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Protocol for constructing subject-specific biomechanical models of kneejointN. H. Yanga; P. K. Canavanb; H. Nayeb-Hashemia; B. Najafic; A. Vaziria

a Department of Mechanical and Industrial Engineering, Northeastern University, Boston, MA, USA b

Department of Physical Therapy, Northeastern University, Boston, MA, USA c Scholl's Center forLower Extremity Ambulatory Research (CLEAR), Rosalind Franklin University of Medicine andScience, North Chicago, IL, USA

First published on: 15 September 2010

To cite this Article Yang, N. H. , Canavan, P. K. , Nayeb-Hashemi, H. , Najafi, B. and Vaziri, A.(2010) 'Protocol forconstructing subject-specific biomechanical models of knee joint', Computer Methods in Biomechanics and BiomedicalEngineering, 13: 5, 589 — 603, First published on: 15 September 2010 (iFirst)To link to this Article: DOI: 10.1080/10255840903389989URL: http://dx.doi.org/10.1080/10255840903389989

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Protocol for constructing subject-specific biomechanical models of knee joint

N.H. Yanga, P.K. Canavanb, H. Nayeb-Hashemia, B. Najafic and A. Vaziria*

aDepartment of Mechanical and Industrial Engineering, Northeastern University, Boston, MA 02115, USA; bDepartment of PhysicalTherapy, Northeastern University, Boston, MA 02115, USA; cScholl’s Center for Lower Extremity Ambulatory Research (CLEAR),

Rosalind Franklin University of Medicine and Science, North Chicago, IL, USA

(Received 22 May 2009; final version received 6 October 2009)

A robust protocol for building subject-specific biomechanical models of the human knee joint is proposed which usesmagnetic resonance imaging, motion analysis and force platform data in conjunction with detailed 3D finite element models.The proposed protocol can be used for determining stress and strain distributions and contact kinetics in different kneeelements at different body postures during various physical activities. Several examples are provided to highlight thecapabilities and potential applications of the proposed protocol. This includes preliminary results on the role of body weighton the stresses and strains induced in the knee articular cartilages and meniscus during single-leg stance and calculations ofthe induced stresses and ligament forces during the gait cycle.

Keywords: knee biomechanics; subject-specific model; motion analysis; finite element method

1. Introduction

The human knee joint is comprised of many elements

including ligaments, menisci and muscles, which are

generally capable of bearing and transferring load during

various physical activities (Donzelli and Spilker 1996).

Understanding and identifying the loads placed on these

various anatomical tissues is critical for understanding and

studying realistic knee mechanics, stresses and strains and

to evaluate the true efficacy of any biomechanical

intervention. Of significant interest is studying the

underlying mechanisms of development and progression

of knee osteoarthritis (OA) – the most common subset of

OA – and subsequently, the development of effective

measures and instructions for prevention or delay of knee

OA. Pathologically, knee OA is associated with the loss of

articular cartilage, changes in the bone beneath the

cartilage, inflammation, muscle weakness and laxity of

associated knee ligaments (Atkinson et al. 1998; Petrella

and Bartha 2000; Felson 2004). These symptoms result in

decreased functional ability and decreased quality of life.

There are currently no effective treatment methods to

prevent or slow the progression of knee OA (Buckwalter

et al. 2004). Ironically, it has been suggested that pain

relieving medication may actually accelerate the rate of

osteoarthritic changes (Schnitzer et al. 1993; Hurwitz et al.

1999). Thus, there is clear evidence for the need to

enhance our understanding of the underlying mechanisms

of development and progression of knee OA. Several

biomechanical factors, in addition to age, gender and

genetics, are known to have a key role in knee OA risk.

These factors include meniscus injury, knee malalignment

and obesity (Sharma et al. 2001; Miyazaki et al. 2002;

Felson et al. 2004; D’Ambrosia 2005). Meniscus injury

and meniscectomy contribute significantly to the devel-

opment of knee OA. Injury to the meniscus, which is

amongst the most frequent injuries in orthopaedic practice,

and the consequent meniscectomy leads to excessive

stresses in the articular cartilage and therefore cartilage

degeneration as observed in several clinical observations

(Englund 2004). Knee malalignment (i.e. valgus and varus

malalignment) is another biomechanical factor that may

considerably increase the risk of OA (Johnson et al. 1980;

Andriacchi 1994; Sharma 2001). Moreover, the risk of

arthritis incidents generally increases with increased body

weight (BW; Felson and Chaisson 1997; Sharma et al.

2000).

The biomechanical models of the knee that estimate

the distribution of stresses and strains in the knee joint

and kinematics and kinetics of knee elements during

various postures (e.g. sitting, standing and lying), postural

transitions (e.g. sit-to-stand or stand-to-sit) and physical

activities (e.g. walking, stair climbing, turning, etc.) can

provide significant insight into the underlying mechan-

isms of knee OA and give an objective evaluation of

knee function. A majority of biomechanical models of the

knee joint are developed based on the finite element

(FE) method. These models provide significant insight

into the stress and strain distribution and contact

kinematics at the knee joint (Li et al. 2001; Haut

Donahue et al. 2002; Andriacchi et al. 2004; Besier et al.

