Analytical Study in Kinematic of the Knee

7/25/2019 Analytical Study in Kinematic of the Knee

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ELSEVIER

Med. Erg. Phys.

Vol. 19. No. 1. 29-36.

p. 1997

Copyright 0 1997 Elsetier Scienc e Ltd for IPEMB. All rights resewed

Printed in Great Britain

PII: S13504533(96)00031-8

1350-4533/97 $17.00 + 0.00

Analytical study on the kinematic and dynamic

behaviors of a knee joint

Zhi-Kui Ling Hu-tJrng Guo and Stacey Boersma

Department of Mechanical Engineering and Engineering Mechanics, Michigan

Technological University, Houghton, MI 49931, USA

Received 1 September 1995, accepted 10 May 1996

ABSTFUCT

A knee model in the sag&al plane is established in this Jtudy. Specafically, the model is used to study the e&s of

inertia, articular surfaces of the knee joint, and patella on the behaviors of a knee joint. These behaviors includt the

joint surfnce contact point, ligament f&-es, instantaneous center and slide/roll ratio between the femur and tibia.

Model results are compawd to experimental cadaver studie.5 available in the literature, as well as between the quasi-

.statir and dynamic mod&. We found that inertia increases the sliding ten&ncy in the latter part of pexion,, and

lengthens the rruciate ligaments. Decreasing the curvature of the femur surface geometry tends to reduce the ligament

forcps and moves the contact points towards the anterior positions. The introduction of the patellar ligament in the

model seems to stabilize the behaviors of the knee joint as reflected by the behavior

qf

the instant centers and the

contact po int pattern on the tibia surface. Furthermore, we ,found that diff,

Prent magnitudes of the external load

applied to the tibia do not alter the qualitative behaviors of the knee joint. 0 1997 Elsevim Science for IPEfiiB. ,411

right,\ re.seroed.

Keywords: Knee kinematics, knee dynamics, knee modeling, geometric modeling

Med. Eng. Phys., 1997, Vol. 19, ?9-36,January

1. INTRODUCTION

A well-defined analytical knee model can be an

effective tool for understanding the functionality

of the largest musculoskeletal joint in the human

body. Statistics show that over two million cases of-

knee injury occur in the United States each year.

This model can provide a scientific explanation as

to the causes of these injuries. Therefore, preven-

tive measure can be taken to avoid them. Further-

more, a well-developed analytical model could

also be used efficiently to determine the effects of

system variables on the performance of the knee

joint, and to guide experimental and clinical

investigations. However, a comprehensive knee

model does not exist in the literature.

Analytical knee models have generally adapted

a four bar linkage methodology, by grounding

either the tibia or femurlm4. In these models, the

two cruciates are assumed rigid links with neutral

ligament fibers staying constant lengths during

flexion or extension. Furthermore, the articular

surfaces o f the femur and tibia are either simpli-

fied, or their effects are ignored completely.

Although these models have provided initial

understanding of the knee kinematics, they can-

Correspondence to: Zhi-Kui Ling.

not accurately portray the actual kinematic and

dynamic characteristics of a knee.

Other studies -

8 have adapted a quasi-static

approach towards the modeling of a knee.

Although the quasi-static models compensate for

the deficiencies of the four bar linkage model by

allowing the cruciate ligaments to change their

lengths, they still cannot take into account the

roles of inertia and other ligaments in the

behavior of a knee.

To consider the effects of inertia, three studies

have attempted to establish the dynamic model

of a knee. The f irst9 proposed a two-dimensional

dynamic model of a tibiofemoral joint. The model

was used to study the contact conditions between

the femur and tibia as well as the characteristics

of the ligament forces. The second study investi-

gated the role of ligaments and muscles as control

elements for a prespecified rolling and slidin

pattern. Recently, Abdel-Rahman and Hefzy

I?

presented a dynamic model which incorporates

additional ligament constraints between the

femur and tibia to the model by Moeinzadeh in

describing the tibiofemoral joint. However, the

entire articular surface of the femur was assumed

to be a circular arc. The contact point positions

and forces between the femur and tibia and the

ligament forces were studied. The existing

dynamic models do provide further understand-


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Analytical study on the behaviors of a knee joint: Zhi-Kui Ling et al.

