Journal of Biomechanics 38 (2005) 241–253

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doi:10.1016/j.jb

In vivo determination of normal and anterior cruciateligament-deficient knee kinematics

Douglas A. Dennisa,b, Mohamed R. Mahfouza,b,*, Richard D. Komisteka,b, William Hoffc

aUniversity of Tennessee, 313 Perkins Hall, Knoxville, TN 37996, USAbOak Ridge National Laboratory, Oak Ridge, TN 37831, USA

cColorado School of Mines, Division of Engineering, Golden, CO 80401, USA

Abstract

The objective of the current study was to use fluoroscopy to accurately determine the three-dimensional (3D), in vivo, weight-

bearing kinematics of 10 normal and five anterior cruciate ligament deficient (ACLD) knees. Patient-specific bone models were

derived from computed tomography (CT) data. 3D computer bone models of each subject’s femur, tibia, and fibula were recreated

from the CT 3D bone density data. Using a model-based 3D-to-2D imaging technique registered CT images were precisely fit onto

fluoroscopic images, the full six degrees of freedom motion of the bones was measured from the images. The computer-generated 3D

models of each subject’s femur and tibia were precisely registered to the 2D digital fluoroscopic images using an optimization

algorithm that automatically adjusts the pose of the model at various flexion/extension angles. Each subject performed a weight-

bearing deep knee bend while under dynamic fluoroscopic surveillance. All 10 normal knees experienced posterior femoral

translation of the lateral condyle and minimal change in position of the medial condyle with progressive knee flexion. The average

amount of posterior femoral translation of the lateral condyle was 21.07mm, whereas the average medial condyle translation was

1.94mm, in the posterior direction. In contrast, all five ACLD knees experienced considerable change in the position of the medial

condyle. The average amount of posterior femoral translation of the lateral condyle was 17.00mm, while the medial condyle

translation was 4.65mm, in the posterior direction. In addition, the helical axis of motion was determined between maximum flexion

and extension. A considerable difference was found between the center of rotation locations of the normal and ACLD subjects, with

ACLD subjects exhibiting substantially higher variance in kinematic patterns.

r 2004 Elsevier Ltd. All rights reserved.

Keywords: Kinematics; Helical axis; Registration; Anterior cruciate ligament deficient; Fluoroscopy; Normal knee

1. Introduction

The ‘‘in vivo’’ measurement of dynamic knee kine-matics is important for understanding the effects of jointinjuries, diseases, and evaluating the outcome of surgicalprocedures. Researchers have used ‘‘in vitro’’ (cada-vers), (Fukubayashi et al., 1982; Goldberg and Hender-son, 1980; Hsieh and Walker, 1976; Mahoney et al.,1994; Markolf et al., 1976, 1979; Muller, 1983; O’Con-nor et al., 1990; Singerman et al., 1994) noninvasive (gaitlaboratories), (Andriacchi et al., 1986, 1994; Andriacchi,1993; Draganich et al., 1987; Lafortune et al., 1992;Murphy et al., 1995; Wilson et al., 1996) and in vivo(roentgen stereophotogrammetry and fluoroscopy)

ing author. Department of Biomedical Engineering,

ennessee, 313 Perkins Hall, Knoxville, TN 37996, USA.

4-4159; fax: +1-865-974-7663.

ess: mmahfouz@rmmrl.org (M.R. Mahfouz).

e front matter r 2004 Elsevier Ltd. All rights reserved.

iomech.2004.02.042

(Chao, 1980; Dennis et al., 1998a,b. Kharrholm et al.,1994; Nilsson et al., 1991; Sarojak, 1998; Stiehl et al.,1995, 1997) approaches to assess human knee motion.Cadaveric and static X-ray measurement methods oftendo not accurately reflect loads encountered duringtypical movements, and often fail to reliably predictoutcome. Therefore, treatments aimed at improvingknee function should be evaluated using data obtainedfrom dynamic measurement methods. This requires thedetermination of six degrees of freedom (DOF) pose(position and orientation) of objects to be measuredduring dynamic activities.The most commonly used methods for assessing

dynamic movement rely upon skin-mounted markersor bone-implemented markers. However, external skin-markers are often unable to accurately represent motionof the underlying bone due to movement of soft tissuerelative to bone. To overcome the inherent inaccuracy ofthe skin-mounted markers, markers have been mounted

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Fig. 1. Volume rendered images (top) showing both knees of a normal

subject. The bottom images show the segmented bone models

superimposed on the volume data.