ISSN 1025-5842 print/ISSN 1476-8259 online

q 2010 Taylor & Francis

DOI: 10.1080/10255840903389989

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*Corresponding author. Email: [email protected]

Computer Methods in Biomechanics and Biomedical Engineering

Vol. 13, No. 5, October 2010, 589–603

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2005; Fernandez and Pandy 2006; Pena et al. 2006) and

have been used to investigate the effect of ligament injury

(Li et al. 2002; Andriacchi et al. 2006; Yao et al. 2006)

and meniscectomy (Pena et al. 2005, 2006, 2008;

Zielinska and Haut Donahue 2006). However, in these

numerical studies, the knee joint was generally subjected

to axial loads with the knee flexion angle fixed at 08 (full

extension) and subject-specific data were not used to

define the joint geometry and loading conditions. To

address these shortcomings, here, we propose a robust

protocol for construction of subject-specific biomechani-

cal model of the human knee joint by combining

magnetic resonance imaging (MRI) of the knee joint, in

addition to motion analysis and force platform data to

construct subject-specific 3D FE knee models. The details

of the proposed protocol are described in Section 2. The

developed protocol is used for several studies to estimate

the stress and strain distribution as well as muscle and

ligament forces in a healthy subject during single-leg

stance and the gait cycle. These studies are discussed in

Section 3. Potential applications of the proposed protocol

and its implications in understanding knee mechanics are

discussed in Section 4.

2. Methods

2.1 3D knee joint geometry

As a starting point for constructing 3D biomechanical

models of the knee joint, a sagittal view MRI of the knee

was acquired of the subject in the supine, non-load bearing

position. This was performed in the early morning to avoid

the day-long weight-bearing of the knee joint. Subjects

were taken to the waiting room and then walked to the

MRI machine. The 2D images were loaded into the solid

modelling program Rhinoceros (Rhinoceros 3.0, Seattle,

WA, USA). The sagittal images were 150 £ 150mm and

were spaced 2mm apart with 256 £ 256 pixel resolution.

The boundaries of the different segments in each 2D MRI

were digitised (Figure 1(A)). Three-dimensional point

clouds were constructed by aligning each 2D segment in

its respective position as shown in Figure 1(B) and used to

define the surface geometry of the individual knee

components. An example of the knee joint geometry is

shown Figure 1(C).

Li et al. (2001) showed that variations of cartilage

thickness occur due to different investigators manually

digitising the cartilage boundaries, leading to different

contact stresses in FE simulations. To minimise variation

and error in the models, a careful procedure was followed

to ensure accurate and precise digitisation. The boundaries

of the different components (i.e. cartilage and bones) were

specified by placing points along the outer surface no

further than 1mm apart on each of the MRI slices. The

bone and cartilage were rigidly attached at their interface.

This method is a time-consuming task and is the reason

why only three subjects were used for comparison in this

investigation. In the current procedure, the thickness of the

cartilage in the 3D models was compared to the MRI; the

maximum difference of the thickness was no greater than

3.76% and the average difference was less than 1.50%.

2.2 Finite element model

The 3D geometry of the knee joint obtained using the

framework described in Section 2.1 was exported to

ABAQUS (Simulia, Providence, RI, USA), commercially

available FE software, to develop subject-specific

biomechanical models of the knee joint. A key challenge

in development of computational models of the knee joint,

and in general in biomechanics, is to select material

models capable of representing the behaviour of tissues

and organs under mechanical stimuli with high fidelity. In

our preliminary results presented in Section 3, the articular

cartilage was modelled as an isotropic elastic material, the

menisci as a transversely isotropic elastic material and the

ligaments as 1D nonlinear spring elements. However,

other material models of the articular cartilage and

meniscus (e.g. viscoelastic, poroelastic, three layer

cartilage model discussed in Vaziri et al. 2008) can be

implemented in the computational models depending on

the application under study.

In studying knee biomechanics, many studies assume

cartilage to behave as a linear elastic material (Li et al.

2001, 2002; Haut Donahue et al. 2002; Andriacchi et al.

2004, 2006; Besier et al. 2005; Pena et al. 2005, 2006,

2008; Fernandez and Pandy 2006; Yao et al. 2006;

Zielinska and Haut Donahue 2006). These assumptions are

justified considering the elastic response of cartilage

during activities involving loading frequencies greater

than 1Hz, such as walking or stair climbing (Besier et al.