ings of a knee joint. However, deficiencies still

exist. Specifically, the role of the patella in the

behavior of a knee joint is usually considered sep-

arately 12*13 Furthermore, the techniques for calcu-

lating the loci of instant centers and the roll/slide

ratio during knee motion are absent in the exist-

ing models. However, these are important factors

in studying the kinematics of a knee. Further-

more, the effects of the joint surface geometry on

the behaviors of the knee have not been carried

out in the literature quantitatively. Finally, the

role of inertia in the overall behavior of the knee

is still not quite clear.

The aim of this study is to address the above-

mentioned three issues in the existing twodimen-

sional analytical knee models. Specifically, a two

dimensional model shown in Figure I is intro-

duced. Formulation for both the quasi-static and

dynamic models of a knee is carried out. The

implicit Euler and Newton-Raphsons numerical

schemes are used to solve the established equa-

tions of motion and nonlinear equations.

Methods to determine the loci of the instant cen-

ters and roll/slide ratio are introduced. In

addition, the effect of the different articular sur-

face geometry of the femur on the behavior of the

knee is also investigated.

Results of the quasi-static model provide the

initial values for the simulation of the dynamic

model. Although no experiments are performed

in this study, results from the analytical model are

compared with the available experimental data in

the literature. These results include the ligament

forces, the contact conditions, the loci of the

instant centers, and the rolling/sliding behavior

between the femur and tibia.

The remainder of this paper is organized into

four sections. First of all, the analytical model and

its numerical solution are introduced. Secondly,

methodologies used to determine the iristant cen-

ters and slide/roll ratio are described. Results and

discussions are presented next, followed by a con-

clusion.

N = Normal Force

I I

Fl = LCL Force

I -t

Fixed Femur Fz = McL Force

F3 = ACL (mhrior) Force

F4 = PCL Qmaior) Force

FS = ACUposterlar) Force

F6= PCL (~tcrtor) Force

F7 = htdlr L@nen t Force

Pa:

Fext = Externd Force

Mext = Extend Moment

\ i $- Moving Tibia

2. ANALYTICAL MODELS

Because the fibula does not make contact at the

articulating surface of the tibiofemoral joint, its

effects are ignored in this study. The contours of

both the femur and the tibia in the sagittal plane

are acquired using the radiographs of a left, unat-

tached leg of a 6%year-old female cadaver, with

carcinoma reported as the cause of death. The

radiograph conditions include an unloaded leg,

horizontally positioned with the lateral side down.

Reference axes for the model are set up with

the Y axes centred along the bones longitudinal

axes, pointed towards the knee joint contact sur-

faces, as shown in Figure 1. In this study, the pro-

file of the femur is described with two segments,

as shown in equations (1) and (2). This reflects

the actual shape of the femur14. A second-order

polynomial in equation (3) is also generated to

describe the two-dimensional profile of the tibia

in the sa ttal plane. The maximum fit errors of

1.8 x10-

P

and 4.497 ~10~~ cm are found for the

two contours of the femur and the tibia, respect-

ively. The three profiles are as follows:

~~(x_=04b0110~~~4637x-0.13492 - 0.0332

(1)

fib(x) = 2.733 + &8144-(x + 2.692)2

(2)

f&d) = 21.34-0.2578x + 0.0477~ 2

(3)

Five major ligaments are represented in the

model. They are the medial collateral (MCL) , lat-

eral collateral (LCL) , anterior cruciate (ACL) , the

posterior cruciate (PCL), and the patellar liga-

ment. Both the ACL and the PCL are represented

by their anterior and posterior bundles. Their two-

dimensional insertion and origin points are

obtained from the literature5, and listed in Table

1. These numbers have been adjusted to the coor-

dinate systems discussed before.

In the following discussion, formulation of the

analytical model is divided into two cases, the

quasi-static model and the dynamic model. Two

constraint equations exist for both models. First,

the tibia surface must be in contact with the femur

surface at one point.

