The biomechanics of the human patella during passive
knee flexion
Heegaard, J.; Leyvraz, P.F.; Curnier, A.; Rakotomanana, L.; Huiskes, H.W.J.
Published in:
Journal of Biomechanics
DOI:
10.1016/0021-9290(95)00059-Q
Published: 01/01/1995
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Citation for published version (APA):
Heegaard, J., Leyvraz, P. F., Curnier, A., Rakotomanana, L., & Huiskes, H. W. J. (1995). The biomechanics of
the human patella during passive knee flexion. Journal of Biomechanics, 28(11), 1265-1280. DOI:
10.1016/0021-9290(95)00059-Q
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Download date: 17. Jun. 2017
Pergamon
J. Biomechwics,
0021-9290(95)00059-3
ESB RESEARCH
THE BIOMECHANICS
J. Hccgaard,*tB
AWARD
Vol. 28, No. 11, pp. 12651279,
1995
Elscvier Scima Ltd
Printed in Great Britain.
0021-9290/95
$9.50 + .oo
1994 (SHARED)
OF THE HUMAN PATELLA
KNEE FLEXION
DURING
PASSIVE
P. F. Leyvraz,* A. Curnier,? L. Rakotomanana*$
and R. Huiskest
*HBpital Orthopedique de la St&se Romande, Lausanne, Switzerland, TDME-LMA, Ecole Polytechnique
Fed&ale de Lausanne, Switzerland; SInstitute of Orthopaedics, University of Nijmegen,
Nijmegen, The Netherlands; and QDP-LGM, Ecole Polytechnique Fed&de de Lausanne, Switzerland
Abstract-The
fundamental objectives of patello-femoral joint biomechanics include the determination of its
kinematics and of its dynamics, as a function of given control parameters like knee flexion or applied muscle forces.
On the one hand, patellar tracking provides quantitative information about the joint’s stability under given
loading conditions, whereas patellar force analyses can typically indicate pathological stress distributions associated for instance with abnormal tracking. The determination of this information becomes especially relevant
when facing the problem of evaluating surgical procedures in terms of standard (i.e. non-pathological) knee
functionality. Classical examples of such procedures include total knee replacement (TKR) and elevation of the
tibia1 tubercle (Maquet’s procedure).
Following this perspective, the current study was oriented toward an accurate and reliable determination of the
human patella biomechanics during passive knee flexion. To this end, a comprehensive three-dimensional
computer model, based on the finite element method, was developed for analyzing articular biomechanics. Unlike
previously published studies on patello-femoral biomechanics, this model simultaneously computed the joint’s
kinematics, associated tendinous and ligamentous forces, articular contact pressures and stressesoccurring in the
joint during its motion, The components constituting the joint (i.e. bone, cartilage, tendons) were modeled using
objective forms of non-linear elastic materials laws. A unilateral contact law allowing for large slip between the
patella and the femur was implemented using an augmented Lagrangian formulation.
Patellar kinematics computed for two knee specimens were close to equivalent experimental ones (average
deviations below 0.5” for the rotations and below 0.5 mm for the translations) and provided validation of the model
on a specimen by specimen basis. The ratio between the quadriceps pulling force and the patellar tendon force was
less than unity throughout the considered knee flexion range (30-1507, with a minimum near 90” of flexion for
both specimens. The contact patterns evolved from the distal part of the retropatellar articular surface to the
proximal pole during progressive flexion. The lateral facet bore more pressure than the medial one, with
corresponding higher stresses (hydrostatic) in the lateral compartment of the patella. The forces acting on the
patella were part of the problem unknowns, thus leading to more realistic loadings for the stress analysis, which
was especially important when considering the wide range of variations of the contact pressure acting on the
patella during knee flexion.
Keywords: Patella; Knee; Kinematics; Stress; Large slip contact.
INTRODUCTION
The main functions of the patella (knee-cap)are to improve the efficiency of the extensor forces through the
entire knee flexion range (Ahmed et al., 1987;Kaufer,
1971),to centralize the forcesof the different quadriceps
musclebelliesand to provide a smooth sliding mechanismfor the quadricepsmusclewith little friction due to
its cartilagecover (Ficat, 1970).It alsoindirectly contributes to the global stability of the knee (Bonnel, 1988).
Finally accordingto someauthors(Fick, 1904;Freehafer,
1962)it provides the anterior aspectof the knee with
a protecting shield.The patellarepresentsthus an important elementof the extensorapparatus;its removal (Patellectomy)leadsto quadricepsatrophy and lossof extenReceived in jinal form 4 April 1995.
l/Author to whom correspondence should be addressed at:
Division of Applied Mechanics, Department of Mechanical Engineering, Stanford University, Stanford, CA 943054040, U.S.A.
sionforce in proportions and this can greatly vary in the
view of various authors (Stougard, 1970;Sutton, 1976).