D.A. Dennis et al. / Journal of Biomechanics 38 (2005) 241–253242

on skeletal pins inserted into the underlying bones(Murphy, 1990), or inserted directly into the bones(Lafortune et al., 1992) to measure skeletal kinematics.Though these studies provide some of the best availablequantitative data during movement, the requirement ofskeletal pins or radio-opaque markers has limitedapplication for human studies. The objective of thecurrent study was to accurately determine the three-dimensional (3D) kinematic patterns of normal andanterior cruciate ligament (ACL)-deficient (ACLD)knees, under in vivo weight-bearing activities using anovel intensity-based 3D-to-2D image registrationmethod, similar to that previously utilized to analyzekinematics of total knee arthroplasty (TKA) (Banks,1992; Banks and Hodge, 1996; Dennis et al.,1996,1998a,b; Hoff et al., 1998; Mahfouz, 2002;Mahfouz et al., 2003; Sarojak 1998; Stiehl et al., 1995;Zuffi et al., 1999.).

2. Materials and methods

2.1. Model creation

The computer software packages used in the currentstudy were developed by the current authors with the aidof software development languages (Open InventorC++ library, and C++ Qt GUI development library;TGS, San Diego, CA). Ten healthy normal volunteerswith an average age of 37 years (range, 22–44 years), andaverage body mass of 76 kg participated in the study.The volunteers were scanned with MRI (Fast Spin EchoT2 FSE) and exhibited no lower extremity pathology orhad any measurable ligamentous instability on clinicalexamination (pivot shift and Lachman exams). Inaddition, five patients with recently isolated ACL tear(4–6 weeks) with an average age 39 years (range, 25–47),and average body mass of 65 kg were also included inthe study to compare their kinematics to the normalsubjects. The five ACLD patients performed a KT-1000test for laxity measurement and the manual max scorewas limited to 3mm or less.Spiral computed tomography (CT) scans of the

subjects’ (normal and ACLD) knees were made at levelsranging from 120mm proximal to the joint to 120mmdistal to the joint. These scans were made at 1–2mmintervals and the volumetric data of the knee joint wasconstructed at 0.5mm interpolation in the transverseplane. Segmentation of the CT-scanned bone wasautomatically performed by applying a thresholdingfilter to the slices which isolated the bone from softtissues. Manual intervention was conducted only whenthe thresholding filter failed. On completion of thesegmentation process, the resulting data were used tocreate full 3D polygonal surface models for the distalfemur (approximately 12 000 polygons), the proximal

tibia and the proximal fibula (combined approximately18 000 polygons) (Fig. 1).

2.2. X-ray fluoroscopy

The subjects were analyzed using a high-frequencypulsated video fluoroscopy unit (Radiographic andData Solutions, Minneapolis, MN). All subjects gaveinformed consent to participate in this study. The studyhas been approved by an institutional research reviewboard (IRRB #0607). Each subject subsequently per-formed weight-bearing deep knee bend activity whileunder fluoroscopic surveillance. During the deep kneebend activity, subjects were asked to begin in fullextension and flex the knee of interest to maximumflexion (Fig. 2). The fluoroscope maximum frame rate is30 frames/s. This frequency puts a limitation on themaximum frequency content of the kinematics data to15Hz to avoid aliasing (Nyquist criteria (Jain, 1989). Inpractice, all the activities that were analyzed with videofluoroscopy in the authors’ research have much lowerbandwidth. The fluoroscopic images of the deep kneebending activity were downloaded to a workstation forprocessing.The fluoroscope is modeled as a perspective projec-

tion image formation model. The perspective projectionmodel treats the fluoroscope sensor as consisting of anX-ray point source and a planar phosphor screen uponwhich the image is formed (Fig. 3). Although imagedistortion and non-uniform scaling can occur, these canbe compensated for by careful calibration (Mahfouzet al., 2003).