Where, x, y0 is the origin location of the tibia

with respect to the femur, fmxc and fmyc, tibxc

and tibyc are the femur and tibia contact points,

respectively.

The second constraint requires colinearity of

Table 1 Ligament insertion and origin coordinates (cm)

Ligament

Tibia X Tibia Y Femur X Femur Y

LCL 3.849 17.579 -2.5 1.9

MCL 2.149 16.079 -2.3 1.4

ACL (anterior) 0.849 21.079 -2.3 1.9

ACL (posterior) 1.149 21.079 -1.9 1.9

PCL (posterior) 3.849 20.579 -3.2 2.4

PCL (anterior) 3.849 20.579 -1.2 2.4

Fii 1 A knee model in its sagittal plane

30


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Analytical study on t/w behaviors of a knee joint: Zhi-Kui Ling et al.

the knee capsular provides negligible resistance,

a zero coefficient of friction is assumed, resulting

in only a normal force component.

The tibia is governed by three equations of

motion, as shown below. In this study, the tibia

mass is estimated at 3.45 kg, the mass for a 50

percentile malel, and the mass moment of inertia

is set to be 392.8 kg cm2.

Table 2 Ligament Stiffnesses

Ligament

k, (kg cm- SK*)

LCL

150 000

MCL 150 000

ACL (anterior) 200 000

ACL (posterior) 100 000

PCL (anterior) 175 000

PCL (posterior) 175 000

the unit normals at the respective contact points

on both the femur and tibia. This can be rep-

resented by the zero cross product of the two nor-

mals. The simplified form of this condition is

shown below.

(5)

In this study, the nonlinearity of ligament forces

for the cruciate and collateral ligaments is mod-

eled with the following expression.

muss

= (F,.), + W + A(nm) (%)y

R

kj( how,- Istart,) *

smagi= o

if howi &art,)

if< lnOWj5 hart,)

(6)

X[fmq$+ fma&l +

Here, j is an index number representing differ-

ent ligament. The stiffness of the collateral and

cruciate ligaments, kj, are taken from the litera-

ture.16, and are listed in Table 2, and lnow is the

current length of a ligament.

In equation (6), hart, the taut length of the

ligament j, is calculated by multiplying the initial

length at full extension of the ligament by its

strain ratio5. This is shown in equation ( { ), and

the strain ratios of the corresponding ligaments

are shown in Table 3.

Istart, = linit ial?q

(7)

In this study, the patellar ligament is assumed

to be in the sagittal plane during flexionl. The

magnitude of the pate110 ligament force is

obtained through the ratio between patellar liga-

ment force and quadriceps force versus flexion

angle . The insertion point of the patellar liga-

ment on the tibia is specified at (-0.251, 17.779)

with respect to the tibia coordinate system. The

angular orientation of the patellar ligament is ref-

erenced from the literature*.

Besides ligament forces, a force at the contact

point exists. Because the synovial fluid present in

Table 3 Ligament strain ratio

Ligament

Strain ratio

LCL 1.02

MCL

1.02

ACL (anterior) 1.05

ACL (posterior)

1.035

PCL (anterior) 1.05

PCL (posterior) 1.05

(9)

where Norm is the magnitude of the normal con-

tact force, and A can be either positive or negative

depending on the curvature.

In this study, the quasi-static model constitutes

equations (l-7), and equations (8-10) with the

left hand sides equal to zero. The dynamic model

consists of equations (l-10). For the quasi-static

model, Newton-Raphsons method is used to solve

for the six nonlinear equations, i.e. equations (S-

10) with zero accelerations, and equations (4) and

(5). The independent variable in these six equa-

tions is the flexion angle, and the six unknowns

are the tibia mass center (3~0, y,J, contact point

with respect to the femur and tibia (femxc, tibxc),

the normal force (nomn), and the required exter-

nal moment (K,,). The solutions starting from

0 with 10 increment up to 90 are calculated.

The solution at each of these positions is found,

if the tolerance is smaller than 0.0001.