2005). The shortcoming of these models is that the linear

elastic models of articular cartilage cannot predict the

temporal variations of the stresses in the knee. Here, the

cartilage was modelled as one layer isotropic elastic

material with an elastic modulus 15MPa and a Poisson’s

ratio of 0.45, which is in agreement with previous

numerical and experimental investigations (Kempson

1980; Blankenvoort et al. 1991; Blankenvoort and Huiskes

1996; Mommersteeg et al. 1996). The meniscus is stiffer in

the circumferential direction and this is due to the Type I

collagen fibres oriented primarily in the circumferential

direction of the meniscus. Various material and structural

models have been used to model the meniscus. For

example, Li et al. (2001, 2002) used nonlinear spring

elements to simulate the equivalent resistance of the

menisci. Pena et al. (2005, 2006a, 2006b, 2008) modelled

the meniscus as a continuous material with isotropic

elastic properties. Various models of the meniscus are

discussed in detail in the complementary article by Vaziri

et al. (2008). In this study, the meniscus was modelled as

N.H. Yang et al.590


transversely isotopic elastic with an elastic modulus of 120

Mpa in the circumferential direction and 20MPa in the

axial and radial directions, respectively, from the

previously published studies based on experimental data

(Haut Donahue et al. 2002; Zielinska and Haut Donahue

2006). The femur, tibia and fibula were modelled as rigid

structures because the bone is much stiffer than the soft

tissue at the knee, and to reduce computational time. The

validity of this assumption is established by Haut Donahue

et al. (2002). The cartilage was rigidly attached to the

bones at the femur and tibia interface.

In the FE models, the ligaments which attach the

meniscus to the tibial plateau at the meniscal horns were

modelled using 10 linear springs, each with a stiffness of

200N/mm and attached to the tibial plateau for a total

stiffness of 2000N/mm (Haut Donahue et al. 2002;

Zielinska and Haut Donahue 2006). The transverse

ligament was modelled as a linear spring with a stiffness

of 900N/mm attached to the anterior horns of the lateral

and medial meniscus (Haut Donahue et al. 2002; Zielinska

and Haut Donahue 2006). The location of the different

ligaments was obtained using MRI. Figure 2(A) shows an

example of the anterior cruciate ligament (ACL) in the

sagittal view MRI. The ACL (Figure 2(B)), posterior

cruciate ligament (PCL) (Figure 2(B)), medial collateral

ligament (MCL) and lateral collateral ligament (LCL)

(Figure 2(C)), were modelled as 1D nonlinear spring

elements according to their functional bundles based on

actual ligament anatomy. The ACL was modelled as

anteromedial and posterolateral bands. The PCL was

Figure 1. 3D geometry of the knee acquired by MRI. (A) Sagittal view MRI of the left knee. The bone geometry and the articulatecartilages are digitised for construction of the entire 3D geometry. (B) Point cloud of the 3D geometry of the tibia. (C) 3D geometry of theleft knee, which includes femur, tibia, fibula, articular cartilage and lateral and medial menisci.

Computer Methods in Biomechanics and Biomedical Engineering 591


modelled as an anterolateral band and a posteromedial

band. The MCL was modelled as a superficial portion and

a deep (inferior) portion. The superficial portion was

subdivided into an anterior portion and a posterior portion,

and the LCL was modelled with a similar method. Each

functional bundle of the ligaments was represented with a

nonlinear spring element with the following piecewise

force–displacement relationship (Blankenvoort et al.

1991)

f ¼

ð1=4Þk12=1l 0 # 1 # 21l;

kð12 1lÞ 1 . 21l;

0 1 , 0;

8>><>>:

where f is the applied force, k is the ligament stiffness

parameter, 1l is the constant nonlinear strain parameter of

0.03 determined from experiment (Blankenvoort et al.

1991) and 1 is the strain in the ligaments calculated from

1 ¼ (L 2 L0)/L0, where L is the ligament length and L0 is

the zero-load length of the ligament. The zero-load length

of the element was estimated from L0 ¼ Lr=ð1r þ 1Þ,

where Lr is the ligament’s initial length from the MRI and

1r is the reference strain that was found by Blankenvoort

et al. (1991). Table 1 shows the values of material

parameters for various ligaments modelled in our 3D FE

analysis (FEA). In Table 1, positive values of 1r correspond

to initial tension while negative values corresponded to

initially slack ligament bundles.

The boundary condition and loading should be also

identified in the FE models, which depend on the

application under study. In the studies discussed in

Section 3, the investigations were performed for static

single-leg stance and during the stance phase of the gait

cycle. In this case, the tibia and fibula were held fixed in all

translations and rotations. In static single-leg stance

simulations, the femur was held fixed at 08 flexion and all

other translations and rotations of the femur were

unconstrained. The loading applied to FE models was

obtained using motion analysis and force platform data

explained in Section 2.3 and was applied to the midpoint

of the transepicondylar axis in the femur. This point was

chosen because studies have shown that there is a fixed

axis in the femur that defines extension/flexion during gait

and this axis is very closely approximated by the

transepicondylar axis (Elias et al. 1990; Hollister et al.

1993; Churchill et al. 1998). The transepicondylar axis is

defined anatomically as the line passing through the

apexes of the medial and lateral femoral epicondyles

(Churchill et al. 1998). This is the location where the knee

joint reactions were calculated in the inverse dynamic

analysis and muscle force reduction models explained in

Sections 2.3.1 and 2.3.2, respectively.

2.3 Motion analysis and force platform procedure

The motion analysis and force platform procedure were

used to determine the kinematics and kinetics of the lower

limb during various activities and to define the loading

Table 1. Material constants used to model the ligaments.