Becauseof the very high mechanicalstresses
to which the
patello-femoraljoint issubjected,it possesses
the thickest
articular cartilage in the human body, but at the same
time is the site of the greatestfrequencyof degenerative
changes(Ficat and Hungerford, 1977).Thesepathologies
have motivated a numberof studies,in order to understand the causesof patellar degeneraciesbetter and
eventually to recommendcorrective treatments[detailed
reviews on these studiescan be found in Hefty and
Grood (1988)and in Hirokawa (1993)]. Thesestudies
have further delineateda number of fundamentaltopics,
the set of which can be genericallyreferredto as patello-femoralbiomechanics.More specifically,patello-femoral biomechanicstypically include analysesof
(1) patellar kinematics(tracking),
(2) extensorforces,
(3) patello-femoralcontact pressure,
(4) stresses
in the patella.
1265
1266
J. Beegaard et al.
The commonapproach to thesestudieshas been to during kneeflexion. Theseloadsare in turn expressedin
assess
only one particular aspectof the abovelist, and in terms of quadricepsand patellar tendon tensions,and
some casesto compare related parametersbetween patello-femoralcontact pressures
which areall intimately
related to the tracking pattern of the patella. Despiteits
healthy and pathologicalknees.
of theseparametershas
Tracking studieshave provided quantitative informa- importance,the interdependence
tion about the patella motion during knee flexion and not yet beeninvestigated:all the availablestudieson the
about thejoint’s stability undergiven loadingconditions patello-femoraljoint did only considerone set of bio(Brossmannet al., 1993;Fujikawa et al., 1983;Heegaard mechanicsparameters(e.g. kinematicsor stressanalyet al., 1994;Reideret al., 1981;Sikorski et al., 1979;van sis)at a time. Hence,the purposeof the presentcontribuKampen, 1987;Veresset al., 1979).Goldsteinet al. (1986) tion is to provide a global description of the patelloreported an analysisof patellar surfacestrains (which femoraljoint biomechanics,in which its kinematicsand
from a mechanicalpoint of view shouldmorecorrectly be dynamics(including analysisof patellar tendon tension,
are assessed
simulconsideredas a kinematics study rather than a stress contact pressureand patellar stresses)
analysis).Force analyseshave highlightedthe doublerole taneously,i.e. taking their interdependence
into account.
played by the patella (as a spacerand as a lever), have
To obtain sucha global description,a comprehensive
thrown light on increaseof the effectivenessof the quad- computer model was constructed, basedon an expeririceps extensorforces (Ellis et al., 1980;Huberti et al., mentalsetupfor providing input data and modelvalida1984;Reilly and Martens, 1972),and have further invali- tion. The essentialfeatures characterizing this model
dated earlier conceptsin which the patello-femoraljoint were to consider three-dimensionaldeformable conwas consideredas a pulley mechanism.Furthermore, tinuum solidsundergoinglargedisplacements
and deforsuccessfulattempts to relate the extensor forcesto the mation, and rigoroustreatment of the large slip contact
patello-femoralcontact forceswere reported (Ahmed et problem typical of joint mechanics.
al., 1987).Patellar cartilage degenerationhas generally
beenassumed
to bedirectly correlatedto someabnormal
METHODS
contact pressuredistribution in the patello-femoraljoint
(Ficat and Hungerford, 1977;Outerbridge and Dunlop, Formulation of the model
1975).Accordingly, severalstudieshave reported patelThe adopted formulation for constructing the threelo-femoral contact characteristics(Ahmed et al., 1983; dimensionalcomputer modelincluded:
Fujikawa et al., 1983;Goodfellow et al., 1976;Goymann
-a material descriptionfor the patellar kinematics,
and Mueller, 1974;Hille et al., 1985;Huberti and Hayes,
-a weak formulation for the linear momentumbal1984;Matthews et al., 1977;Seedhomet al., 1979;Shoji,
1974; Soslowskyet al., 1990):near extension there is ance,
-elastic constitutive lawsfor the different mediaconvirtually no contactbetweenthe patellaandfemurbut durstituting
the joint,
ing progressiveflexion a bean-shapedcontact area in-a unilateral contact formulation allowing for large
creasesover both the lateral and medialfacetsand migratesproximally. Contact pressureswere generallyde- slips.
ducedfrom calculatedforcesand measuredcontact areas Inertial and gravitational effects were neglectedin the
(Goodfellowet al., 1976;Matthews et al., 1977).However, presentmodel reflecting its quasi-static nature. Consisin a few studies,contact forces(Bandi, 1970)or contact tently, time did not appearexplicitly in the formulation.