For the dynamic model, there are also six equa-

tions: the three equations of motion, equations

(S-10)) and three algebraic constraints, equations

(4) and (5). Smce the six equations are a mix of

nonlinear and differential equations, this study

uses both the implicit Euler and Newton-

Raphsons in simulating the dynamic model. The

advantage of implicit Euler is its stability, however,

the method only provides a first order accuracy.

Iterations are performed at each time step until

convergence occurs. In this study, the time

increment for the implicit Euler method is set at

0.0001 s. The solution for each time step is achi-

eved if the tolerance of 0.0001 is achieved.

31


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Analytical study on the behaviors of a knee joint: Zki-Kui Ling et al.

2.1. Instant centers and slide/roll ratio

The instantaneous centers of rotation for both the

quasi-static and dynamic models are found by

using the instantaneous velocities of two points on

the moving tibia. They are the tibia contact point

and the tibia mass center. For the quasi-static

model, the line representing the velocity at the

tibia contact point is approximated by the line

connection between the contact point, and the

same contact point at its next position. The same

technique is used to determine the velocity line

at the mass center of the tibia. The instant center

is then determined by constructing two perpen-

dicular lines to the two velocity lines at the contact

point and the mass center. The intersection point

of the two perpendicular lines is the instant center

location at that particular instance.

For the dynamic model, the velocity at the mass

center of the tibia is known from model simul-

ation. However, the velocity at the contact point

is unknown. In this study, the x and y components

of the contact point velocity,

(dXtitml/&

df2( tibxc,)/dt), are found by using the contact

point and the same point at the next time step.

A first order forward difference approximation

scheme is used in the determination of the con-

tact point velocity. The same two perpendicular

lines to the velocities at the mass center of the

tibia and the contact point are constructed to

determine the corresponding instant centers.

To calculate the slide/roll ratio, the arc lengths

travelled on the surfaces of the tibia and femur

between two consecutive simulation times are

determined with the following numerical inte-

gration.

(11)

For either contour surface (fi or f2), the lower and

upper limits are the X components of two adjac-

ent contact points. The slide/roll ratio is defined

as the difference. between the larger distance (D)

and the smaller distance (d) travelled on the

femur and tibia over the smaller of the two arc

lengths travelled (d) .

3. RESULTS AND DISCUSSIONS

The already-established analytical models are used

to provide a comparison between the kinematic

and dynamic results in terms of the following

characteristics: the contact points on the femur

and tibia; the ligament forces; the instant center

locations; the slide/roll ratio, all with respect to

the flexion angle.

The effects of the articular surface geometry on

the dynamic behavior of a knee are also investi-

gated. This is accomplished with the reduction of

the curvatures of the femur surface. Specifically,

the coefficient for the linear term, 0.4637, of the

first femur profile in equation (1) is reduced to

0.4137 and 0.3637, respectively. The second pro-

file of the femur is also changed. In this case, the

Profun --

Pnnnoz.---

P lww3 - . .

KiMllUllO.0

I

x

-3.5 -

-4-

Figure 2 Femur contact points

radius of the circular arc in equation (2) is

reduced by 0.1 cm and 0.22 cm, respectively. The

original and the two new profiles of the femur are

identified as profile 1, 2 and 3 hereafter. Finally,

the effect of the patella on the knee behavior is

also studied with the model.

A constant impulse force with a magnitude of

20 N is applied along the x axis of the tibia with

a duration of 0.1 ..s in the analytical model.

Although the effect of different external loads on

the behavior of the knee is not the focus of this

study, a qualitative study is performed. In the

remainder of this section, the results of the afore-

mentioned studies are presented.

The contact point with respect to the femur and

tibia travels posteriorly with flexion as shown in

Figures 2 and 3. This is in aFeement with the

results reported by othersgs, . In Figure 2 it can

be observed that as the curvature of the femur

surface profile becomes smaller, contact points

shift towards the anterior direction. The inertia

has a greater impact on the contact point

behavior towards the latter part of the knee

flexion. From Figure 2, it is also shown that the

transition from the firs t to the second profile of

the femur is not perfectly smooth. This is due to

the fact that the slope at the connection point of

0

0 10 M 30 40 50 60 70 30 so

Flaxion Dqrn

Figure 3 Tibia contact points


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the two profiles is not completely continuous in

the modeling of the femur contour. The rest of

the results are also affected by this phenomenon.