2.3. 3D-to-2D registration

Registering 3D models using 2D fluoroscopy imageshas been the subject of much research. Generally,

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Fig. 2. Single plane fluoroscopy allows the patient-free motion during deep knee bend activity in the plane between the X-ray source and the image

intensifier.

Fig. 3. The imaging model for the fluoroscope is perspective projection. X-rays are emitted from a point source, pass through the object, and strike

the image plane.

D.A. Dennis et al. / Journal of Biomechanics 38 (2005) 241–253 243

3D-to-2D registration methods fall into two differentcategories: feature-based methods and intensity-basedmethods (Penney et al., 1998). Feature-based methodsrely on matching features of the model to observedfeatures in the image. In X-ray images, this isusually done by matching points on the surface of themodel to the observed silhouette in the image (Banksand Hodge, 1996; Hoff et al., 1998; Lavallee andSzeliski, 1995; Gueziec et al., 1998). Accuracy can beimproved by using bi-plane (i.e., stereo) fluoroscopy(You et al., 2001) instead of single-plane (mono)imaging, but it would unacceptably constrain themotion of the patient.Intensity-based methods match the image values

directly to a predicted image of the object. A predictedimage is generated of the object in a hypothesized pose,and the pixel values are compared directly to the valuesin the actual input image, without trying to pre-segmentthe object from the image. A variety of image differencemeasures can be used, such as pattern intensity (Weeseet al., 1997), gradient difference (Penney et al., 1998) andcross-correlation (Lemieux et al., 1994). Since the

measures are global in nature, they are robust to smallamounts of clutter and occlusions.Whatever similarity measure is used, either in feature-

or intensity-based methods, it is still necessary to searchover the space of possible poses for the pose with thebest value of the similarity measure. This can be viewedas finding the minimum of a 6D function (threerotational, three translational DOF). A potentialdifficulty is that the function may contain local minima,or false solutions, which represent ambiguities in thepose of the model. For example, Fig. 4 shows a plot ofthe objective function for the registration of a femoralimplant model to an image. The similarity measure wasa type of cross-correlation (Mahfouz, 2002, Mahfouzet al., 2003). One of the two deep minima corresponds tothe correct solution; the other corresponds to a falsesolution that is a rotated version of the correct solution.The silhouettes are very similar.There are a number of strategies for dealing with the

problem of local minima. Pre-processing of the image tosegment the object from the background can reduce thenumber of local minima (Banks and Hodge, 1996;

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Fig. 4. Objective function values as the femoral implant model is rotated about two axes. The two poses shown correspond to the two large minima.

Note the many smaller local minima.

D.A. Dennis et al. / Journal of Biomechanics 38 (2005) 241–253244

Lavallee and Szeliski, 1995; Gueziec et al., 1998; Zuffiet al., 1999). However, it may be difficult to know inadvance which image points belong to the object insteadof the background (Mahfouz et al., 2004). Convergenceto the correct solution is more likely if one has a goodinitial guess (Mahfouz et al., 2003; Lavallee et al., 1996).The investigators have developed a 3D-to-2D regis-

tration intensity-based method that has been applied toknees, hips, shoulders, and other joints (Hoff et al.,1998; Mahfouz et al., 2001, 2003). The method isapplicable to implant components as well as normalbones. It registers a surface model of the object to asingle-plane fluoroscopy image, using a direct image-to-image similarity measure. Prior segmentation of thefluoroscopic image is not performed due to the inherenterror created (Mahfouz et al., 2004). The disadvantageof this method is that it can result in numerous localminima that make it difficult to find the correct solution.This problem is avoided through utilization of a robustoptimization algorithm (simulated annealing) that canescape local minima and find the global minimum (truesolution) (Mahfouz et al., 2003).Accuracy tests were performed on cadaver images

that were very similar to in vivo clinical X-rayfluoroscopy images, to allow a fair assessment of thealgorithm. A completely independent method using anoptical sensor was used for determining the groundtruth, unlike other work that uses ground truth derivedfrom X-ray data (Banks and Hodge, 1996; Lavallee andSzeliski, 1995; Penney et al., 1998; Weese et al., 1997).The registration method is highly accurate for measur-ing relative pose. The average errors in X, Y, and Ztranslations were �0.023, �0.086, and 1.054mm respec-tively (standard deviations are 0.473, 0.449, and3.031mm). Likewise, the average errors in x, y, and zrotations were �0.068, 0.001, and 0.253 degrees(standard deviations are 0.942, 0.771, and 0.841degrees). These numbers represent the errors in ourprocess plus the errors in the independent measurement

system as well (i.e. the upper bound) (Mahfouz et al.,2003). If we view the knee joint in the sagittal (XY)plane, then the relative translational motion in the Zdirection should be small and is not of interest in ourapplication.