Ligament BundleStiffness parameter, k

(N) 1r

Anterior cruciate Anterior 5000 0.06Posterior 5000 0.1

Posterior cruciate Anterior 9000 20.24Posterior 9000 20.03

Lateral collateral Anterior 2000 20.25Superior 2000 20.05Posterior 2000 0.08

Medial collateral Anterior 2750 0.04Inferior 2750 0.04Posterior 2750 0.03

Note: The stiffness parameters for different ligaments are from Butler et al. (1986)and Danylchuck (1975). The reference strain values are from Blankenvoort et al.(1991).

Figure 2. (A) Sagittal view of the left knee showing the ACL and its insertion locations into the femur and tibia. (B) Posterior view of theleft knee FE model. The springs represent the ACL and PCL. (C) Sagittal view of the knee FE model. The springs represent the LCL.

N.H. Yang et al.592


conditions in the FE models. Eight passive reflective

markers were placed at the bony landmarks to minimise

error associated with the large movement of muscle during

gait, as shown in Figure 3(A). These markers were used in

the motion analysis experiments. The position of the knee

and ankle joints was defined as the average position of the

lateral and medial femoral condyles and malleoli,

respectively, and defined with virtual markers in the

motion analysis software (Figure 3(B)). Virtual markers

also defined the position of the centre of mass (COM) of the

foot and lower leg section. The locations of the COM of the

different segments were defined as a function of the length

of the segment from statistical anthropometric data (Winter

2005).

Kinematics and kinetic data were collected with a six

camera motion analysis system (EVaRT 5.0, Motion

Analysis Corporation, Santa Rosa, CA, USA) and two

force plates (Models OR6-6-2000, OR6-7-2000,

Advanced Mechanical Technology, Inc. (AMTI),

Waltham, MA, USA; (Figure 3(C)). The motion cameras

recorded the position, velocity and acceleration of the

passive reflective markers while the force plates recorded

the ground reaction forces (GRF). The motion analysis

cameras recorded at a frequency of 120Hz over a capture

volume of 1.15 £ 2.00 £ 2.00 m. This frequency is

adequate for recording since the frequency of walking

gait is approximately 10–30Hz (Winter 2005). The global

coordinate system was defined with the z-axis normal to

the ground platform and the subjects’ frontal plane in the

x–z plane. This coordinate system was used rather than a

local anatomical coordinate system because it is stationary,

not subject dependent and not influenced by markers

which could move with the skin (Schache et al. 2008). The

force platform configuration consisted of two force plates

recording the ground reactions at a frequency of 1200 Hz

and time synchronised with the motion analysis system.

The ground reactions and centre of pressure (COP) were

determined by the force platforms. The coordinate system

of the force platform was aligned with the motion analysis

system. An inverse dynamic analysis was developed to

Force plates

A B

C

Figure 3. Motion analysis. (A) The marker set used in the motion analysis experiment. Passive reflective markers are placed atanatomical locations at (1) anterior superior iliac spine (ASIS), (2) the greater trochanter, (3) the lateral femoral condyles, (4) the medialfemoral condyles, (5) the lateral malleoli, (6) the medial malleoli, (7) the head of the second and (8) the second sacral vertebra (S2) of thespine. (B) The marker set including virtual markers representing the joint location and the position of centre of mass. (C) Motion cameraand force plate configuration used for gait analysis.



calculate the reaction forces and moments at the knee joint

from the motion analysis and the force platform trials; this

is described in Section 2.3.1.

2.3.1 Inverse dynamic analysis

First, the lengths of each leg segment were defined using

the marker set and the COM and mass moment of inertia

about the COM for each segment were estimated using

statistical anthropometric data (Winter 2005). The mass

for the foot and leg is assumed to be a percentage of the

total body mass Mb as 0.0145Mb and 0.0465Mb,

respectively. The mass moment of inertia I0 about the

COM for each segment about the x-axis (which is normal

to the sagittal plane as shown in Figure 4) is mr2o, where m

is the mass of the segment and r0 is the radius of gyration

of the respective segment. The radius of gyration for the

foot and leg can be estimated from 0.475Lfoot and

0.302Lleg, respectively. In general, for an arbitrary

segment, the forces and moments can be related to the

segment kinematic data, measured here using the motion

analysis, using classical 3D dynamics relationships.

Figure 4 shows the segment link models and the sagittal

view of the reflective markers placed at the foot

(Figure 4(A)) and lower leg (Figure 4(B)). In the coordinate

system shown, the z-axis is normal to the force platform and

the y-axis is normal to the frontal plane of the subjects. The

ankle joint reactions were first calculated in order to

determine the knee joint reactions using inverse dynamics

analysis. Then, the reactions at the knee were calculated

considering the reactions at the ankle and the accelerations at

the COM of the leg. Figure 5 shows the knee reaction forces

and moments calculated using the method outlined above.

These forces and moments were used to define part of the

loading conditions at the knee in the FE models.