In Figure 3, the contact point with respect to the

tibia travels posteriorly as well with flexion. How-

ever, the contact point moves in a faster rate

towards the posterior direction for the dynamic

model than for the kinematic model as flexion

exceeds 40. The effect of the patellar ligament

on the contact point pattern of both the femur

and tibia is barely noticeable.

Ligament forces in both LCL and MCL exhibit

maximum magnitudes at the full extension pos-

ition. As flexion starts, the ligament forces start to

decrease and are faded to zero before the full

flexion of 90 is reached. It is found that the effect

of inertia on the collateral ligaments is not appar-

ent. Furthermore, the ligament forces abate as the

curvature of the femur surface decreases. Intro-

duction of the patella in this study does not make

any difference in the behavior of the MCL, but

stretches the LCL ligaments. This is shown in Fig-

ure 4.

It may not be appropriate to compare the

results of the ligament forces from the analytical

models with those of experiments quantitatively,

since different boundary conditions were used in

those experiments and the analytical studies. Fur-

thermore, different opinions exist with regard to

the functions of ligaments20-24. Finally, most of the

experimental studies are quasi-static in nature.

Therefore, the discussion here on the ligament

forces is confined to qualitative comparison. For

the collateral ligaments, the results from this study

using both the quasi-static and dynamic models

qualitatively agree with those found in the exyeri-

mental studies. The experimental results21.2 ,24.L5

indicated that from full extension, the collateral

ligament forces have the maximum values, and

decrease as the knee flexes.

While the ligament force of the anterior PCL

increases and then decreases with respect to the

flexion angle, the force in the posterior PCL

decreases very rapidly, as shown in Figure 5. Fur-

thermore, while the force in the anterior fiber of

the PCL from the dynamic model is larger than

300

I

h

250 \

\

MCL with Pateliar l igament = -

I

I\

MCL without PaWar l igament =.-.-.-

200

LCL wkh Paldlar liint =

(

\

LCL withoul Patek tiit = +

f 1

T+

0 +++&

0

10 20 30 40 50 M) 70 80 90

Ftexion Degree

Figure4 h~fluence of the patellar ligament over the forces in the

collateral ligaments

Analytical study on the behaviors of a knee joint: %hi-Kui Ling et al.

Figure 5 The behaviours of the PCI. ligament force iu terms of in

antrrior and posterior bundles

that from the quasi-static model during the early

part of flexion, the trend reverses after flexion

angle passes 42.

As the curvature of the femur declines, the liga-

ment forces in both fibers of the PCL decrease.

While the patellar ligament has no effect on the

posterior fiber of the PCL, it stretches the anterior

fiber of the PCL in the early part of the flexion,

and provides relief for the fiber in the later part

of the flexion. From the modeling standpoint, this

can be explained with the fact that the posterior

fiber is used to provide the moment to balance

that produced by the patellar ligament in the early

part of flexion.

The analytical results of the PCL match with

those from experiments22. The difference exists in

the anterior fiber, where the maximum ligament

force occurs during the early part of flexion in the

analytical modeling. However, the overall trend of

the ligament forces follows the experimental

results. The effect of inertia on the posterior fiber

of PCL is difficult to observe, since the ligament

force becomes zero before 10 of flexion. The

small value of posterior PCL force was also indi-

cated by others 26,2. The anterior portion of PCL

exhibits a decrease in the ligament force when the

inertia force is considered in the later part of

flexion. This is probably due to the fact that the

inertia force acts along the same direction as the

ligament force in the anterior PCL during flexion

of the knee.