2.4. Analysis of knee kinematics

On the workstation screen, perspective projectionimages of the model of the femur and the tibia (with thefibula) were processed according to their positionsrelative to the X-ray source and to the image intensifier.When both models of the femur and tibia are properlyregistered (overlaid), the relative pose between the twomodels should be the same as it was between the tibio-femoral joint at the time the X-ray image was created.Registering the models in the selected frames will resultin calculating the relative pose between the two bonesover the entire range of activity.The pose of each bone model is represented by a 4� 4

homogeneous transformation matrix H that is com-prised of rotation matrix and translation vector (see theAppendix). The rotation matrix Model

fluoroRxyzðg; b; aÞ is givenby (Craig, 1990)

ModelfluoroRxyzðg;b; aÞ ¼ RzðaÞRyðbÞRxðgÞ

¼

CosðaÞ �SinðaÞ 0

SinðaÞ CosðaÞ 0

0 0 1

264

375

�

CosðbÞ 0 SinðbÞ

0 1 0

�SinðbÞ 0 CosðbÞ

264

375

�

1 0 0

0 CosðgÞ �SinðgÞ

0 SinðgÞ CosðgÞ

264

375;

ARTICLE IN PRESSD.A. Dennis et al. / Journal of Biomechanics 38 (2005) 241–253 245

ModelfluoroRxyzðg;b; aÞ ¼

r11 r12 r13

r21 r22 r23

r31 r32 r33

264

375;

g ¼ A tan 2ðr32=CosðbÞ; r33=CosðbÞÞ;

b ¼ A tan 2ð�r31;ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffir211 þ r221

qÞ;

a ¼ A tan 2ðr21=CosðbÞ; r11=CosðbÞÞ;

where g, b, and a are the model’s angles ofrotations about x, y, and z axes, respectively, andthe translation vector is Modelpfluoro ¼

Modelxfluoro;

Modelyfluoro;ModelzfluoroÞT: Therefore, the relative pose of

the femur with respect to the tibia is then calculatedusing the equation Tib

FemH ¼TibFluoro H

FluoroFem H:

Two methods were utilized for analyzing the relativemotion of the femur and tibia. The first was the methodof screw axis decomposition, or the helical axis ofmotion (HAM). In this method, the axis in space aboutwhich the moving body rotates is determined. The

Fig. 5. Helical axes of motion corresponding to nine increments of knee flexio

geometrical center of the femur rotating around the screw axis is demonstra

location and orientation of the axis is defined withrespect to the coordinate system of the tibia. If the kneewas a simple hinge joint, with pure rotation about themedial axis, then the screw axis would be a stationaryline perpendicular to the sagittal plane. This is not thecase due to the complexity of knee motion whichincludes translation as well as rotation about other axes,resulting in the screw axis being not exactly perpendi-cular to the sagittal plane, and not fixed in space(Fig. 5). The screw axis was calculated betweensubsequent recordings of motion frames (i.e. betweenframes 1–2, 2–3, 3–4 etc.).The second method utilized to analyze the relative

motion of the knee was to track the contact paths of thefemur on the tibia. The minimum point on the surface ofthe medial and lateral condyles for each respectiveflexion angle was calculated automatically and projecteddown (vertical) onto the tibial plateau (Fig. 6). Theintersection of these projected lines from the femoralcondyles with the tibial plateau surface (in every flexionangle) are considered the contact paths of the femur on

n from 0� to 120�, shown in the sagittal (A) and frontal (B) planes. The

ted in the sagittal (C) and frontal (D) planes.

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Fig. 6. (A) A frontal view of the knee joint showing the minimum points on the femoral condyles projected on the tibial plateau. (B) A sagittal view

of the knee joint showing three flexion angles and their corresponding lowest points projected on the tibial plateau (C) Anteroposterior view of the

knee joint showing the three lowest points on the femur projected on the frontal plane. (D) Axial view of the minimum points projected on the tibial

plateau.