stresses
(Fergusonet al., 1979)weredirectly measuredat
Material description. A material description was addiscretelocationsof the articular surface,usingadequate opted to describethe kinematicsof the different compotransducers.Finally, the few publishedstressanalysesof nents of the joint as it proved to be more suitablefor
the patella typically attemptedto correlate stresstrajec- cohesive solids undergoing moderate deformations
toriesto cancellousbonearchitecturein a way to charac- (Truesdell, 1977;Gurtin, 1981):the actual position of
terize further the pathogenesisof articular lesions(Da a material particle x wasexpressedby the map y = y(x)
Silva and Bratt, 1970;Haasters,1974;Hayeset al., 1982; calledthe motion or the deformation.The deformationof
Maquet, 1984;Minns et al., 1979;Muller et &1980).
an infinitesimalmaterial fiber wascaptured by the (unAssessingthese various aspects of patellar bi- symmetric)deformation gradient F (i.e. dy = Fdx). The
omechanicsbecomesespeciallyrelevant when facing the (right Cauchy-Green) material metric tensor
problem of evaluating surgicalproceduresin terms of
C = C(x) = F’F
(1)
standard(i.e.non-pathological)kneefunctionality (classical examplesof such proceduresinclude total knee re- and the (Green-Lagrange)material strain tensor
placement(TKR) and elevation of the tibia1 tubercle,
E=E(x)=~(~-~=~(F~F-I)
(2)
Maquet’s procedure.)It must be noted, however,that in
the real patello-femoraljoint, unlike any currently avail- wereusedasalternative objective strain measures
(indifable analysis,many variables usedto describeits bio- ferent to rigid body motion).
mechanicscannot be consideredasindependentbut are
The only forcesassumedto act on a material particle
in fact related to each other in a way which is still not x were the contact forcesper unit of referencesurface,
fully understood.For instance,it is clear that the stresses representedby the nominal stressvector p = p(x). The
occurringin the patellawill largely depend(amongother state of stressat a material point was recordedby the
things)upon the loadsto which the patella is submitted (unsymmetricfirst Piola-KirchholI) nominal stress tensor
The biomechanics
of thehumanpatelladuringpassive
kneeflexion
P = P(x) implicitly definedby the fundamentalformula
P(X)= P(x)W,
(3)
where fi referred to the unit outward normal to the
referencesurfaceon which thesetractions were applied.
The (symmetricsecondPiola-KirchhofQ material stress
tensor
S=S(x)=F-‘P
(4)
was used as an alternative objective stressmeasurein
order to match the preferenceof the material strain
E over the deformation gradient F.
General principles. Since conservation of massand
balanceof angularmomentumaresatisfieda priori in the
material description, the balanceof linear momentum
wasthe only principle of mechanicswhich remainedto be
satisfied.A weak form of this principle, suitablefor discretization, is the principle of virtual work:
1267
geneousmaterial(Heegaard,1993):
s = S(F)= op(F* I
l)[W])
if F 2 0 (tension),
if F < 0 (compression),
(8)
where F = l/l0 is the stretch ratio betweenthe fiber’s
initial length lo and its actual length 1 along the fiber’s
axis f.
Large slip contact. The contact problem betweenthe
patella(consideredasa deformablestriker body) and the
femoralgroove (consideredasa smoothrigid target obstacle)wassolvedby usingfrictionlesslarge slip contact
elements:starting from a three-dimensionalfinite element discretizationof the striker body into N nodesand
of the interfaceinto M node-on-surface contact elements,
a non-lineargap vector and its first variation werederived
in terms of the nodal displacements.The gap vector
d betweena ‘striker’ point y’ belongingto the patellar
tr(Vw’P)dV =
wT@
dA, Vw,
(5) articular surfaceand a ‘target’ surfaceY (Hermite bisn
I OzIP
cubic patch),locally fitting the femoralgroove, could be
where w = w(x) is an arbitrary test function satisfying expressedby
w(x) = 0, Vx E aa,, bestinterpretedasa virtual displaced = WA
= (Y= - Y'),
19)
ment. and whosematerial gradientis VW. X& represents
the part of the boundary where displacements
are pre- where 5 = (r’, r*) represent parametric coordinates
scribedand 8R, the complementarypart where stresses of Y defined over a simply connecteddomain $3 and
y’ = y( 5,) standsfor the projection of y’ on Y, i.e.
are applied.
Constitutive laws. The principle of virtual work (5)
governsthe equilibriumof all deformablebodies,regardSC= arcy>llY(S) - YlII.
(10)
lessof their constitutive materials.Constitutive lawswere
neededto characterizethe behavior of specificmediaas The associatedsignedcontact distanced, could then be
different asbonesand tendons.The patellawasassumed definedas:
to be elastic,isotropic, and homogeneous.
Moreover, the
d, = d,(y’,{,) = d-ii,
(11)
patella beingsubjectedto large displacements
(finite rotations)during kneeflexion, an objectiveconstitutive law where B= f@J representthe inward normal vector to
had to beimperatively employed.The linear elasticlaw of the target surfaceat yc. In this way, d, becamenegative
Kirchhoff-St. Venant (objective form of Hooke’s law) whenevera striker point penetratedthe target surface.
was usedto simulatethe elasticbehavior of the cortical The extremum (ti, 5:) satisfyingequation (10) was oband cancellousbones:
tained by solving the following minimization problem:
S = S(E) = Itr(E)I + 2pE,
(6)
min(+&d) = id.‘(g).