While the ligament force of the posterior ACL

increases with respect to the flexion angle, force

in the anterior ACL decreases and then increases

for the dynamic model. Yet, while the posterior

ACL increases and then decreases with respect to

the flexion angle, the anterior ACL decreases in

the kinematic model. They are shown in Figure 6.

The difference between the kinematic and

dynamic model is due to the presence of the iner-

tia, which changes the contact pattern on both

the femur and tibia as discussed in the previous

section. Consequently, the insertion point of the

ACL on tibia changes its pattern of motion at the

later stage of flexion, which causes itself to be

stretched in the process. As the curvature of the

33


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Analytical

on the behaviors

o?a

nee joint: Zhi-Kui Ling et al.

350,

250

ACL(a) Profile 1

ACL(a) P rofile 2

ACL(a) P rofile 3

ACL(a) Kinematic

ACL(p) Profile 1

ACL(p) Profile 2

ACL(p) Profile 3

.ACL(p) Kinematic

it

1 20 30 40 50 60 70 60 90

Flexion Degree

Figure 6 The behaviours of the ACL ligament force in terms of its anterior and posterior bundle

femur decreases, the ligament forces in both fib-

ers of the ACL decrease.

While the patellar ligament has little effect on

the pattern of the ligament forces in either fiber

of the ACL, the forces in ACL increase as flexion

increases. From the modeling standpoint, this can

be explained with the fact that the ACL is used

to provide the moment to balance that produced

by the patellar ligament in the later part of

flexion. For the anterior cruciate ligament, results

from the analytical model without the patella do

match with those from experiments, such as

France et a~ . The ligament force in the anterior

portion of the ACL decreases from 0 to 90 of

flexion, while the posterior portion increases from

0 to 50 and then decreases towards the 90

flexion.

The instantaneous centers obtained from the

quasi-static model follow a circular path, begin-

ning at around 20 of flexion anteriorly on the

proximal femoral condyle, and ending at 90 pos-

teriorly closer to the joint surface also on the

proximal femoral condyle. When the inertia is

considered, the instantaneous centers have the

same pattern as demonstrated by the quasi-static

model, nevertheless, they are located in the pos-

terior side of the instant centers from the kinem-

atic model. The effect of changing the surface

geometry of the femur on the instant centers is

minimal. However, the instant centers shift

anteriorly as the curvature of the femur decreases.

Patellar ligament constraint makes a big differ-

ence in the locus of the instant centers for the

dynamic model. Figure 7 demonstrates this differ-

ence where the model without the patella locates

the instant centers in the tibia side of the joint at

the higher degrees of flexion, while the instant

centers of the model with the patella are located

on the femur side.

The loci of the instant centers from experi-

mental studies are only available with the quasi-

static approach. These instant centers were found

using X-rays at incremental degrees of rotation**.

Therefore, comparison between the results from

34

x am (a)

Figure 7 Influence of the patellar ligament over the loci of the

instant centres

this study and the experimental results may not

be appropriate, as the methods used to determine

each individual instant center are different. How-

ever, for the quasi-static model, the analytical

results display the trend of the instant center locus

which is similar to the experimental results.

It can be concluded that rolling is dominant at

the beginning of flexion, and sliding becomes the

dominant factor as the flexion increases. There is

very little difference between the slide/roll ratio

of the quasi-static and the dynamic model during

flexion from 0 to 60. However, the ratio dips

lower for the quasi-static model when the flexion

angle exceeds 60. The change of femur curvature

has very little effect on the slide/roll ratio. Fur-

thermore, the patellar ligament facilitates the

increase of rolling in the latter part of knee

flexion, as illustrated in Figure 8. Although there

are no experimental results available, the

slide/roll ratio obtained from this study matches

with the consensus as related to the

ing versus rolling in the literature

r

attern of slid-

.

Although no graphs are presented in this paper

to illustrate the behaviors of the knee joint under


7/8

5

with patekr l igament = -

4.5 -

without pstellar lQa"Mt = - - -

/

4-

, -

/

3.5

-

, -

1-

Slii


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Analytical study on the behaviors of a knee joint: Zhi-Kui Ling et al.

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Analytical Study in Kinematic of the Knee

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