3-d anatomically based dynamic modeling of the human knee to include tibio-femoral and...

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Dumitru I. CaruntuPh.D.

e-mail: [email protected]

Mohamed Samir Hefzy1

Ph.D., P.E.e-mail: [email protected]

Biomechanics and Assistive TechnologyLaboratory

Department of Mechanical, Industrial andManufacturing Engineering

The University of ToledoToledo, Ohio, USA 43606

3-D Anatomically Based DynamicModeling of the Human Knee toInclude Tibio-Femoral andPatello-Femoral JointsAn anatomical dynamic model consisting of three body segments, femur, tibia and pahas been developed in order to determine the three-dimensional dynamic responsehuman knee. Deformable contact was allowed at all articular surfaces, which were mematically represented using Coons’ bicubic surface patches. Nonlinear elastic spwere used to model all ligamentous structures. Two joint coordinate systems werployed to describe the six-degrees-of-freedom tibio-femoral (TF) and patello-femoraljoint motions using twelve kinematic parameters. Two versions of the model were doped to account for wrapping and nonwrapping of the quadriceps tendon aroundfemur. Model equations consist of twelve nonlinear second-order ordinary differeequations coupled with nonlinear algebraic constraint equations resulting inDifferential-Algebraic Equations (DAE) system that was solved using the DI ifferential/AI lgebraic SIystem SIolIver (DASSL) developed at Lawrence Livermore National LaboratoModel calculations were performed to simulate the knee extension exercise by apnon-linear forcing functions to the quadriceps tendon. Under the conditions tested,‘‘screw home mechanism’’ and patellar flexion lagging were predicted. Throughoutentire range of motion, the medial component of the TF contact force was foundlarger than the lateral one while the lateral component of the PF contact force was foto be larger than the medial one. The anterior and posterior fibers of both anteriorposterior cruciate ligaments, ACL and PCL, respectively, had opposite force patternsposterior fibers were most taut at full extension while the anterior fibers were mostnear 90° of flexion. The ACL was found to carry a larger total force than the PCL atextension, while the PCL carried a larger total force than the ACL in the range of 7590° of flexion.@DOI: 10.1115/1.1644565#

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Introduction

Mathematical knee-joint models have been used to obtabetter understanding of the complicated mechanical behaviothe substructures, which comprise the human musculoskeletaltem including the knee joint. Three survey papers@1–3# haveappeared during the last decade to review mathematical kmodels, which can be classified into either phenomenologicaanatomical based models. The later models are more sophisticand are used to study the behavior of particular structures cprising the human knee. Most of the three-dimensional anatombased models that were developed to study knee behaviorstatic or quasistatic, and therefore did not predict the effectsdynamic inertial loads, which occur in many locomotor activiti@4–14#. To the best of our knowledge, the model developedAbdel-Rahman and Hefzy@15# is the only three-dimensional anatomical dynamic model of the knee joint available in the literture. However, Abdel-Rahman and Hefzy’s model did not incluthe patello-femoral joint, nor did it account for deformation of tarticular surfaces. The only anatomical dynamic models thatclude both tibio-femoral and patello-femoral joints are twdimensional~Tumer and Engin@16# and Ling et al.@17#!. Thecurrent-state-of-the-art for dynamic knee models differs slighthan that presented in the review conducted by Hefzy and Co

1Corresponding author: Phone:~419! 530.8234; Fax:~419! 530.8206; e-mail:[email protected]

Contributed by the Bioengineering Division for publication in the JOURNAL OFBIOMECHANICAL ENGINEERING. Manuscript received by the Bioengineering Divsion August 20, 2002; revision received August 25, 2003; Associate Editor: CVaughan.

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n ar ofsys-

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by-a-deein-

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@3#, and can be summarized as follows:‘‘A single 3-D anatomicaldynamic model that includes both tibio-femoral and patelfemoral joints does not yet exist.’’

Anatomical based models require an accurate description oarticular surfaces in order to solve the contact problem. Sincedynamic model we propose to develop is by itself an elaborand computationally demanding model, we will use a simplificontact theory to model the deformable contact at the articsurfaces@18–20#. In this simplified theory, the normal stress btween two contacting surfaces is proportional to the shortest petration distance between these two surfaces.

In this work, we present for the first time the 3-D dynamresponse of the knee joint using an anatomical based modelincludes three body segments involving both tibio-femoral apatello-femoral joints. The model allows for deformable contacthe articular surfaces and allows for the wrapping of the quaceps tendon around the femur, which occurs at large flexangles. Model equations consist of twelve nonlinear second-oordinary differential equations coupled with nonlinear algebrconstraint equations. To solve this system of equations,second-order differential equations were transformed into atem of first-order differential equations and then were combinwith the algebraic equations to produce a system of DifferenAlgebraic Equations~DAE!. The DAE system is solved by usina DAE solver, namely the DI ifferential/AI lgebraic SIystem SIolIver~DASSL! developed at Lawrence Livermore National LaboratoModel calculations are performed to simulate the knee extenexercise, a commonly prescribed rehabilitation regimen@21#.Nonlinear quadriceps forcing functions with different charactertics were obtained from the experimental data available in

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2004 by ASME Transactions of the ASME

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literature@22# and used as input to the model. Results are repoto describe the knee response including tibio-femoral and patefemoral motions and contact forces and anterior and posteriorciate ligament forces. A comparison of model predictions wrelated data available in the literature is then presented.