2.3.2 Muscle force reduction model

The contribution of the muscle forces increases the total

compressive forces at the knee three to six times the BW

during the gait cycle (Kuster et al. 1997). Internal muscle

forces in the knee joint cannot be calculated using the

inverse dynamic method discussed above. To resolve this

issue, we used the ‘muscle force reduction method’ to

estimate the internal muscle forces during both single-leg

stance and walking according to the procedures developed

in previous studies (Morrison 1968, 1969, 1970;

Schipplein and Andriacchi 1991). To explain this method,

we have re-plotted a typical external flexion/extension

moment pattern, Mx, from heel strike to toe-off, in

Figure 6. The inset shows the schematic of the knee with

the corresponding muscle groups that act to oppose the

external moment. During the initial loading response, the

hamstrings (Hams) and the quadriceps (Quads) contract to

5

7

Foot COM

5

3

Leg COM

A

B

Figure 4. 3D segment link modelling of the lower leg used to determine the knee joint reactions. (A) Ankle joint reaction. The dottedline which attaches points 5 and 7 represents the foot segment. (B) Knee joint reaction. The dotted line which attaches points 3 and 5represents the lower leg. The arrows show the reaction forced and moments applied at each point.

N.H. Yang et al.594


provide stability. As the moment becomes an external knee

flexion moment, the quadriceps (Quads) muscle group will

act to oppose this moment. This assumption is valid since

the electromyography (EMG) studies have shown that the

quadriceps muscles group supplies the majority of the

muscle force to oppose the external knee flexion moment.

During the late stance phase, the gastrocnemius (Gast)

group will create an ankle plantar flexion moment for

propulsion. Directly prior to toe-off, the quadriceps group

acts to extend the knee. In contrast, the hamstrings act at

the beginning of the stance phase to counteract the external

hip flexion moment and provide stability at the knee and

the gastrocnemius acts during late stance phase, which is

the extension period (Morrison 1969). In general, the

moment arm and line of action of the muscles at the knee

vary with the sagittal plane knee flexion angle. To account

for the change in muscle direction with knee flexion, data

were taken from a previously published study by Kellis

and Baltzopoulos (1999) that gave the moment arm of the

patella tendon to the centre of rotation of the knee, the

angle of the patella tendon with the tibial plateau and

the moment arm of the hamstring muscle with respect to

the centre of rotation of the knee as a function of the knee

flexion angle (Table 2). The line of action of the hamstring

Figure 5. Knee reactions forces and moments calculated using the inverse dynamic analysis during gait for a healthy subject with BW of725N and no history of knee injury or prior knee OA. The force Fx is the medial/lateral force, Fy is the anterior/posterior force and Fz isthe axial compressive force. The moment Mx is the flexion/extension moment, My is the varus/valgus moment and Mz is theinternal/external moment.



muscle was assumed to be parallel to the femur during

knee flexion. The moment arm of the gastrocnemius

muscle to the knee joint centre was taken as 25 mm based

on the data in O’Connor (1993). The line of action of the

gastrocnemius ran parallel to the tibia and created no

additional shear forces.

In order to determine the force of the muscles, muscle

forces and moments are balanced in the model assuming

that there was no co-contraction of the flexors and

extensors. This led to a conservative, but overall more

realistic, estimation of the force at the knee joint. Figure 7

shows examples of muscle force calculated using the abovemethod based on the external flexion/extension moment

shown in Figure 6. The peaks in the model are due to the

action of the different muscles. The troughs are due to the

absence of the co-contraction of the antagonistic muscles

which would decrease the magnitude of the troughs if co-

contractions were included in the analysis (Morrison et al.

1969). Similar results were obtained from previous studies

(Morrison 1968, 1969, 1970; Schipplein and Andriacchi

1991). Figure 7 also shows the joint reaction forces

obtained from inverse dynamic analysis. The total

compressive and shear force plotted in Figure 7 are the

summation of the joint reaction forces and muscle forces

and were used to define the loading in our FE calculations.

3. Results

In this section, we provide several examples of the

applications of the proposed protocol for studying knee

biomechanics at different body postures and during

Figure 6. External flexion/extension knee moment from theinverse dynamic analysis used to determine the additional muscleforce contributions. The inset shows the sagittal view of the kneeand the location and line of action of the muscle groups whichoppose the external flexion/extension moment, Mx, and includethe hamstrings (Hams), gastrocnemius (Gast) and the quadriceps(Quads).

Table 2. Data used to define the moment arm and line of actionof the quadriceps muscle group and moments arm to the centre ofrotation of the hamstring muscle group from Kellis andBaltzopoulos (1999).

Knee flexionangle (8)

Quadricepsmuscle moment

arm (mm)Line ofaction (8)

Hamstringsmuscle moment

arm (mm)

0–10 36.9 135.7 29.911–3 39.3 126.7 25.421–30 40.9 118.2 26.631–40 42.5 112.8 28.241–50 42.6 107.5 27.951–60 41.7 101.0 28.361–70 41.7 96.8 27.871–80 40.5 94.8 24.381–90 39.5 92.6 20.5

Note: The line of action of the hamstrings muscle group was parallel to the femurbone during flexion. The gastrocnemius muscle group had a constant moment armand line of action.

Figure 7. The muscle force contribution, joint reaction forceand total knee force during the stance phase of the gaits cycle forthe (A) axial force and (B) anterior/posterior force. BW denotesthe subject body weight.