D.A. Dennis et al. / Journal of Biomechanics 38 (2005) 241–253246

the tibia because the articular surfaces and meniscus inCT are difficult to segment or model. These contactpoints are also located in the coordinate space of thetibia. This method is more standardized, and hencemore consistent than other methods that use the centersof the posterior femoral condyles (Asano et al., 2001;Iwaki et al., 2000) because the process for obtainingthese minimum points in our approach is automatedand reproducible, thus eliminating human errors. On theother hand, the other methods require averaging ofrepeated measurements conducted by human operatorswith a period of time between each measurement whichcan introduce errors to the process.All anterior–posterior measurements are made with

respect to a plane (frontal) that is located at thegeometric center of the tibia the geometric center iscalculated automatically. If the AP contact position ofthe condyle is anterior than this plane, the AP position isdesignated as positive. If the contact position is poster-ior than this plane, the AP position is designated as anegative value. For each selected frame of the X-rayvideo sequence, femorotibial contact paths were deter-mined for the medial and lateral condyles and plottedwith respect to knee flexion angle.The pivot position was determined by analysis of

medial and lateral condylar contact positions at full

extension and 90� flexion. Initially, the contact positionsof the medial and lateral condyles were determined. Aline was constructed between these two points at fullextension and at 90� knee flexion. An angle (y) then wasdetermined between these two lines. If the full extensionand 90� flexion lines converged on the medial half of thetibial insert in the coronal plane (apex of angle y, it wasdenoted that a medial pivot location occurred. If thelines converged on the lateral half of the tibial insert, itwas denoted that a lateral pivot location had occurred.If these lines remained parallel to each other and theangle appeared to approach infinity, no convergencewas found and it was denoted that there was no pivotlocation (Dennis et al., 2003).

3. Results

3.1. Normal knee

While performing a deep knee bend, all 10 normalknees experienced posterior femoral translation of theirlateral condyle and minimal change in the position ofthe medial condyle (Fig. 7, Table 1). The averageamount of posterior femoral translation of the lateralcondyle from 0� to 120� flexion was 21.07mm (standard

ARTICLE IN PRESSD.A. Dennis et al. / Journal of Biomechanics 38 (2005) 241–253 247

deviation 9.30), whereas the average medial condyletranslation was 1.94mm (standard deviation 1.86), inthe posterior direction. The majority of the posteriorfemoral translation of the lateral condyle seemed tooccur in the first 75� knee flexion (Fig. 8). Posteriorfemoral translation was not always continuous with

Fig. 7. The average normal subjects medial and lateral condyle contact

positions are plotted during a deep knee bend activity.

Table 1

Average anteroposterior femorotibial translation during deep knee

bend activity

Medial condyle (mm) Lateral condyle (mm)

Average STD Average STD

Normal 1.94 1.86 21.07 9.30

ACLD 4.65 3.99 17.00 7.31

Lateral and Medial Anterior Posteri

-20.00

-15.00

-10.00

-5.00

0.00

5.00

10.00

15.00

0 20 40 60Flexion A

Tra

nsl

atio

n in

mm

Average LAP

Average MAP

Fig. 8. Anteroposterior translation of the contact points b

increasing knee flexion because small amounts ofparadoxical anterior translation of femorotibial contactwere observed in mid-to-deep flexion which was morecommon medially (90%) than laterally (30%).All 10 knees experienced a normal axial rotation

pattern during a deep knee bend (tibia internallyrotating with increased knee flexion, i.e., screw homemechanism), because the posterior translation of femor-otibial contact laterally was greater than that observedmedially (Fig. 7, Table 2). The average amount of axialrotation for the 10 subjects from 0� to 120� flexion was23.67� (standard deviation, 9.56) in the normal direc-tion. The majority of this rotation occurred in the first30� knee flexion (average, 9.35�, Fig. 9). Rotationalmovements are best represented by describing HAM forthe motion. This axis is an imaginary line in space,around which the femur rotates. Because of the out ofplane motion of the knee (six DOF), the axis is almostnever perpendicular to the sagittal plane The helical axisroutinely translated posteriorly during progressive kneeflexion (Fig. 10). Therefore, posterior femoral transla-tion observed was due to a combination of pure APlinear motion and rotation of the femur relative tothe tibia.

or Translation For Normal Patients

80 100 120 140ngles in Degrees

etween the femur and the tibia during knee flexion.