(12)
where 1 and p stand for the Lame’s coe&ients. This
SE3
formulation wasadequateto filter the large rotations of
The contact interaction is governedby the principle of
the patella while keepingtrack of its deformations.
action and reaction which took the discretelocal form
The major drawback of Kirchhoff-St. Venant’slaw is
its lack of monotony under axial or triaxial compresp: =p== -pl,
(13)
sion.This‘instability’ can becomea seriouslimitation in
where the contact force p was defined as the force
the modelingof the articular cartilagewhich works primarily in compression.Most recentmodelsof the cartilage p’ exerted by the striker node y’ on the rigid obstacleat
(Suhet al., 1991)took both its incompressible
liquid and yc, and p1the reaction at y’. This force, which depended
viscoelasticsolid phasesinto account. In the current on d, wasdistributedon the striker nodeaccordingto the
discreteprinciple of virtual work. More specifically
model an alternative law (Rajaonah,1990)was adopted
to representthe cartilageasan elasticnearly incompress6W, = p[d(u)]-ad(u) = f(u)-& = SW,
(141
ible neo-Hookeanmaterial:
i.e. the virtual work 6 W, developedby the contact force
s = S(C) = A(J - l)c-’ + p(C - I),
(7) p [d(u)] through a variation of the contact distanced(u)
must be equal to the virtual work SW developedby the
whereJ = det F. The patella is connectedto the tibia
elementnodal force f’ through the correspondingnodal
through the patellar tendon, which is composedof paraldisplacementsvariation &I. In the frictionlesscase,the
lel, flexible collagenfiber bundlesthat can only support
contact force is directed along the target facet normal
tensile stresses. This latter characteristicwastaken into
fi and could be written as
account through a unilateral Kirchhoff-St. Venant’smaterial law which wasapplied to a uni-directional homop =fnP.
(15)
1268
J. Heegaard et al.
The gap distance was finally related to the conjugate provide exact solutionswith respectto contact, in conpressure by a (multivalued non-differentiable) unilateral
trast to penalty methods.
contact law. At the interface,the frictionlesscontact Iaw
was locally characterized by three complementary Model specijications
Signorini conditions
The ground for the computer model input was pro4 20, L GO, d,f, = 0,
(16) vided by a preliminary experimental study (Heegaard
respectively,expressingthat the contacting bodiescould et al., 1994)on two cadaverknees.In that study, Roentnot penetrateeachother, could not pull on each other gen stereophotogrammetricanalysis (RSA) techniques
(Selvik, 1989)were usedto preciselymeasurepatellar
and were either separatedor pressingon eachother.
Problemformulation.
The principle of virtual work (5), tracking during knee flexion in which the femur was
the constitutive laws(6), (7) or (8) the principle of action rigidly fixed whilethe tibia wasflexed from full extension
of reaction (13) and the contact law (16) together with (0”) to full flexion (150”) in 15” increments,with a conproper boundary condition form a well posedboundary stant 40N force pulling on the quadricepsmuscle.The
value contact problem; yet it is difficult to solve in this correspondingnumericalmodelwasbuilt by generating
form. By assumingthat all the appliedforceswere con- a finite elementmeshof the extensorapparatusand by
servative,a first alternative form wasfound by consider- specifyingproper boundary conditions.
Finite element mesh characteristics. From a structural
ing the total energy n(u) of the system,given by the
to consistof three
differencebetweenthe internal elasticstrain energy 4(u) point of view, the patellawasassumed
constitutive
regions:
(1)
a
core
of
cancellous
bone, (2)
stored in the patella and the external potential energies
a shellof cortical bone and (3)a cartilage layer delimiting
q(u) = @*IIdevelopedby the appliedforcesII
the articular surface(Fig. 1).The geometryof theseparts
44 = 4(u) - v(u).
(17) was obtained by meansof computerized tomography
Adding the first Signorinicondition to this last equation, (Somatron DR3, SIEMENS); the shape and internal
the large slip contact problemreducedto finding a con- structure of each patella were digitized from a set of
strainedlocal minimumof n(u):
4 mm spacedsagittal slices.Hereby, the three-dimensionalmeshgenerationproblemwasreducedto a simpler
min,a(u): = c%(u)
- q(u),
two-dimensionalproblem. The three-dimensionalmesh
(18)
of the patella was obtained by assemblingthesetwoI s.t. d,(u) E Iwy
dimensional meshes.The corresponding mechanical
whered,(u) representedan M-dimensionalvector, whose properties were taken from published values and are
componentscontained the gap distancesof the M con- listed in Table 1.
tact elementsdiscretizingthe interfacebetweenthe conThe femur beingconsideredasa rigid obstacle,only its
tacting bodies.
geometrywastakeninto account:it wasaccuratelymeasThe resulting inequality constrained minimization ured using a stereophotogrammetriccurve reconstrucproblem was then transformed into an unconstrained tions (SCR) system(Meijer et al., 1989)and discretized
saddlepoint problemusingaugmented
Lagrangian multi- into a regularly spacedset of 250points fitting the fempliers,which at equilibriumhold the contact forces.A de- oral groove surface.