N.H. Yang et al.596


various physical activities. The provided examples include

studying the distribution of stresses and strains in the knee

joint during static single-leg stance and the role of BW in

stress distribution in the knee joint. Moreover, we used the

proposed method to estimate the stress and strain

distribution at the cartilage and the forces in the ligaments

during the stance phase of the gait cycle. The results of

these studies are described in the following sections. For

all of the following examples, a 23-year-old male subject

with no history of knee injury or prior knee OA with a BW

of 725N was recruited after Institutional Review Board

approval and signed consent from the subject was obtained

for the experimental procedure.

3.1 Distribution of stresses in knee joint

As the first example, we calculated the stress and strain

distributions in the knee joint during static single-leg

stance. The FE model of the left knee of the subject was

constructed using MRI as explained in Sections 2.1 and

2.2. We determined the knee joint forces and moments

and internal muscle forces using the inverse dynamic

analysis and muscle reduction model explained in

Section 2.3. The axial forces Fz, the posterior force Fy

and varus moment My were applied to the femur while

constraining the tibia and fibula. As discussed before, the

varus knee moment is a key factor in the overall

distribution of the force at the knee joint (Andriacchi

1994; Chao et al. 1994; Zhao et al. 2006, 2007; Schache

et al. 2008), which is neglected in most of the existing

3D FEA knee models. The material models used to

represent the behaviour of articular cartilages and

meniscus are linear elastic as discussed in Section 2.2.

In Figure 8, we show the distribution of normal stresses

and Tresca stress (i.e. maximum shear stress under

multi-axial state of stress) in the articular cartilages and

meniscus of the knee joint as obtained from our FE

model. Experimental results have related cartilage

damage with the magnitude of the normal stress and

strain (Repo and Finlay 1977; Kerin et al. 1998; Chen

et al. 1999, 2003; Zhang et al. 1999; Clements et al.

2001; Quinn et al. 2001; Borrelli et al. 2004; Morel and

Quinn 2004). Other studies have shown that the shear

stress is associated with increase in catabolic factors and

decrease in cartilage biosynthetic activity (Bachrach et al.

1995; Lee and Bader 1997; Andriacchi et al. 2004;

Heiner and Martin 2004). The results presented in

Figure 8 show that the stresses are concentrated on the

medial compartment of the femoral and tibial cartilages.

Furthermore, the percentage of the total normal force

distributed to the medial knee compartment for each

subject was approximately 81% for the varus subject,

79% for normal aligned subject and 78% for the valgus

subject. Thus, the total normal force distributed to the

lateral compartment was 19, 21 and 22% for the varus,

normal and valgus aligned knee, respectively. This

illustrates the importance of including the varus knee

moment, when studying knee biomechanics and may

explain why knee OA occurs more frequently in the

medial compartment of the knee compared to the lateral

compartment (Sharma et al. 2000; Engh 2003).

3.2 Preliminary results on the role of BW

Clinical longitudinal studies have related obesity to

increased progression of knee OA (Felson and Chaisson

Figure 8. Subject-specific FEA results of the normal stress distribution and maximum shear stress distribution on the femoral cartilage,meniscus and tibial cartilage for the left knee during single-leg stance.



1997; Felson 1999; Cooper et al. 2000; Sharma et al. 2000;

Sowers 2001; Cerejo et al. 2002; Englund and Lohmander

2004). The increased risk of knee OA is most likely due to

the increased mechanical loading at the knee joint. Here,

we used the proposed protocol to carry out a preliminary

study on the role of increased BW on the stresses in the

knee during static single-leg stance. To simulate change in

BW in this study, the GRF and moments were increased

linearly with the BW while assuming the same mass

distribution throughout the body. Then, the knee joint

reactions were recalculated with the inverse dynamics

equations and considering the muscle force contributions

estimated from the muscle force reduction method.

Although the muscle forces and line of action of

individuals may change with increase in weight, they

were kept constant in this analysis. Furthermore, the

lengths of the different segments of the lower leg were

kept constant as measured using the motion analysis

techniques. The location of the COP and the COM of the

different segments was also kept constant. BW increases

were only performed for static stance due to the change in

kinematics during gait that could occur with increased

weight and may produce inaccurate results. Similar to the

calculations performed in the previous section, the axial

forces, the posterior force and varus moment were applied

to the femur while constraining the tibia and fibula. Figure

8(A) shows the maximum value of the Tresca stress at the

medial femoral cartilage for a subject with simulated

change in BW. Figure 9(B) shows the Tresca stress

distribution at the subject’s normal BW (725N) and at a

simulated weight increase of 800N. Similar results were

also obtained for the normal stress and normal strain. This

preliminary investigation provides some quantitative

insight into the role of BW in the magnitude of the stress

and strain at the knee joint.