Table 2

Average axial femorotibial rotation during deep knee bend activity

Axial rotation (deg)

Average STD

Normal 23.67 6.09

ACLD 9.56 3.23

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3.2. ACLD knee

While performing a deep knee bend, all five subjectswith an ACLD knee experienced posterior femoraltranslation of their lateral condyle with increasedtranslation of the medial condyle when compared tonormal knees (Fig. 11, Table 1). The average amount ofposterior femoral translation of the lateral condyle from0� to 120� of knee flexion was 17.00mm (standarddeviation, 7.31), while the medial condyle translationwas 4.65mm (standard deviation, 3.99), in the posteriordirection. The majority of the posterior femoraltranslation of the lateral condyle occurred in the first30� of knee flexion (Fig. 12). Similar to the normal knee,posterior femoral translation was not always continuouswith increasing knee flexion as small amounts ofparadoxical anterior translation of femorotibial contactwere observed in mid-to-deep flexion which was morecommon laterally (80%) than medially (20%); oppositeto the normal knee observation.All subjects experienced an axial rotation pattern

similar to that of the normal knee subjects only duringthe first 30� (Fig. 13). After 30� the axial rotationpattern of the ACLD knees exhibited different behaviorthan normal knees (changing slope from positive tonegative in Fig. 13). The average amount of axialrotation for the five subjects from 0� to 120� of flexionwas 9.87� (standard deviation=3.23). Similar to normal

Fig. 10. Helical axis of the normal knees showing the axis does not stay fix

motion. Each column represents different view, each row is a different subje

Average Tibio-Femoral Axial Rotation

-10.00

-5.00

0.00

5.00

10.00

15.00

0 15 30 45 60 75 90 105 120

Flexion Angle (Degrees)

Axi

al R

ota

tio

n A

ng

le (

Deg

rees

)

Fig. 9. Axial rotation of the femur during knee flexion for normal

subjects.

knees, the helical axis translated posteriorly duringincreasing knee flexion (Fig. 14).

4. Discussion

The current study determined that accurate 3Dmotion of normal and ACLD knees, under in vivo,weight-bearing conditions, can be determined usingvideo fluoroscopy and a 3D-to-2D image registrationprocess. The results differed from previous kinematicanalyses of total knee arthroplasty (Dennis et al., 1996,1998a,b) which typically demonstrated reduced magni-tudes of both posterior femoral translation and axialrotation and an increased incidence and magnitude ofparadoxical anterior femoral translation when com-pared to normal knees.Normal and ACLD knee subjects demonstrated

similar pattern of posterior femoral translation duringprogressive knee flexion but they exhibited differentaxial rotation pattern after 30� of knee flexion (Fig. 15).Reduced mean magnitudes of posterior femoral transla-tion laterally (normal, 21.07mm; ACLD, 17.0mm) andaxial rotation (normal, 23.67.4�; and ACLD 9.9�) were

ed in space during an entire movement through the available range of

ct.

Fig. 11. Axial (top) view of the minimum points projected on the tibial

plateau of the five ACLD knees. Clear medial pivoting motion of the

femur relative to the tibia is seen in all five knees.

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-20.00

-15.00

-10.00

-5.00

0.00

5. 00

10 .00

15 .00

0 20 40 60 80 100 120 140

Average LAPAverage MAP

Fle

xio

n in

Deg

rees

Lateral and Medial Anterior Posterior Translation For ACLD Patients

Translation in mm

Fig. 12. The average ACLD medial and lateral condyle contact positions are plotted during a deep knee bend activity.

-15.00

-10.00

-5.00

0.00

5.00

10.00

0 15 30 45 60 75 90 105 120

Axi

al R

ota

tio

n A

ng

le (

Deg

rees

)

Average Tibio-Femoral Axial Rotation For ACLD

Flexion Angle (Degrees)

Fig. 13. Axial rotation of the femur during knee flexion.

Fig. 14. Helical axis of the ACLD knees showing that axis does not stay fixed in space during progressive knee flexion.