Giled description of the method has been discussedin
The joint’s interfacewasmodeledby a setof 100large
(Heegaardand Cumier, 1993).The main featuresof this slip contact elements. Typical valuesfor cartilage-cartimethodare to leadto well conditionedproblemsand to lage friction coefficientsare below 0.01 (e.g. Mow and
Fig.1. Schematic
representation
of thepatello-femoral
three-dimensional
model(left)andits correspondingFE mesh(right).
1269
The biomechanics of the human patella during passive knee flexion
knee 1
knee 2
90
60
30
-f-
numerical
-e-
experimental
e,B’
0
0.
30
60
90
30
60
90
120
-101
150
0
30
60
90
120
15( I
0
30
60
90
120
150
2:
1.
o. e-0
/
9
'
'a
-1.
/
-o-
-2.
-9
\
\
/
F'
i
\ I'
-3. J-4:
30
60
90
120
150
30
60
90
120
150
-4’
0
30
60
90
120
,J
150
30
60
90
120
150
-6:
-9i.
I
0
knee
flexion
[deg.]
-'
1.
0
1
knee
flexion
[deg.]
Fig. 2. Patellar three-dimensional tracking, including (from top to bottom) flexion, tilt, rotation and shift,
as a function of knee flexion for knee 1 (left column) and knee 2 (right column). Solid line curves (filled dots)
represent numerical results and dashed curves (hollow dots) represent experimental results (from Heegaard
et al., 1994).
J. Heegaard et a/.
1270
Table 1. Material constants assumed for the extensor apparatus
elements
Cortical bone
Cancellous
bone
Cartilage
Pattelar
tendon
1 (MPa)
P OfPa)
E
(MPa)
8.65 x lo3
1.73 x lo2
5.76 x lo3
1.15 x 104
1.5 x lo4
3.0 x lo2
1.065 x 10’
0
6.8 x 1o-3
5.0 x 10’
Table 2. Finite element mesh specifications of the two knees
used in the model
v
0.30
0.30
Problem dimension
DOF permode
Total number of nodes
Total DOE
Displacement BC
Force BC
Number of sets
*Taken from Reilly et al. (1972), Townsend et al. (1976) and Elements
Mow et al. (1991). I and p represent the Lam6 coefficients, E the Cortex (eight node isoparam)
Young modulus and v the Poisson ratio.
Cancellous bone (eight node
isoparam)
Cartilage (eight node isoparam)
Tendon (two-node line)
Soslowsky,1991)so that contact could be assumedfric- Contact (two-node large slip)
tionless.The striker nodesbelongedto the external sur- Femur (16~nodes Hermite)
faceof the patellar cartilagelayer. Adjacent nodesfrom Flexion increment
the femoral groove were connectedtogether into 16- Number of increments
nodesbi-cubic Hermit target patches.AssociatedLag- Convergence criteria
rangian nodes holding the unknown contact force were
2.0
lo2
0.47
0.00
addedand connectedto their respectivestriker nodes.
The patellar tendon(PT), connectingthe patella to the
tibia1 tuberosity, was discretized into three string-like
elements(Fig. 1) representinglateral, central and medial
fiberswhoseinsertionsitesweremeasuredusingthe RSA
system.
The characteristicsof the final meshfor both kneesare
summarizedin Table 2. The foregoingcontinuum mechanicsproblem with unilateral contact wasimplemented
and solvedwith the program TACT (Curnier, 1985).
Boundary conditions. The three tibia1 insertion points
of the PT were subjectedto a prescribedmotion which
controlled the flexion process.The successivepositions
of thesepoints were recorded during the preliminary
experimentusing the RSA system.In the model, knee
flexion angleswereonly consideredin the rangebetween
30 and 135”for stability reasons(Heegaardet al., 1994).
A constant40 N pulling force, representingthe rectus
femoris (RF) actions, was equally distributed on three
nodesof the patellar basecentral part. The direction of
theseforces varied during flexion (due for example to
muscleinteraction with the femoralgroove).An estimate
of thesedirectionswasobtainedduring the experimental
measurements
of patellar kinematics.
Model sensitivity. When the articular surfacetopography or the patellar structure geometry were measured,
the reconstructionerrors were small(x l/10 mm) when
comparedto the uncertainty on localization of PT and
RF insertionpointson the patella(x l-2 mm).Therefore
the influenceon computedpatellar kinematicsof a 2 mm
posteriorshift of the PT insertionsand of a 2 mm posterior or anterior shift of the RF insertions, was also
analyzed.