Model Formulation

1 Kinematic Analysis. Three local coordinate systems oaxes were identified on the fixed femur and moving tibia apatella as shown in Fig. 1. The tibial and patellar systems wcentroidal principal systems of axes, (y1

T ,y2T ,y3

T) and(y1

P ,y2P ,y3

P), respectively; both of them were parallel to the femral coordinate system at full extension. The femoralx1 axis, hav-ing i¢ as a unit vector along it, was directed medially for a left knand laterally for a right knee, and the femoralx2 andx3 axes were

Fig. 1 Tibio-femoral and patello-femoral joint coordinatesystems.

Journal of Biomechanical Engineering

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directed anteriorly and proximally, respectively. The twelve dgrees of freedom describing the tibio-femoral joint~TFJ! and thepatello-femoral joint~PFJ! motions were defined using two joincoordinate systems@11,15,23#, and include 3 rotations and 3 translations for each of the tibial and patellar moving systems. The T( i¢,e¢2

T ,k¢T) and the PFJ (i¢,e¢2P ,k¢P) coordinate systems are identifie

in Fig. 1; k¢T andk¢P are two unit vectors along the tibialy3T and

patellary3P local axes, respectively.

The position vectors of any point on bodyg @for tibia: g5Tand for patella:g5P] with respect to the femoral coordinate sytem and the local body coordinate system,R¢ g(X1

g ,X2g ,X3

g) andr¢g(x18

g ,x28g ,x38

g), respectively, are related according to the folowing transformation:

R¢ g5R¢ Og 1@Rg#r¢g (1)

whereR¢ Og (x1

g ,x2g ,x3

g) is the position vector of the origin of thegth

body coordinate system with respect to the femoral coordinsystem. Equation~1! is written using tensor notation in the following form: Xi

g5xig1Ri j

g xj8g , i , j 51,2,3, and the (333) rota-

tional matrix @Rg# that describes the orientation of thegth bodywith respect to the femoral coordinate system is given as:

@Rg#5F s2gc3

g s2gs3

g c2g

2c1gs3

g2s1gc2

gc3g c1

gc3g2s1

gc2gs3

g s1gs2

g

s1gs3

g2c1gc2

gc3g 2s1

gc3g2c1

gc2gs3

g c1gs2

gG , g5T,P

(2)

whereskg5sinak

g , ckg5cosak

g , k51,2,3. The rotation vectors,u¢T

andu¢ P describing the orientation of the tibia and patella, resptively, with respect to the femoral coordinate system are thus wten as:

u¢T52a1Ti¢2a2

Te¢2T2a3

Tk¢T, u¢P52a1Pi¢2a2

Pe¢2P2a3

Pk¢P (3)

wherea1T ~knee flexion!, a3

T ~tibial internal rotation!, a2T5(p/2

6Abduction), a1P ~patellar flexion!, a3

P ~lateral patellar tilt! anda2

P5(p/26Patellar lateral rotation); a positive sign is used f

Fig. 2 3-D model of the knee joint „tibio-femoral and patello-femoral joints …

showing the collateral and cruciate ligaments

FEBRUARY 2004, Vol. 126 Õ 45

46 Õ Vol. 126, FEBRUARY

Fig. 3 Initial and final positions during knee extension exercisesimulation

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the right knee and a negative sign for a left knee@11,15,23#. Therelative locations of tibia and patella with respect to the femurany position are thus described byy, a vector of dimension (n512) consisting of twelve independent kinematic parameters

y5~x1T ,x2

T ,x3T ,a1

T ,a2T ,a3

T ,x1P ,x2

P ,x3P ,a1

P ,a2P ,a3

P! (4)

2 Mathematical Representation of the Articular Surfaces.The articular surfaces of the distal femur and the posterior patalong with the planar tibial plateaus were mathematically repsented using Coons’ parametric bicubic surface patches as shin Fig. 2. The cartesian coordinates of any point on a surface pare expressed as bicubic functions of two local parametric cdinates,w1 andw2 , in the range of 0 to 1 over the patch. Thefunctions were determined in terms of the coordinates of thener points, which were obtained by digitizing a cadaveric spemen. Details of this mathematical procedure are given by Heand Yang@11#.

3 Joint Loads

Quadriceps Tendon Force.Two cases were considered in thanalysis to allow for the wrapping~Fig. 3a! and nonwrapping~Figs. 2 and 3b! of the quadriceps tendon around the femur. Fthe non-wrapping case, the direction of quadriceps tendon fwas assumed parallel to a line of lengthLq, joining the patellarbasis, pointb, and the attachment of the quadriceps muscle, pq. The components of the quadriceps force expressed in the feral system,Fi

q , i 51,2,3, can thus be written in terms of the skinematic parameters describing PFJ motions as follows:

Fiq5FqLi

q/Lq (5)

where Fq is the magnitude of the quadriceps tendon force,Liq

5(Xiq2xi

P2Ri jPxj

b) are the femoral components of the positiovector of pointq with respect to pointb, andXi

q and xjb are the

components of the local coordinates of pointsq and b, respec-tively.