3.3 Stress variation and ligament forces during gaitcycle

In this part of the study, we calculated the stress and strain

distributions in the knee joint of a healthy subject during

stance phase of the gait cycle from heel strike to toe-off for

a single leg. Figure 5 shows the knee reaction forces and

moments calculated using the inverse dynamic analysis

during gait. Figure 7 shows the muscle force contribution,

joint reaction force and total knee force during the stance

phase of the gaits cycle obtained using the muscle force

reduction model. The forces and moment applied to the

femur were calculated using the method discussed in

Sections 2.3.1 and 2.3.2. Elastic material models were

used for the cartilage and meniscus and the knee ligaments

were modelled as 1D nonlinear elastic springs as discussed

in Section 2.2. Figure 10 shows the varus/valgus knee

moment during the stance phase of the gait cycle and the

corresponding normal stress distribution at different times

of the stance phase. At 10% of the gait cycle, the normal

stress distribution showed that a majority of the load was

carried on the lateral compartment due to the initial valgus

moment at heel strike. At 25% of the gait cycle, when the

compressive load and the varus moment were greatest, the

stress distribution showed that the majority of the load was

distributed to the medial compartment. At 65% of the

stance phase (the single-leg support phase), the load

appeared to be more evenly distributed between the medial

and lateral compartment due to the decreased varus

moment during the single-leg support phase. However, the

maximum values still occurred on the medial compart-

ment. At 75% of the gait cycle when the second peak axial

load and varus moment occurred, the majority of the load

occurred on the medial compartment. At toe-off (95% of

the stance phase), the stress distribution showed a majority

of the load on the lateral compartment due to the valgus

Figure 9. (A) The effect of BW on the maximum shear stresses in the articular cartilage. The calculations were performed using adetailed 3D subject-specific biomechanical model of the knee joint for a subject with a BW of 725N. (B) Maximum shear stressdistributions in the femoral cartilage of the right knee for simulated changes in BW.

N.H. Yang et al.598


moment. These results further emphasise the role of the

varus/valgus knee moment in determining the location of

the maximum stress at the knee cartilage.

An additional application of the developed model is

the calculation of the ligament forces. As discussed in

Section 2.2, the location of the different ligaments in the

FE model was obtained from the MRI data. The forces in

the respective ligaments during the stance phase of the gait

cycle are shown in Figure 11.

4. Discussions and conclusions

Human knee joint is comprised of many elements

including ligaments, menisci and muscles. Each of these

structures is capable of bearing and transferring load, and

their orientation and properties determine the extent of

load transfer to the articular cartilage. While various

interventions and surgical procedures, which are per-

formed for preventing knee OA or reducing its detrimental

effects, are based on redistributing the loading across the

knee joint, still there is a lack of fundamental under-

standing of the biomechanical factors that contribute to the

development and progression of knee OA. There are

currently no truly effective tools for functional assessment

of patients with knee laxity disability and for the outcome

of knee ligaments surgery. What exists nowadays are tens

of semiquantitative scoring systems that are more or less

valid, reliable and responsive, all differing from one

institution to another. Comparing or interpreting results is

very difficult. All these scores are subjective and static

Figure 10. Varus/valgus knee moment during the stance phase of the gait cycle determined from inverse dynamic analysis and the FEAresults of the normal stress distribution of femoral cartilage, meniscus and tibial cartilage of the left knee corresponding to different timesof the stance phase.



since they are based on questionnaires. In addition, knee

surgeons now require more subtle comparisons between

two potentially efficacious treatments (e.g. two types of

positioning or types of grafts for ligament reconstruction).

Therefore, the use of instruments that have increased

sensitivity and specificity in evaluating knee functioning

(e.g. laxity) compared to traditional scoring systems or

devices is needed to enhance the surgeon’s ability to assess

the overall outcome in patients after knee ligament

surgery. Recently, a few investigators have addressed

objective assessment of knee functioning pre- and post-

ligament reconstruction during activity of daily living

(Lewek et al. 2002; Gokeler et al. 2003; Knoll et al. 2004;

Favre et al. 2006). However, in these studies they mainly

focus only on kinematics data to assess knee laxity. While

kinetics of knee motion may provide more accurate and

sensitive outcome for assessing knee functioning pre- and

post-operation.

During single-leg support, approximately 70–75% of

the load passes to the medial compartment of the knee

joint due to the varus moment (Hsu et al. 1988; Andriacchi

1994; Andriacchi et al. 2004). Each subject exhibited

greater than 75% of the load to the medial compartment of

the knee including the muscle forces in the frontal plane

that may decrease the distribution of the total knee force to

the medial knee compartment. Haut Donahue et al. (2002)

calculated an even force distribution between the medial

and lateral knee compartments when applying an axial

compressive load of 800N to a 3D FE knee model. This

comparison shows the importance of applying the varus

knee moment in FE knee models.

The application of the varus knee moments led to each

subject demonstrating a larger magnitude of stress and

strain on the medial cartilage compared to the lateral

cartilage. Pena et al. (2006) applied only an axial

compressive load of 1150N and an anterior tibial load of

134N to a 3D knee model and found maximum normal

stress of 3.11MPa on the lateral femoral cartilage and

2.68MPa on the medial cartilage. The magnitude of the

normal stresses on the medial knee cartilage doubled with

the application of the varus knee moment in the current

model. This illustrates the importance of including the

varus knee moment when studying knee biomechanics and

may explain why knee OA occurs more frequently on the

medial compartment of the knee compared to the lateral

compartment (Sharma et al. 2000; Engh 2003). It is

difficult to define a specific value of stress and strain that

leads to cartilage damage and experimental values vary

based on the conditions of the experimental set-up (i.e.

loading rate, strain rate, specimen type, etc.). Morel and

Quinn (2004) showed that at strain rates of 7 £ 1024 s21,

no damage occurred to cartilage up to 80% axial strain.