Knee 1
Knee 2
3
3
1445
3452
903
9
6
3
3
1371
3120
993
9
9
280
252
320
180
3
99
148
7.5”
14
1o-4
288
162
3
90
169
7.5”
14
1O-4
Kinematics
The motion of the patella is expressedby three Eulerand three translations relative to a fixed
coordinate frame Wr attached to the femur (Fig. 1).
Adopting the Tait-Bryan convention, the rotations are
carried first around the x-axis (patellar Jlexion), then
around the y-axis (patellar tilt) and finally the z-axis
(patellar rotation). Patellar shift denotestranslation along
the x-axis.
Model validation. The computedkinematicsof the patella (Fig. 2; solidline, black dots)matchalmostperfectly
the correspondingexperimentalcurves(dashedline, hollow circles).The averageroot meansquare(RMS) errors
between numerical and experimental curves are less
than 0.5” for both specimens.
Maximum deviationsand
RMS errors are listed in Table 3 for both knees.The
maximum errors occur at high flexion anglesin both
specimens(except for the secondknee patellar flexion).
Patellar shift RMS error is below0.5 mm for both specimens.
Sensitivityanalysis. Patellarkinematicswerenot significantly altered after slightly shifting the tendon insertions, as shownon Table 4 which summarizesthe maximum and RMS tracking deviationsbetweenalteredand
original tendon insertions.In all casesthe RMS deviations are below 0.2” for the rotations and 0.03mm for
the translations.
ian rotations
Patellar
tendon tension
The computedtensionllfprll in the PT (representingthe
magnitude of the three fibers resultant) is reported in
Fig. 3 asa function of knee flexion for both knees.This
RESULTS
tension remainsbelow 40 N for both specimens.The
The output from the computermodelincludespatellar quadricepspulling force magnitudellfoll being constant
kinematics,PT tension,joint contact pressureand patel- (40N), it is readily seenthat the ratio llfPrll/llfoll is below
lar stresses,
as a function of knee flexion.
unity throughout the consideredflexion range(30-135”).
The biomechanics of the human patella during passive knee flexion
1271
Table 3. Maximum absolute error (E,,.) and RMS error (8) between experimental and computed patellar
tracking parameters during knee flexion
Flexion
Tilt
Shift
B
%hux
B
4mx
B
Klux
B
0.78
0.60
1.12 (105.00”)
1.21 (119.80”)
0.36
0.42
1.20 (89.06”)
1.64 (138.00”)
0.39
0.55
0.74 (139.20”)
1.61 (42.08”)
0.21
0.49
hm%
Knee 1 2.51(139.20”)
Knee 2 1.91 (42.08”)
Rotation
Note: Errors for the rotations are in degrees and in mm for the translations. The tibia1 flexion angle
corresponding to the maximal error is indicated between parentheses.
Table 4. Maximum absolute error (E,..) and RMS error (3 between the standard simulation and perturbed
simulations: (a) PT 2 mm posterior shift, (b) RF 2 mm posterior shift and (c) RF 2 mm anterior shift.
Flexion
(4
Tilt
4M.
B
0.47 (89.06”)
0.68 (9.48”)
0.20 (89.06”)
0.13
0.19
0.06
Rotation
L
0.12 (72.23”)
0.17 (72.23”)
0.27 (9.48”)
Shift
H
Gil..
z
hn..
z
0.05
0.05
0.07
0.38 (89.06”)
0.20 (9.48”)
0.13 (9.48”)
0.15
0.07
0.04
0.09 (9.48”)
0.09 (25.25”)
0.06 (9.48”)
0.03
0.03
0.02
Note: Errors for the rotations are in degree and in mm for the translations. The tibia1 flexion angle
corresponding to the maximal error is indicated between parentheses.
articular surface,whereasin the secondknee,the medial
and lateral zonesmergetogetherthrough a narrow contact zone acrossthe central ridge. In the latter specimen,
a secondmedial contact zone appearson the mediodistal edge.Except in the mid-flexion range, the lateral
facet bears more pressure (mean peak pressure:
prnax
= 0.51MPa for knee 1, and 0.52MPa for knee 2)
than the medial one (ji,,,.. = 0.4 MPa for knee 1, and
40
z
$
2
s
‘0
d=
i
g
36
32
26
0.48 MPa for knee 2) (Fig. 5).
24
20
--A-
knee
-
knee2
Cancellous bone stresses
1
The stresses
occurring in the patellar cancellousbone
are expressedin termsof hydrostatic and deviatoric (von
Mises)invariants of the stresstensor. Throughout knee
knee flexion
[deg.]