Wrapping occurs at large flexion angles, normally greater t70° of knee flexion as shown in Fig. 3a. In this situation, thedirection of the quadriceps force was assumed parallel to a linlengthLw, joining the patellar basis, pointb, and the most distafemoral point where wrapping occurs, pointw, as shown in Fig.3a. The components of the quadriceps force are then expresse

Fiq5FqLi

w/Lw (6)

whereLiw5(Xi

w2xiP2Ri j

Pxjb) are the femoral components of th

position vector of pointw with respect to pointb, andXiw are the

femoral coordinates of pointw. Since pointw is a point on afemoral patch, its cartesian coordinates,Xi

w , are bicubic functionsof its two parametric coordinates,w1

w andw2w . To specify pointw,

two conditions are imposed. First, the plane containing the po

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q, b andw is assumed parallel to the femoralx2 axis. Using thescalar triple product this condition is expressed as:

« i2kLiqLk

w50 (7)

where« i2k is the alternating tensor. Second, the lineLw ~the di-rection of quadriceps force acting on patella! must be tangent tothe femoral surface. This condition is expressed mathematicas:

LiwNi

f~w1w ,w2

w!50 (8)

whereNif are the femoral components of unit vector normal to t

femoral patch at pointw.The components of the moment vector of the quadriceps fo

around the centroid of the patella,Miq , are expressed for both

nonwrapping and wrapping cases as follows:

Miq5 xi j

b Rk jP Fk

q (9)

whereFkq are given by Eq.~5! or ~6!, andxi j

b are the componentsof the antisymmetric tensor@ xb# of the local position vector of thepatellar basis whose components arexj

b .

Contact Loads. Friction forces were neglected because of textremely low coefficient of friction of the articular surfaces@24#,and a simplified model was used to allow for a deformable conat the articulating surfaces of both TFJ and PFJ@18#. While thesubchondral bone was assumed to be rigid, the articulating clage was considered to be a thin layer of isotropic and lineelastic material. The normal stress,s, between two contactingpatches located on the moving and fixed surfaces was expreas s5Ku, whereu is the penetration, i.e. the total deformatio~of both patches! at a contacting point in a direction perpendiculto the moving surface. The contact stiffness,K, was calculated asK5$@(12n)E#/@(11n)(122n)t#% whereE, n andt are the elas-tic modulus, Poisson’s ratio and the thickness of the contaccartilage layers. By assuming E55 MPa, n50.45 andt52 mm@18#, the contact stiffness was calculated asK55 N/mm3. In thisanalysis, a uniform stress distribution over each patch wassumed.

An iterative procedure was employed to determine all pairscontacting patches. Each pair consisted of a source patch loceither on tibia or patella and a target patch located on femur.penetration distance,u, was calculated as the projection of a linST onto the normal to the source patch from its center, poinS.PointT, the target point, was identified as the point of intersectof a line drawn from pointS, and parallel to a specified preferredirection, with the target patch. To calculate TF and PF penetions, the preferred direction was assumed parallel to they3

T tibialaxis and they2

P patellar axis, respectively. Accordingly, the femoral coordinates of pointT were calculated using the following tworelations:

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« i j 3~XiT2Xi

S!Njd50, «1 jk~Xj

T2XjS!Nk

d50 (10)

where« i jk is the alternating tensor,Nid are the femoral compo

nents of a unit vector parallel to the preferred direction, andXiT

andXiS are the femoral coordinates of pointT andS, respectively.

In this iterative procedure, the coordinates of pointS are known.Using the parametric nonlinear equations of the target patcEqs.~10! become a nonlinear algebraic system in two unknowthe parametric coordinates of the target point,w1

T and w2T . The

penetration distance between two patches is then calculated uthe following relation:

u5~XiT2Xi

s!Ni (11)

whereNi are the femoral components of the outward unit vecnormal to the source patch at its center. If u is positive, no conoccurs. The components of the total TF and PF contact forcesFi

ct

andFicp , respectively, are expressed as:

Fict5 (

m51

M1

@smAm~Ni !m#ct; Ficp5 (

m51

M2

@smAm~Ni !m#cp

(12)

where the subscriptm refers to themth source contacting patch,Aand s are the patch area and the patch normal contact strrespectively, andM1 and M2 are the total numbers of the tibiaand patellar contacting patches, respectively; superscriptsct andcp refer to cIontacting tIibial patches within the TFJ and cIontactingpIatellar patches within the PFJ, respectively. The componentthe total moment vectors of the tibial and patellar contact foraround the tibial and patellar centroids, respectively, are expreas follows:

Mict5Rk j

T (m51

M1

@~ xi j !msmAm~Nk!m#ct;

Micp5Rk j

P (m51

M2

@~ xi j !msmAm~Nk!m#cp (13)

where xi j are the components of the antisymmetric tensor oflocal position vector of pointS of the mth contacting patch de-scribed by its componentsxi . Equations~10!–~13! show that thecontacting forces and their moments are expressed in terms otwelve kinematic parameters describing TFJ and PFJ motions,the parametric coordinates of the target points located onfemoral patches that are in contact with either the tibia orpatella.

Ligamentous Forces.Along with the posterior capsule, thlateral and medial collaterals, and the anterior and posteriorciates were modeled as shown in Fig. 2. Using 12 discrete fibundles, these ligamentous structures were represented usinglinear spring elements whose force-elongation relationshipscluded quadratic and linear regions@15,18# as follows:

~F,!n5H 0, «n<0

~kq!n~Ln2L0n!2, 0,«n,2«0

~k,!n@Ln2~11«0!L0n#, «n>2«0

n51,2, . . . ,12 (14)

where «n, (kq)n, (k,)n, Ln and L0n are the strain, the stiffnes

coefficients for the quadratic and linear regions, and the curlength and slack length of thenth element, respectively. The linearange threshold was specified as«050.03. The coordinates of thligamentous insertion points, the different slack lengths andstiffness coefficients were obtained using Abdel-RahmanHefzy’s data@15#. The components of the resultant of the ligametous forces acting on the tibia,Fi

, , expressed in the femoral coordinate system are written as:


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Fi,5(

n51

12

~F,!nLin/Ln (15)

and the components of the resultant moment about the tibialter of mass of the ligamentous forces,Mi

, , expressed in the tibiacoordinate system are expressed as follows:

Mi,5(

n51

12

xi j8nRk j

T ~Fk,!n (16)

whereLin are the femoral components of the position vector of

nth ligamentous structure’s femoral insertion with respect totibial insertion point and expressed as:

Lin5Xi

n2xiT2Ri j

T xj8n (17)

where Xin and xj8

n are the local coordinates of the femoral antibial insertion points, respectively, andxi j8

n are the components othe antisymmetric tensor of the local position vector of the tibinsertion of thenth ligamentous structure. Equations~15!, ~16! and~17! show thatFi

, andMi, are explicitly written as functions of the

six tibial kinematic parameters.

