.

R. Pettersson et al. / Journal of Biomechanics 45 (2012) 461–468 465

due to acting muscles about the hip joint. Therefore it is oftenoptimal to strive for more vertical orientation of the pelvis segment(resulting in large anterior tilt). The differences between experi-mental and simulated knee flexion angles, while significant, arerelatively small (3–51) and may still be within the experimentalmeasurement error. Furthermore no hyperextension was allowed inthe control models, which accounts for the most of the knee angleerrors. One possible improvement to the model would be to includepassive stretch resistance with a subject-specific neutral position.

The simulation in case 1 agreed in the subject’s strategy ofhyperextending the left knee. Because of passive resistance (not

included in cost function) at the end of the knee ROM, the modelmoved the COM left to unload the right leg such that weight wasalmost equally distributed between the feet. The subject howeverrested most of his weight on his left leg. A possible explanation isthat the subject experiences better control by leaning on thestronger side. In case 2 the good agreement can partially beexplained by the use of shank orientation to limit the ankleinstead of measured passive dorsiflexion ROM. In case 3 with lessimpairment, the results were similar to the control study; themodel overestimated the pelvic tilt (also shown in hip angle),while the knee and ankle angle were well-predicted.

Since no foot width or height was modelled, the prediction offoot placement in the lateral direction was primarily dependenton the GRF orientation to avoid hip ab/adductor activation, whichcould explain some differences in observed and simulated hipab/adduction. As a consequence the inversion/eversion momentabout the ankle has not been considered.

The postural changes observed and simulated between thesubjects’ different conditions (Table 4) differ by sign in somemeasurements, specifically in pelvic orientation and thereby affect-ing the relative hip angle, which was also indicated in the controlstudy. Furthermore, subjects 1’s and 2’s postures are more affectedby their more severe disabilities. In these cases a wider analysis oflocal optima, in addition to the global optimum presented here,could provide more understanding of alternative postures.

The choice of optimization criterion can and should be ques-tioned. The subjects’ choice of posture and the cost function aremost likely not on the same scale; i.e. the experienced effort isprobably not doubled when the cost function is doubled. It is alsomost likely subjective and person-specific. The inclusion ofpenalties to represent discomfort or to consider stability (e.g.Fig. 4) should also be considered for future developments.Furthermore, the muscle strengths used as constraints in theoptimization were rough estimates based on clinically availabledata; directly measured strength would increase applicability,and could be directly input into the model.

Parameter studies (e.g. Fig. 6) could give an overview of effectsand tendencies of specific treatments. They could also revealdifferent solutions depending on the goal of the treatment, sinceminimizing the overall activation vs. minimizing a specific mus-cle’s activation can lead to qualitatively different solutions.

Besides the control group, only three persons with motiondisorders were included in this paper. Our main objectives wereto present the method and discuss its potential and complicationsin more detail rather than to perform a full validation, whichwould require more subjects.

Table 4Quantitative data describing both the measured and the simulated postures.

Case 1 Case 2 Case 3

0 mm heel lift 30 mm heel lift Postural changesa Barefoot AFO Postural changesa 0 mm heel lift 30 mm heel lift Postural changesa

Posture

measure

Optimal

Fig 2a

Measured

Fig 2bb

Optimal

Fig 2c

Measured

Fig 2db

From

optimal

solutions

From

measured

postures

Optimal

Fig 3a

Measured

Fig 3bb

Optimal

(penalty

representing

discomfort)

Fig 4bc

Optimal

Fig 3c

Measured

Fig 3db

From

optimal

solutions

From

measured

postures

Optimal

Fig 5a

Measured

Fig 5b

Optimal

Fig 5c

Measured

Fig 5d

From

optimal

solutions

From

measured

postures

Cost function

value

0.0363 n.a. 0.1811 n.a. 0.1448 n.a. 0.1028 n.a. 1.2208 0.0050 n.a. �0.0978 n.a. 0.0050 n.a. 0.0537 n.a. 0.0487 n.a.

HAT’s COM

ant. pos. (cm)d

�1.1 �2.9 �1.0 �4.9 0.0 �2.0 �3.8 �0.9 �11.7 �1.0 �2.5 2.8 �1.6 �0.6 �5.4 �1.9 �5.8 �1.2 �0.4

HAT’s COM

lat. pos. (cm)e

�0.2 �2.6 1.0 �3.0 1.2 �0.4 �1.7 �0.8 4.6 0.1 �0.7 1.8 0.2 �0.7 �1.5 �0.8 �0.9 �0.1 0.6

Pelvis tilt,

sagittal (1)f

11.1 16.2 16.8 10.8 5.7 �5.4 30.9 16.0 �0.9 37.8 17.5 6.8 1.5 33.5 12.7 16.1 11.5 �17.4 �1.2

Pelvis

obliquity,

frontal (1)g

�11.3 �7.6 �7.6 �4.0 3.7 3.6 �11.5 �5.7 �13.3 0.7 �1.0 12.1 4.7 �5.5 �4.5 �7.6 �3.9 �2.0 0.5

Hip flexion (1)

R 15.5 22.1 25.3 13.6 9.9 �8.5 30.9 31.8 14.7 38.0 12.4 7.1 �19.4 31.6 3.0 16.7 9.1 �14.9 6.2

L 0.0 5.3 0.0 �3.2 0.0 �8.5 38.8 37.4 19.7 37.0 19.5 �1.7 �17.9 35.1 7.9 19.7 10.8 �15.3 3.0

Hip ad-/

abduction (1)

R �1.8 �0.6 �5.4 �8.6 �3.6 �8.0 11.0 �0.3 22.8 0.4 �4.8 �10.6 �4.5 8.3 0.9 6.9 0.3 �1.4 �0.6

L 13.4 2.4 8.1 �2.4 �5.4 �4.8 12.5 2.4 15.4 �1.9 1.7 �14.4 �0.7 3.2 4.7 10.4 9.7 7.2 5.0

Knee flexion

(1)