flexion, stresses
are concentratedmainly at the periphery
Fig. 3. Patellar tendon tension as a function of knee flexion, for of the cancellousbone,near the cortical shell,whereasin
knee 1 (triangles) and for knee 2 (squares). The dotted line the bulk the stresses
remainlow. In the sagittalplane,the
corresponds to the applied 40 N quadriceps force.
compressivehydrostatic pressuresare primarily located
beneaththe patello-femoralcontact region(Fig. 6).In the
frontal plane,the higheststresses
are locatedbeneaththe
lateral contact area (Fig. 7). Tensilehydrostatic stresses
The tension llfpTlldecreases
until 90” of flexion whereit
reaches a minima (33.3 N for specimen 1 and 22.5N for are mainly concentratedat the distal subchondralbone,
beneaththe patellar tendon insertion region (Fig. 8). In
specimen2), and then increasesagainduring the last 40
the frontal plane, tensilestresses
are found beneaththe
of kneeflexion.
lateral facet (knee 1) or the medial one (knee2) (Fig. 9).
Von Misesare primarily locatedat the distalpole (patelContact pressures
lar apex) beneath the patellar tendon insertion region
Both knees present similar contact pressure distribuand along the anterior face of the cancellousbulk
tions up to 90” of kneeflexion: contact areasare divided
(Fig.
10).
into a smallermedial zone displaying higher pressure
gradientsthan over the lateral facet (Fig. 4). Theseareas
shift proximally during flexion. However, between90
DISCUSSlON
and full flexion, inter-specimendifferencesappear:for the
first knee, the contact area is still split over the medial
The purposeof this researchwas to adopt a global
and lateral facet and is located in the upper third of the approachto the analysisof patello-femoralbiomechanics
(’
45
.
.
60
’
*
75
’
’
90
s
’
105
’
’
120
*
’
135
.
150
1
1272
J. Heegaard et 01.
60”
proximal
medial
t
+--/----
lateral
distal
90”
120”
knee 1
knee 2
Fig. 4. Patello-femoralcontact patterns at (from top to bottom): 45,60,90 and 120”of knee flexion,for knee
1 (right column) and knee 2 (left column).
by simultaneously assessing its kinematics and dynamics.
Such an approach became essential when realizing
how the loads acting on the patella vary during its
motion.
The computed three-dimensional motion of the patella
during knee flexion could be characterized by an increasing flexion, a wavy tilt, a small lateral rotation and
a me&o-lateral shift. Such trends were already found in
previous experimental studies (Heegaard et al., 1994;
van Kampen and Huiskes, 1990). Patellar tracking could
thus be used to validate the model by comparing computed results with the corresponding experimental ones.
Validation of the model rested on a specimen-related
comparison: predicted numerical results were analyzed
and compared to equivalent experimmtal ones. Here,
‘equivalence’ means that the model used the same articu-
The biomechanics of the human patella during passive knee fledon
MEDIAL
LATERAL
60
45
60
75
90
105
120
135
45
knee flexion [deg.]
1273
60
75
90
105
120
135
knee flexion [deg.]
Fig. 5. Peak pressures on the medial (left) and lateral (right) contact surface as a function of knee flexion, for
knee 1 (dark gray) and knee 2 (light gray). The dashed horizontal line represents the average of the peak
pressures over the full flexion range.
Anterior
120”
Fig. 6. Evolution of compressive hydrostatic stressesin the cancellous bone (knee 1) at 45,90 and 120” of
knee flexion, as viewed across a mid-sag&al section.
lar surfacegeometriesand boundary conditionsasthose
usedin preliminary experimentalstudy (Heegardet al.,
1994).During this experimentalstudy, it wasshownthat
a simplified description of the extensor apparatus, in
which all soft tissuestructures(exceptfor the PT and RF)
wereremoved,is still ableto predict meaningfulresultsin
the knee flexion range from 30” to full flexion. The
presentnumerical resultshave in turn shown that the
proposedmathematicalmodelrepresentingthis simplified extensorapparatusaccurately reproducedthe experimentaltracking within the specifiedflexion range.
Smallvariationsof the tendon insertionlocationsproduced only smallchangesin patellar tracking (Table 4),
reflectingthe good stability of the modelwith respectto
positioningof the extensorelements(patellartendon and
rectusfemoris).Thus, sometolerancein the positioning
of the bony insertion sites of thesestructureswas allowed. This was of particlilar importancein the case of
the patella,whereit isdifficult (indeedevenimpossible)to
locateRF and PT insertionsitesprecisely,asboth structuresseemto blend togetherover the anterior margin of
the patella. Furthermore, the effectsof quadricepstension characteristicson the patellar trajectory have been
shown to be small (Ahmed et al., 1989; van Kampen,
1987)especiallyin the consideredflexion range.