Patellar Ligament Force. Connecting the tibial tuberositypoint t of local coordinatesxj

t , to the patellar apex, pointa oflocal coordinatesxj

a , the patellar ligament is modeled as a linespring whose tensile force is expressed as:

Fp5kp~Lp2L0p! (18)

wherekp, Lp andL0p are the stiffness, the current length and t

slack length of the patellar ligament. The stiffnesskp was assumedto have a value of 200 N/mm@25#, and the slack lengthL0

p wasspecified to allow a ratio of 0.6 between the patellar ligamforce and the quadriceps force at 90° of flexion@11#. The femoralcomponentsLi

p of the position vector of pointa with respect topoint t are expressed as:

Lip5xi

P1Ri jp xj

a2xiT2Ri j

T xjt (19)

The components of the patellar ligament force acting on the tand the patella,Fi

pt and Fipp , respectively, with respect to th

femoral coordinate system, are then written as;

Fipt5FpLi

p/Lp, Fipp52FpLi

p/Lp (20)

Also, the components of the moments of the patellar ligamforce around the tibial and the patellar centroids, expressed incorresponding local coordinate system are written as

Mipt5 xi j

t Rk jT Fk

pt , Mipp5 xi j

a Rk jP Fk

pp (21)

wherexi jt andxi j

a are the components of the antisymmetric tensof the local position vectors of the tibial tuberosity and the pateapex. Equations~18!–~21! indicate that the patellar ligament forcand its moments around the patellar and tibial centroids areplicitly written in terms of the local coordinates of pointst anda,and the twelve motion parameters,xi

T and a iT , and xi

P and a iP ,

describing PFJ and TFJ motions, respectively.

4 Equations of Motion. Three Newton and three Euleequations are written for each of the moving tibia and pateresulting in a system of 12 differential equations of the secoorder that describes patello-femoral and tibio-femoral motioNewton equations are written with respect to the fixed femocoordinate system of axes as~superscript T for tibia and P forpatella!:

WiT1Fi

ct1Fipt1Fi

,2mTxiT50, i 51,2,3 (22)

WiP1Fi

cp1Fipp1Fi

q2mPxiP50, i 51,2,3 (23)


tc

t

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gs

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e

where xiT and xi

P are the components of the accelerations oftibial and patellar centers of masses, respectively, with respethe fixed femoral coordinate system; (mT,Wi

T) and (mP,WiP) are

the masses and weights’ components of the tibia and the parespectively;Fi

ct , Fipt and Fi

, are the components of total TFcontact force, the patellar ligament force and the resultant ofligamentous forces, respectively, all acting on the tibia;Fi

cp , Fipp

and Fiq are the components of the total PFJ contact force,

patellar ligament force and the quadriceps force, respectivelyacting on the patella.

Euler equations are written in terms of the local tibial and ptellar centroidal principal coordinate systems, (y1

T ,y2T ,y3

T) and(y1

P ,y2P ,y3

P), respectively. Using tensor notation, the componeof the tibial and patellar angular velocities and angular acceltions vectors, denoted byu i8

g and u i8g , respectively (g5T for

tibia and g5P for patella!, are expressed, with respect to threspective local principal system of axes as follows:

u i8g52Rj i

g ~Ujmg 1ak

gUjk,mg !am

g (24)

u i8g52Rj i

g ~Ujmg 1ak

gUjk,mg !am

g 2Rj ig ~Ujm,p

g 1Ujp,mg

1akgUjk,mp

g !apgam

g (25)

where Ujk,mg 5]Ujk

g /]amg and Ui j

g are the components of a tranformation matrix@Ug# which are defined using the nomenclatuof Eq. ~2! as:

@Ug#5F 1 0 c2g

0 c1g s1

gs2g

0 2s1g c1

gs2gG (26)

Euler equations of motion are thus written as:

Mict1Mi

pt1Mi,2~ I Tu8T! i2

1

2« i jk@~ I Tu8T!ku j8

T2~ I Tu8T! j uk8T#

50, i 51,2,3 (27)

Micp1Mi

pp1Miq2~ I Pu8P! i2

1

2« i jk@~ I Pu8P!ku j8

P2~ I Pu8P! j uk8P#

50, i 51,2,3 (28)

where« i jk is the alternating tensor;Mict , Mi

pt , Mi, are the tibial

components of the moments about the tibial center of mass oTFJ contact force, the patellar ligament force and all ligamentforces, respectively;Mi

cp , Mipp and Mi

q are the patellar components of the moments about the patellar center of mass of thecontact forces, the patellar ligament force, and the quadricforce, respectively;I i

T andI iP are the principal moments of inerti

of the leg and patella about their respective centroidal princaxes, respectively. The inertial tibial parameters were specaccording to the anthropometric data available in the literat@15# as mT54.0 kg, I 1

T50.0672 kg m2, I 2T50.0672 kg m2 and I 3

T

50.005334 kg m2. The inertial patellar parameters were assumas follows: mP50.1 kg, I 1

P50.000015625 kg m2, I 2P

50.00003125 kg m2 and I 3P50.000015625 kg m2.