D.A. Dennis et al. / Journal of Biomechanics 38 (2005) 241–253 249

observed in ACLD knees. Additionally, increasedvariability in knee kinematic patterns were observed inACLD knees as compared with normal knees. Fig. 16shows the point of intersection of the helical axis withthe sagittal plane passing through the center of the tibia,

for each of the patients in both normal and ACLDgroups. As can be seen, the dispersion of the intersectionpoints is greater for the ACLD group than the normalgroup. Reduced magnitudes of both anteroposteriorfemorotibial translation (Brandsson et al., 2001) and

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Axial Rotation Normal Vs. ACLD

-15.00

-10.00

-5.00

0.00

5.00

10.00

15.00

0 20 40 60 80 100 120 140

Tra

nsl

atio

n in

mm

Axial Rotation Normal

Axial Rotation ACLD

Roration in Degrees

Fig. 15. Normal vs. ACLD axial rotation of the femur relative to the tibia during knee flexion.

Fig. 16. The intersection of the helical axes of motion with the sagittal

plane, passing through the center of the tibia. The ellipsoids represent

the covariance of each group of points, at a one-sigma level of

significance.

D.A. Dennis et al. / Journal of Biomechanics 38 (2005) 241–253250

axial rotation (Brandsson et al., 2001; Jonssonet al.,1989; Karrholm et al., 1988) after ACL disruptionhave similarly been observed in other reports ofknee kinematics of ACLD knees. These studies, how-ever, have typically reported lesser amounts of bothanteroposterior translation and axial rotation than in

the present study. This may be related to the factthat most of these studies evaluated the knee in only alimited flexion range (i.e. 0–60�) and/or testedunder nonweight-bearing conditions. Posterior femoraltranslation was substantially greater laterally thanmedially in both normal and ACLD subjects, creatinga medial pivot type of axial rotation pattern in whichthe tibia internally rotates relative to the femur asflexion progresses and externally rotates as the kneeextends. Typically, as the knee approaches deepflexion, the medial femoral condyle translates a smallamount anteriorly (o5mm) while the lateralfemoral condyle continues to translate posteriorly.However, in previous study of the authors’ (Komisteket al., 2003) one individual with a ‘‘normal’’knee demonstrated substantial medial translationand exhibited a lateral pivot kinematic pattern (i.e.,reverse screw home axial rotation pattern) in chair sitactivity.Anteroposterior translation was not always unidirec-

tional as flexion or extension progressed, particularly atthe medial femorotibial articulation. For example, someindividual subjects had initial posterior femoral roll-back, then demonstrated paradoxical anterior transla-tion of femorotibial contact, followed by a return toposterior femoral rollback in deep flexion. Therefore, aforward and back translational pattern occasionally wasseen as flexion progressed. Both the incidence andmagnitude of paradoxical anterior femoral translationobserved progressive knee flexion was greater in ACLDknees (especially on the lateral articulation >5mm,80%).

ARTICLE IN PRESSD.A. Dennis et al. / Journal of Biomechanics 38 (2005) 241–253 251

The paradoxical anterior femoral translation ob-served in the present investigation, particularly inACLD knee subjects, may have undesirable conse-quences. This phenomenon results in more anterior axisof flexion and can reduce maximum knee flexion(Dennis et al, 1998a,b). Second, the quadriceps momentarm is decreased, resulting in reduced quadricepsefficiency. Lastly, articular cartilage shear forces willlikely be increased, enhancing the risk of prematuredegenerative change, which is commonly observed inthose with chronic ACL injuries (Beynnon et al., 2002).With knee flexion, the normal tibia typically internally

rotates relative to the femur and conversely, externallyrotates with knee extension (i.e., screw home mechan-ism) (Nordin and Frankel, 1980; Rosenberg et al., 1994;Shaw and Murray, 1974; Smillie, 1962; Van Dommelenand Fowler, 1989). Hypotheses advanced to explain thescrew home mechanism include the length and tensionof the cruciate and collateral ligaments (Brantigan andVoshell, 1941; Haines, 1941; Lewin, 1952) as well asdifferences in dimensions of both the medial and lateralfemoral and tibial condyles (Nordin and Frankel, 1980;Roesnberg et al., 1994). The reduced amount of axialrotation and increased incidence of a reverse screw homeaxial rotation pattern observed in ACLD knee subjectsmay be detrimental, enhancing the risk of patellofemor-al instability.Individual variances in the pattern and magnitudes