R 18.8 21.1 34.3 30.9 15.5 9.8 21.8 37.1 35.0 17.0 10.2 �4.8 �26.9 5.9 �2.6 22.0 14.7 16.1 17.3

L �20.7 �14.6 �20.5 �12.5 0.2 2.1 39.5 52.1 49.6 15.0 20.2 �24.5 �31.9 13.6 6.5 29.1 20.4 15.5 13.9

Ankle dorsi-/

plantarflexion

(1)

R 15.2 18.1 15.5 17.5 0.3 �0.7 22.2 13.4 21.1 16.8 2.6 �5.4 �10.7 7.9 8.6 8.2 5.5 0.3 �3.1

L �8.9 1.4 �14.3 �6.7 �5.4 �8.1 32.0 20.5 30.6 15.8 �0.5 �16.3 �21.0 12.1 11.3 12.3 7.5 0.2 �3.8

Weight

distribution

R 0.51 0.63 0.11 0.44 0.55 0.50 0.06 0.47 0.34 0.47 0.45 0.00 0.11

L 0.49 0.37 �0.11 0.56 0.45 0.50 �0.06 0.53 0.66 0.53 0.55 0.00 �0.11

GRF position

(cm)h

R 4.3 7.2 2.9 10.5 1.4 8.9 �1.6 3.8 5.1 9.3 6.8 5.5 1.7

L 1.0 2.9 1.9 12.1 1.4 8.7 �3.4 4.5 5.1 7.5 5.5 3.0 0.4

a Postural changes when adding 30 mm heel lift in case 1 and 3, and for with and without orthoses in case 2.b No force plate data available.c Including penalty term.d COM position relative to midpoint between hip joint centers in sagittal plane. Positive indicates anterior position.e COM position relative to midpoint between hip joint centers in frontal plane. Positive indicates positioning to the right.f Positive indicates anterior tilt.g Positive indicates right tilt.h The distance from ankle to the ground reaction force at each foot.

R.

Pettersso

net

al.

/Jo

urn

al

of

Bio

mech

an

ics4

5(2

01

2)

46

1–

46

84

66

SIMULATED, GRFunconstrained

SIMULATED,PENALTY included

Fig. 4. Simulation of subject 2. (a) Posture when not considering the stability of

the solution, i.e. allow the GRF to be placed under the ankle joint. (b) Posture when

considering discomfort.

SIM

SIM with HEEL LIFT MEASURED with HEEL LIFT

MEASURED

Fig. 5. The resulting posture in Case 3: (a) without heel lift simulation, (b) without

heel lift measured, (c) 30 mm heel lift simulation, (d) 30 mm heel lift measured.

510

1520

25

510

1520

250

0.5

1

1.5

2

2.5

Cos

t fun

ctio

n, s

quar

e ro

ot v

alue

510

1520

25

510

1520

250

5

10

15

20

25

30

Mom

ent [

Nm

]

510

1520

25

510

1520

250

0.1

0.2

0.3

0.4

0.5

Act

ivat

ion

leve

l

Right AFO angle [°]

Right AFO angle [°]

Right AFO angle [°] Left AFO angle [°]

Left AFO angle [°]

Left AFO angle [°]

Fig. 6. Results of parameter study with varying AFO-neutral angles of the right

and left sides separately. The figures show (a) the square root of the resulting cost

function value (purpose: to clarify the variations with low value), (b) required left

knee moment, (c) activation level of left leg rectus femoris.

R. Pettersson et al. / Journal of Biomechanics 45 (2012) 461–468 467

The presented model can be valuable in the early stage ofprescription or investigation of possible orthopeadic treatmentsto improve static posture strategy. The promising results withsubject-specific simulations, some even rather severely affectedby their disabilities, also demonstrates the model’s applicability,especially the model’s ability to predict postural changes due tocommon orthopeadic interventions. Probably the most compar-able study to ours is that of Amankwah et al. (2006). However,unlike theirs, the present model can be tailored to suit a widevariety of biomechanical constraints, and allows comparison toreal subjects; our simulations can be performed without theartificial posture constraints such as fixed stance width orenforced symmetry applied in the referred study. Also, the

present model applied minimization of muscle activations,instead of joint moments followed by a separate optimizationfor load-sharing, making the actual posture strategy based on thesubject’s individual muscle groups’ capabilities.

5. Conclusions

A model for studying posture strategies has been created andits design and potential have been explored. Moments or

R. Pettersson et al. / Journal of Biomechanics 45 (2012) 461–468468

activation of separate muscle groups can be studied in combina-tion with varying subject properties. The model suggests posturesbased on lower extremity contractures, weaknesses, heel lifts,orthoses and anthropometry for each specific case. Input data wasscaled according to reported pediatric anthropometry and eachindividual’s physical assessment. The simulation was able tosuccessfully predict postural changes due to heel elevation in acontrol group. We also found good qualitative agreement withcaptured data among the persons with movement disorders, butfurther investigation of the cost function, and more detailedanthropometric descriptions are required for more biofidelity.We are optimistic that the method has potential to providemechanical theory in clinical treatment decision-making.

Conflict of interest statement

There are no conflicts of interest present in the work.

Acknowledgements

The authors gratefully acknowledge the main financial supportfrom the Swedish Research Council. The financial supportfrom Skobranchens fond is also acknowledged. We thank alsoMarie Eriksson, CPO and Eva Brostrom, PhD, PT for their partici-pation in data collection.

Appendix A. Supplementary data

Supplementary data associated with this article can be foundin the online version at doi:10.1016/j.jbiomech.2011.11.053.

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