Two cases have been identified: wrapping and non-wrappinthe quadriceps tendon around the femur. In the non-wrappingation, the system of equations to be solved consists of 12 nonear second-order differential equations~Eqs.~22!, ~23!, ~27!, ~28!andk1 nonlinear algebraic equations in (121k1) unknowns wherek152 (M11M2) and M1 and M2 are the numbers of the tibiaand patellar contacting patches, respectively; for each contacpatch, there are two nonlinear algebraic constraints given by~10!. The unknowns in this DAE system of (121k1) equations arethe 12 kinematic parameters defined in Eq.~4! and (M11M2)pairs of parametric coordinates (w1

T ,w2T) defining the target points

48 Õ Vol. 126, FEBRUARY 2004

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on the femoral patches that are in contact with the tibia andtella. This DAE system consists thus of 12 nonlinear differenequationsF(y,y,y,w,t)50 that can be decomposed into two paas follows:

F~y,y,y,w,t !5F~y,y,y,t !1F~y,w!50 (29)

and k1 nonlinear algebraic constraintsG(y,w)50 wherey is avector of dimension 12 containing the 12 kinematic parame@see Eq. 4#, and w is a vector of dimensionk1 containing theunknown parametric coordinates defining contact. In Eq.~29! y5dy/dt andy5dy/dt.

In the wrapping situation, two additional nonlinear algebraequations@Eqs. ~7! and ~8!# are added to the DAE system. Thcorresponding two additional unknowns are the parametric codinates of the most distal point on the femur where wrappingthe quadriceps tendon occurs, (w1

w ,w2w).

Solution AlgorithmThe second-order DAE system is transformed to a first-or

system by relating joint motions~kinematic parameters! to theirvelocities (n i

g ,v ig), i 51,2,3. The resulting DAE system contain

24 first-order ordinary differential equations andk1 nonlinear al-gebraic constraints@k152(M11M2) for non-wrapping situationandk152(M11M211) for the wrapping situation#, and can bewritten as:

F~y,y,w,t !5F~y,y,t !1F~y,w!50 @24 equations# (30a)

G~y,w!50 @k1 equations# (30b)

wherey is a vector of dimension 24 containing the 12 kinemaparameters and their velocities. FunctionsF(y,w) represent thecontribution of the contact forces~and the quadriceps force in thwrapping situation! to the equations of motion.

In order to solve the DAE system the time span is divided intime steps. At each time station,tn11 , components ofyn11 areapproximated in terms ofyn11 andy at previous time steps usina Backward Differentiation Formula~BDF!. The DAE system,defined in Eqs.~30a! and ~30b!, is thus transformed to the nonlinear algebraic system:

F~yn11 ,wn11 ,tn11!50 (31a)

G~yn11 ,wn11!50 (31b)

A solution of the resulting system foryn11 and wn11 is thusobtained evaluating iterativelyyn11

(m11) and wn11(m11) in two steps.

First, wn11(m) are calculated by solving the following system

equations:

G~yn11~m! ,wn11

~m! !50 (32)

using the Newton-Raphson iteration method. Next,yn11(m11) , the

approximation ofyn11 in the (m11)th iteration of a modifieddifferential form of the Newton-Raphson method, is calculatusing the following relation:

yn11~m11!5yn11

~m! 2c@K ~yn11~m! ,wn11

~m! ,tn11!#21F~yn11~m! ,wn11

~m! ,tn11!

(33)whereF is defined by Eq.~31a!, @K # is defined as@26#:

@K #5F]F

]yG1bF]F

] yG (34)

andb andc are two constants that change with the step sizeorder of BDF to speed up the rate of convergence.

The DI ifferential/AI lgebraic SIystem SIolver ~DASSL!, developedby Petzold@27#, was used to solve the DAE system given by Eq~30a!. This solver uses a Predictor-Corrector algorithm whestarting at time stationtn , a predictor polynomial extrapolates thvalues ofyn11

(0) and yn11(0) at time stationtn11 based on the values


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a

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of y at earlier time stations. Then a corrector utilizes a BDFtransform~30a! into ~31a!. Input to the DASSL includes the initiaand final times,to andtF , respectively, and the initial values ofy,w and y. These values satisfy the DAE system~Eqs. 30! at timet5t0 . This was done by assuming initial values ofy at time t5t0 , and calculating the initial values ofw andy by solving Eqs.~30a! and ~30b!.

User-supplied sub-routines evaluate the load vectorF(y,y,w,t)and the stiffness matrix@K (y,y,w,t)#. The stiffness matrix@K # isdivided into two parts:

@K ~y,y,w,t !#5@K ~y,y,t !#1@K ~y,w!# (35)

The stiffness matrix@K #, which does not depend onw, was cal-culated using Eq.~34! employing closed-form analytical expressions. On the other hand, the stiffness matrix@K #, which dependson w but does not depend ony, was determined by approximatinthe partial derivatives using a backward differentiation formulafollows:

K i j ~y,w!5]F i

]yj~y,w!>

F i~y,w!2F i@y2~Dyj !ej ,w#

Dyj(36)

where the vectorw is found by solvingGby2(Dyj )ej ,wc50 andej is the j th unit vector of dimensionn524, which is expressed aej5(0,0, . . .,0,1,0, . . . 0).