of AP translation and axial rotation were observedamong subjects in both groups. This likely is attribu-table, at least in part, to variances in ligamentoustension, muscle contraction patterns, anatomic condylargeometry variations and effort intensity among differingsubjects.An advantage of the present experimental model is it

allows analysis under in vivo, weight-bearing conditionsthroughout the entire range of knee flexion. The 3Dkinematic process reduces potential error caused bymotion between external skin markers and the under-lying bone (Andriacchi et al., 1986). The importance ofweight-bearing in kinematic evaluation of the knee issupported by the work of Hsieh and Walker (1976), whodetermined that in an unloaded knee joint, laxity isprimarily determined by soft tissue constraints, whereasin the loaded knee joint, the geometric conformity of thejoint surfaces plays the major role in controlling jointlaxity. Additionally, abnormal anteroposterior femor-otibial translation patterns following ACL disruptionhave been shown to be accentuated when tested underweight-bearing conditions (Beynnon et al., 2002;Brandsson et al., 2001; Kannus and Jarvinen, 1989;Roos et al., 1997). Methods which allow assessmentduring dynamic muscle contraction provide a superiorestimation of true knee kinematics. Markolf et al.(1978), in an in vivo study of knee stability of 49healthy subjects, found laxity measurements were

reduced by as much as 50% with the addition of musclecontraction.In summary, the present analysis has demonstrated

progressive posterior femoral translation and internaltibial rotation occurs in both normal and ACLDsubjects during deep knee flexion. Reduced magnitudesof anteroposterior translation and axial rotation as wellas increased variability in kinematic patterns wasobserved in ACLD subjects. Future studies will includeevaluation of a larger number of subjects and reanalysisof subjects at sequential intervals to assess changes inkinematic patterns over time, particularly in thosesubjects with ACL injuries. In the present study, allACLD subjects were evaluated relatively soon (o6months) following their ACL injury. We suspectdifferences in kinematic patterns between normal andACLD subjects may increase in chronic ACLD subjectsdue to attenuation of secondary soft tissue stabilizingstructures over time.

Acknowledgements

This work was supported by National ScienceFoundation, Grant #9729255, Radiographic and DataSolutions, Minneapolis, MN, USA.

Appendix. Background on pose estimation

The pose of a rigid body {A} with respect to anothercoordinate system {B} can be represented by a six-element vector B

Ax ¼ BxAorg; ByAorg; BzAorg; a;b; g T

;where BpAorg ¼

BxAorg; ByAorg; BzAorg

Tis the origin of

frame {A} in frame {B}, and (a,b,g) are the angles ofrotation of {A} about the (z, y, x) axes of {B}. Analternative representation of orientation is to use threeelements of a quaternion; the conversion between Eulerangles and quaternions is straightforward (Markolfet al., 1979; Craig, 1990). Equivalently, pose can berepresented by a 4� 4 homogeneous transformationmatrix

BAH ¼

BAR

BpAorg

0 1

� �; ðA:1Þ

where BAR is the 3� 3 rotation matrix corresponding to

the angles (a,b,g). In this paper, we shall use the letter xto designate a six-element pose vector and the letterH todesignate the equivalent 4� 4 homogeneous transfor-mation matrix.Homogeneous transformations are a convenient and

elegant representation. Given a homogeneous pointAp ¼ AxP; AyP; AzP; 1

T; represented in coordinate

system {A}, it may be transformed to coordinate system{B} with a simple matrix multiplication Bp ¼B

A HAp: The

homogeneous matrix representing the pose of frame {B}

ARTICLE IN PRESSD.A. Dennis et al. / Journal of Biomechanics 38 (2005) 241–253252

with respect to frame {A} is just the inverse of the poseof {A} with respect to {B}, i.e., A

BH ¼BA H

�1: Finally, ifwe know the pose of {A} with respect to {B}, and thepose of {B} with respect to {C}, then the pose of {A}with respect to {C} is easily given by the matrixmultiplication C

AH ¼CB H

BAH:

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