However, multiple studies observed damage in the

cartilage at approximately 30% strain (Repo and Finlay

1977; Kerin et al. 1998; Zhang et al. 1999). Other studies

have shown cartilage damage under impact loading of

14MPa (Quinn et al. 2001; Morel and Quinn 2004) while

others have shown cyclic loading at stresses of 5–6MPa

decreases cell viability associated with the early signs of

cartilage damage and OA (Clements et al. 2001; Chen et al.

2003; Heiner and Martin 2004).

The magnitude of the ligament forces calculated in the

current model agreed with the previously published

studies. The maximum value in the ACL computed by

Morrison (1970) was 156N in a mathematical model,

303N calculated by Shelburne et al. (2005) in a 3D

computer model and 411N by Harrington (1976) in a

mathematical model. The maximum load in the LCL

computed by Morrison (1970) was 262N and Shelburne

et al. (2005) calculated as 150N. Our maximum LCL force

was greater compared to the previous studies but this could

be attributed to the varus moment generated during the gait

cycle or due to a difference in walking velocity. The

previous studies used normal healthy individuals and did

not consider varus alignment. Morrison (1970) observed

that the forces in the knee ligaments varied significantly

between individuals due to subject-specific gait charac-

teristics and knee joint geometry, which is in agreement

with our analysis. Woo et al. (1991) found an ultimate

tensile stiffness of 2160 ^ 157N from cadaver knees (25–

35 years old specimens) and found the ultimate strength to

decrease with increased age. The current investigation was

not designed to determine when ligament injury would

occur but could be used in future investigations to research

kinematics and kinetic conditions that put the knee

ligament at risk.

In this study, we described a robust protocol for

construction of subject-specific biomechanical models of

human knee joint, which can be used to determine the

stress and strain distribution in the knee joint during

various activities. Although many of the techniques in this

Figure 11. Ligament forces determined from FEA during thestance phase of the gait cycle.

N.H. Yang et al.600


methodology have been described by previous authors,

there has been no FE study that has attempted to

incorporate subject-specific joint geometry with loading

and boundary conditions based on subject-specific data. In

the proposed procedure, the loading and boundary

conditions at the knee were defined through an inverse

dynamics analysis and muscle force reduction model with

data from a motion analysis and force platform

configuration. Subject-specific knee joint geometry was

created by digitising sagittal view MRI and included the

bones, cartilage, meniscus and ligaments at the knee joint.

The proposed protocol offers an innovative and robust

approach to assess 3D kinetics of knee and the stress and

strain distributions in the knee-based subject-specific

biomechanical models of the human knee joint, MRI

imaging and measured kinematic data. This may open new

avenues for objective assessment of knee functioning pre-

and post-operation. We have used the proposed protocol to

study the stress and strain distribution in the knee cartilage

and meniscus during static stance, simulated changes in

BW and during the stance phase of the gait cycle.

Additionally, the forces within the knee ligaments were

obtained during the stance phase of the gait cycle. Future

studies will investigate the effect of different types of total

and partial meniscectomy on the stresses and strain at the

knee joint, as well as investigating other athletic activities

such as drop-landings and side-step manoeuvres. The

method described in this study is expected to provide

significant new insight into the underlying mechanisms

and biomechanical factors of knee OA. This, in turn, can

lead to development of better preventive and treatment

procedures to avoid knee OA and its detrimental effects.

The outcome of such investigations may assist clinicians

and medical doctors identify individuals that may be at

high risk of knee OA and provide guidelines and

preventive measures to reduce risk of knee OA. Moreover,

this study can help clinicians decide about the necessity of

the meniscectomy surgery and its extent as well as provide

better instructions to patients post-surgery.

Limitations of the current model include using

material properties based on experimental investigations.

High loaded regions of cartilage show increased

thickness and enhanced mechanical properties (Andriacchi

et al. 2006). Furthermore, with increased weight, the

proportions and location of the COM and COP may

change due to change in the weight distribution throughout

the body. However, predicting the exact weight distri-

bution in the lower extremities with increased BW is

impossible. Only static FE models were simulated with

increased BW due to change in kinematics during gait that

may occur with increased BW that would translate to

inaccurate FE results. However, this demonstrates

the ability of the current model to investigate the effect

of different biomechanical factors on the stress at the

knee joint.

The inverse dynamics analysis was based on subject-

specific kinematics and kinetics but the location of the

COM and the mass of the different segments were based

on statistical anthropometric data. Defining the COM from

anthropometric data is a limitation of the study but the

results showed that inertia effects had a small contribution

to the overall knee-joint reactions.

The methods used to determine the internal muscle

forces are general and are not based on subject-specific

data. Determining the moment arm of the muscles is

difficult during the entire stance phase of gait and

involves taking functional MRI. Data from weight

bearing or MRI taken at different knee flexion angles

provide data that could be used to define muscle moment

arms. Furthermore, use of EMG-driven models may

provide improved data for individual muscle forces when

applied to the FE model. However, existing FE studies do

not consider the muscle forces which significantly add to

the overall joint loading and neglecting these muscle

forces may severely underestimate the cartilage stress

and strain.

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