Model CalculationsIn a test situation a forcing function was applied to simulat

knee extension exercise activity. This dynamic loading wasplied to the patella through the quadriceps tendon causing theand patella to undergo 3-D motions while the femur was fixeda horizontal position. The forcing functions, shown in Fig. 4, wespecified according to Grood et al.’s experimental data@22# wherethe quadriceps force was measured during knee extension. Get al. @22# reported that this force had a constant average valu200 N in the range of 10 to 50 deg of knee flexion. This forreached 345 N at full extension, and was 75 N at 90 deg. Inanalysis, this forcing function will be referred to as the 200quadriceps force. Two other forcing functions with a similar ptern were employed to simulate higher levels of quadriceps ctractions: 400 N and 600 N quadriceps forces. The 400 N andN quadriceps forcing functions were specified such that they hvalue of 75 N at 90 deg that increased to 400 N and 600respectively, as the knee extended to 50 deg remained conuntil 10°, and increased to 530 N and 735 N, respectively, atextension.

In this simulation, the tibia and the patella were assumedbegin their motions from rest. The initial position is depictedFig. 3a and was defined according to the experimental data avable in the literature data@10,11,13# by specifying the following

Fig. 4 Forcing functions applied to the quadriceps tendon tosimulate knee extension exercise.


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kinematic parameters: tibial flexion of 89.80 deg, patellar flexof 71.26 deg, varus~adduction! angle of 4.61 deg, tibial internarotation of 22.96 deg patellar medial rotation of 0.25 deg, apatellar medial tilt of 0.46°. For each forcing function, and as tsimulation progressed from the initial position, the quadriceps tdon continued to wrap around the femur until the flexion andecreased to a point when wrapping stopped. It was found thanonwrapping started at 74 deg, 74.14 deg and 74.50 deg of kflexion for the 200 N, 400 N and 600 N quadriceps forcing fun

Fig. 5 Duration of knee extension exercise for different quad-riceps forcing functions

Fig. 6 Internal-external tibial rotations and varus-valgus rota-tions versus knee flexion angle for different quadriceps forcingfunctions

Fig. 7 Patellar flexion angle versus knee flexion angle for dif-ferent quadriceps forcing functions


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tions, respectively. The corresponding patellar flexion angles w57.63, 57.91 and 58.22 deg, respectively. Figure 3b shows thepredicted final position when the 200 N quadriceps forcing fution was used. Figures 3a and 3b show that with knee extensionthe tibia rotated externally~the ‘‘screw home mechanism’’! whichprovides verification for the dynamic simulation. Additionmodel validation is obtained by studying the changes in the pdicted positions of the TF and PF contact with extension. Figu3a and 3b show that as the knee was extended, the location of

Fig. 8 Medial and lateral components of the tibio-femoral con-tact force versus knee flexion angle for different quadricepsforcing functions

Fig. 9 Lateral and medial components of the patello-femoralcontact force versus knee flexion angle for different quadricepsforcing functions

Fig. 10 Forces in the anterior and posterior fibers of the ACLversus knee flexion angle for different quadriceps forcing func-tions


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TF contact moved posteriorly on the medial tibial plateau aanteriorly on the lateral plateau, which reflects an internal rotatof the femur with respect to the tibia~an external rotation of thetibia with respect to the femur!. Also, and as expected, the location of the TF contact on the femur moved distally on tcondyles. These figures also show that as the knee was extethe PF contact area moved proximally on the femur. It was afound, as shown in Figs. 2 and 3b, that the PF contact moveddistally on the patella with knee extension. Fig. 5 shows thatduration of the knee extension exercise decreases as the levthe quadriceps forcing function increases, which also verifiesdynamic analysis. Figure 6 shows that as the tibia was extenfrom 90° to full extension, it underwent an average of 18.5 degexternal rotation and 4 deg of valgus rotation~abduction!. ‘‘Patel-lar lagging,’’ which occurs with knee flexion, was also predictas shown in Fig. 7.

Figure 8 shows that the medial component of the TF conforce was larger than the lateral component for all tested sitions. This figure also shows that the medial component of thecontact force doubled during the last 30 deg of knee extensreaching a maximum at full extension. Figure 9 shows thatlateral component of the PF contact force was much larger thamedial component for all tested conditions. Both medial anderal components of the PF contact force had their largest valuethe range of 45 to 60 deg of knee flexion.

Figure 10 shows that as the knee was extended from the 90position to around 35 deg of flexion, the tension in the ACL wgreatest in its anterior fibers. As the flexion angle decreased,tension decreased while tension in the posterior fibers increaand became dominant. Figure 11 shows that the anterior fibethe PCL carried large forces as the knee was in a flexed posiThese forces decreased as the knee was extended. The posfibers of the PCL were in tension in the last 10 deg of knextension. At full extension, the forces in the posterior fibers wgreater than those in the anterior fibers, but much smaller thanlarge forces that were found in the anterior fibers near 90 deflexion.

Discussion and ConclusionsA review of the literature reveals that most of the publish

anatomically based dynamic knee models are 2-D@1–3#. Abdel-Rahman and Hefzy’s@15# model is the only anatomical modeavailable in the literature that predicts the three-dimensionalnamic response of the joint under impact loads. Yet, this mowas limited in that it was only for the tibio-femoral joint, assumea rigid contact formulation, and used a spherical representationthe femoral articular surfaces. In this paper, a three-dimensioanatomically based dynamic model of the knee that includes b

Fig. 11 Forces in the anterior and posterior fibers of thePCL versus knee flexion angle for different quadriceps forcingfunctions


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tibio-femoral ~TF! and patello-femoral~PF! joints is presented.The model allows for deformable contact and for a piecewmathematical representation of the femoral articular surfaces.model also allows for wrapping of the quadriceps tendon arothe femur that occurs at large flexion angles.

The system of equations forming anatomically based dynamodels is a system of Differential-Algebraic Equations~DAE!.Several techniques have been proposed to solve the DAE sythat describes the dynamic response of the knee@3#. Most of thesetechniques were limited in that they could not solve the comcated DAE system that represents the three-dimensional situaIn this paper, the DI ifferential/AI lgebraic SIystem SIolIver, DASSL~developed at Lawrence Livermore National Laboratory!, wasused to solve the complex DAE system to obtain the thrdimensional response of the TF and PF joints under dynamic loing. In order to use this solver, user-supplied subroutines wdeveloped to evaluate the stiffness matrix of the system. Cloform analytical expressions were written in terms of the kinemaparameters to determine the contributions of the ligamentouspatellar tendon forces, and the quadriceps force in the nwrapping situation to the stiffness matrix. On the other hand,contributions of the contact forces and the quadriceps force inwrapping situation to the stiffness matrix were determinedmerically using backward differentiation formulas.

The dynamic response of the knee joint is much different frthe static response@28#. Hence, it is hard to compare the presemodel predictions with similar data reported elsewhere becaalmost all of the data available in the literature that describesbehavior of the human knee joint are static or quasi-static inture. In the following, and within these qualifications, model pdictions will be discussed and compared with those availablethe literature.

The tested condition was used to simulate the knee extenexercise, a common rehabilitation regimen@21#. The forcing func-tions applied to the quadriceps tendon were specified accordinthe experimental data available in the literature that quanquadriceps forces during a knee extension exercise@22#.

The classic ‘‘screw-home’’ pattern described by many invegators@29# was observed from model predictions. As the knee wextended from 90 deg of flexion to full extension, the tibia rotaexternally an average of 18.5 deg. This predicted amount ofternal rotations is consistent with the experimental values repoin the literature~10 deg by Shoemaker et al.@29#, 14.5 deg byBiden et al.@30#, 20 deg by FitzPatrick@31#, and a range of 14 to36 deg by Wilson et al.@32#!. Model calculations also show thaas the knee was extended from 90 deg to about 55 deg of flexit did not rotate either in valgus or in varus. As the flexion angfurther decreased, the tibia went into valgus~abduction!. The dy-namic simulation indicated that the tibia was ab-ducted an aveof 5 deg as the knee was extended from 55 deg of flexion toextension. These results are consistent with those reported byson et al. @32# where the authors used two testing conditio~fixed tibia and fixed femur! and 13 knees to describe the knmovements that are coupled with passive knee flexion~nointernal-external torques, no varus-valgus torques, no anteposterior loads!. They reported that as the knee was extendedflexed from a 50 deg of flexion position, it was ab-ducted aad-ducted on average 5 deg and 1 deg, respectively. Theyreported that when the femur was fixed~similar to our numericalsimulation!, the knee was neither abducted nor adducted as itflexed from 50 deg to 90 deg of flexion, and was abducted 5as it was extended from 50 to 0 deg of flexion, which is consistwith our model predictions.

Model calculations have also shown that the anterior and pterior fibers of both anterior cruciate ligament~ACL! and poste-rior cruciate ligament~PCL! had opposite force patterns. The poterior fibers of the ACL were slack in the range of 90 to 60 degflexion, and tightened progressively as the knee was extenreaching a maximum at 0 deg of knee flexion. The anterior fib


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of the ACL were most taut at 90 deg; the tension decreased aknee was extended. The forces in the anterior fibers of the Pwere large at 90 deg of flexion, increased slightly with kneetension to a maximum around 75 deg, and then decreased prosively until they vanished at 0 deg. On the contrary, the forcesthe posterior fibers of the PCL were much smaller: they vanisin the range of 90 to 30 deg and then increased slightly to reamaximum at 0 deg. These results are in agreement with threported in the literature to describe the function of the anteand posterior fibers of both ACL and PCL@33–36#.

Model calculations also show that the total force in the AC~anterior and posterior bundles! reaches a maximum at full extension during the knee extension exercise. At this position, the foin the PCL is comparatively small, which indicates that the ACcarries a larger total force than the PCL at full extension durinknee extension exercise. Also, and in the range of 75 to 90 deknee flexion, the forces in the anterior fibers of the PCL~whichare maximum! are greater than the forces in the anterior fibersthe ACL ~which are also maximum!. The posterior fibers of thePCL and the posterior fibers of the ACL do not carry a load inrange of 75 to 90 deg of flexion. These data thus indicate thattotal force in the PCL is larger than the total force in the ACLthe range of 75 to 90 deg of flexion. These results are in agment with Wilk et al.’s data@37# reporting that maximum posterior and anterior tibio-femoral shear forces occurred aroundand 0 deg of flexion, respectively. This is in agreement with omodel calculations since posterior and anterior tibio-femoral shforces are resisted by the PCL and ACL, respectively. Also,model calculations have shown that the maximum forces carby the ACL at full extension are larger than the maximum forccarried by the PCL near 90 deg of flexion. These findingsconsistent with other investigators’ data reporting that the greaamount of tibial displacement occurs within the last 30 degknee extension during knee extension exercise@22,37–39#.

It was also found that the medial component of the tibfemoral contact force was always larger than the lateral comnent. It is hard to compare these results with those available inliterature since, and to the best of our knowledge, no data hasreported to compare the medial and lateral components oftibio-femoral contact force during knee extension exercise. Nertheless, it has been reported that the medial condyle camore load than the lateral condyle during a closed chain exer@40–42#.

Model calculations also show that the ratio of the total patefemoral~PF! contact force to the quadriceps force decreased frnearly 1.1 at 80 deg to 0.45 near full extension as shown in F12. This is in agreement with other model predictions and w

Fig. 12 Ratio of total patello-femoral contact force to quadri-ceps force versus knee flexion angle: comparison betweenmodel predictions and published experimental data and otherquasi-static models’ predictions


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published data from in vitro studies. Figure 12 shows a compson between the present model predictions and those ofEijden et al.’s@43# and Gill and O’Connor’s@44# models, and theexperimental data of Ahmed et al.@45# and Buff et al.@46#. Figure12 shows that the present model predictions are more in agment with Ahmed et al.’s@45# in vitro study than that of Buffet al. @46#.

Model calculations also show that the component of thecontact force on the lateral side is always greater than the comnent on the medial side. These results are in agreement with tobtained from the 3-D static models developed by Hefzy aYang @11# and Hirokawa@14# who has also reported larger contaforces on the lateral side. Also, our model calculations areagreement with the experimental data available in the literadescribing patello-femoral contact. The present results indicthat the contact area on the lateral patellar facet is larger thanon the medial facet at all flexion angles~this is because contacforces are based on contact areas in the present model!. This is inagreement with Hille et al.@47#, Hefzy et al. @48#, and Hehneet al. @49# who have reported larger contact areas on the latside. Model calculations have also shown that with knee exsion, the location of the PF contact moves proximally on themur and distally on the patella. These results are in agreemwith those reported in the literature describing the locations ofPF contact areas@Hefzy and Yang@11#, Hefzy et al.@48#!.

Our model calculations also indicate that the pattern thatscribes how the patello-femoral~PF! contact forces changes witknee flexion depends on the level of quadriceps activation.large quadriceps forces, the PF contact forces decrease as theis extended from 60 deg to full extension. For small quadricforces, the PF contact forces remain nearly constant in this raof motion. These data are in agreement with those reported inliterature by Cohen et al.@50# and Takeuchi et al.@51#. During anunloaded open kinetic chain knee extension exercise, Cohen@50# used a quadriceps force with a value of 62 N at 90 dincreasing to 137 N at 60 deg and remaining nearly constant f60 to 20 deg~no data were reported for lower flexion angles!.They reported that the associated PF contact force increased120 N at 90 deg to 150 N at 70 deg and remained nearly consuntil 20 deg of flexion. Our model predictions show that forforcing function ofFq5200 N ~quadriceps force was 75 N at 9deg!, the total PF contact force increased slightly from 70 N atdeg to 175 N at 60 deg then remained nearly constant with furknee extension. A higher and constant quadriceps force of 30was used by Takeuchi et al.@51# who reported that for physiologically normal Q-angles, the PF contact forces decreased fromN at 60 deg to 143 N at 15 deg of knee flexion. At 90 degflexion, they reported a PF contact force of 241 N. Our mocalculations show that for a forcing function ofFq5300 N, thePF contact force increased from 70 N at 90 deg to 255 N atdeg, then decreased almost linearly to 200 N at 20 deg. Thedicted PF contact force has a lower value at 90 deg than thaTakeuchi et al.’s probably because all dynamic quadriceps forfunctions used in this study have a value of only 75 N at tposition.

These results suggest that 3-D anatomically based dynamodels of the human musculoskeletal joints are a versatile toostudy the internal forces in these joints. These models are museful than those less sophisticated quasi-static models, becthey can account for the dynamic effects of the external loaHowever, the formulation of these dynamic models is criticwhen it comes to obtaining a solution. The simpler 3-D dynammodel that accounts only for the tibio-femoral joint could notsolved using Moeinzadeh et al.’s formulation@52#. In the formu-lation presented here, all coordinates of the ligamentous attment sites were dependent variables that were expressed inof the independent kinematic parameters. In addition, these ipendent kinematic parameters that describe tibio-femoralpatello-femoral motions were used to formulate the wrapping


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the quadriceps tendon around the femoral surface and to mthe deformable contact at the articular surfaces. As a result,possible to introduce more ligaments, to split each ligament iseveral fiber bundles, to model the wrapping of the quadricaround the femur that occurs at large flexion angles, and to afor more or fewer surface patches, as appropriate, to comecontact. This formulation allowed the solution of this intricate 3dynamic model.

The present 3-D model can be used to analyze knee respondynamically applied load, which has particular application tojury mechanics. Since most injuries to the knee involve dynamloads, this model, which accounts for the inertial effects of thodynamic forces, will provide a valuable tool to study the undering mechanisms for these injuries. In this model, the menisci wnot included and the definition of the tibial articular surfaces wnot considered. No distinction was made between the mediallateral tibial articular surfaces. However, further developmentsincorporating the menisci, differentiating between the medial alateral articular surface geometry, particularly in terms of posteslope and medio-lateral inclination, and modeling muscularcontractions will allow the model to be used to study daily livinactivities where dynamic axial compressive forces act on the jas in walking and running.

AcknowledgmentsThis work was supported by grant BES-9809243 from the B

medical Engineering Program of the National Science Foundat

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3-d anatomically based dynamic modeling of the human knee to include tibio-femoral and...

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