The predicted patellar tendon force magnitudeswere
consistentwith publishedexperimentalresults(e.g. Ahmedet al., 1987;Ellis et al., 1980;Huberti et al., 1984)or
numericalresults(e.g.Hirokawa, 1991;van Eijdenet al.,
1986;Yamaguchi and Zajac, 1989),all of which found
a decreasingratio up to about 90” of knee flexion. According to Ellis et al., (1980)the differenceof tensionsin
the PT and quadricepsis not dueto frictional forcesbut
due to the geometry of the patello-femoraljoint. This
could befurther deducedin the presentfrictionlessmodel
by consideringthe equilibriumof momentsof the pulling
forcesf, and fo actingon the patella:the ratio /f,ll/llf,-J
reflected,in first approximation, the ratio betweenthe
J. Heegaard et al.
1274
Wal
0
Lateral
Anterior
+
+-/-+
Medial
+
Posterior
Fig. 7. Evolution of compressioe hydrostatic stresses in the cancellous bone (knee 2) at 45,!N and 120” of
knee flexion, as viewed across a transverse section.
[MPal
5 410 -2
Proximal
Posterior
Anterior
+
45"
Distal
90”
120"
Fig. 8. Evolution of tensile hydrostatic stressesin the cancellous bone (knee 2) at 45,90 and 120” of knee
flexion, as viewed across a mid-sagittal section.
area proximal and distal to the contact region, which
decreased
until 90” of flexion asthe contact regionmoved
from distal to proximal.
The motion of the contact area evolved during knee
flexion along the sametrends asthose reported by pre-
vious investigations(e.g. Ahmed 1983;Fujikawa et al.,
1983)and could be characterizedby a proximal shift of
the contact region during the first part of flexion (up to
90” of flexion) and by a backward distal shift during the
last phaseof knee flexion. Noticeable differencesap-
The biomechanics of the human patelladuringpassive
kneeflexion
Lateral
Anterior
+
1275
w4
Medial
0
> 410-2
Posterior
Fig.9. Comparison
betweentensile hydrostaticstresses
in the cancellous
boneknee1 (top) and knee
2 (bottom) at 45,90 and 120” of knee flexion, as viewed across a transverse section.
WW
> 410-2
Proximal
4
Anterior
Posterior
45"
Distal
90”
120"
Fig. 10. Evolution of von Mises stressesin the cancellous bone (knee 1) at 45,90 and~l20” of knee flexion, as
viewed across a mid-sagittal section.
pearedbetweenboth specimens,
stressingthe influenceof
individual anatomy on thesepatterns.This descriptionof
the contact patternsevolution yieldedto a global view of
the pressuredistribution acrossthe articular surface,but
did not provide precisequantitative information about
the pressuremagnitudes.Hence,to completethe description, peak pressures on the medialand lateral facetswere
also analyzed as a function of knee flexion (Fig. 5) and
showedthat the pressuredistribution betweenthe medial
and lateral facetsis characterizedby higher lateral peak
pressuresnear extension and full flexion, and by an
almost even distribution in the mid-flexion range. This
could further confirm the tendency to find more worn
fibrils on the superficiallayer of the lateral facet in patellar chondropathy (Mori et al., 1993).
Patellar trabecular bone architecture has been describedasa non-homogeneous
stacking-upof sheetsand
struts (Townsendet al., 1976),resulting from an optimizedremodelingprocess.Moreover, the loadssustained
by the patella dependon the location of the contact areas
which vary during flexion. It follows that the mechanical
propertiesof cancellousbone will be spatiodependent.
It can thus be hypothesizedthat the highesthydrostatic
compressivestresses
found primarily beneaththe lateral
1276
J.
Heegaard et al.
facetare relatedto the highersubchondralbonedensities expressingequilibrium in termsof rigid body kinematics
found in the proximal part of the lateral facet(Ecksteinet parameters. Suchmodelshave beenwidely usedfor anaal., 1992).In the presentmodel,however,cancellousbone lyzing kneejoint kinematics(e.g.Andriacchi et al., 1983;
wasconsideredasan isotropic and homogeneous
mater- Blankevoort and Huiskes, 1991; Essingeret al., 1989;
ial, suggestingsomecare in interpreting the computed Wismanet al. 1980),with emphasisbrought to the tibiofemoral joint (a comprehensivereview can be found in
stresses.
It is, nevertheless,
instructive to relatethesepatternsto (Huiskes,1992)).However, lessattention hasbeenpaid in
available resultsin the literature. Using a simplethree- modeling the patello-femoraljoint (Hefty and Grood,
dimensionalplanar beam model, Minns et al. (1979) 1988).Van Eijden et al. (1986)publishedthe first mathcomputed surfacestresspatterns along transversesec- ematical model to compute patellar kinematicsby detions, which were characterizedby compressionat the scribingthejoint asa two-dimensionalmechanismacting
posteriorface(articular surface)and tensionat the oppo- in the sagittal plane. Since then only a few more twosite anterior surface.They further observedthat high dimensionalpatellar models have been published up
tensilestresson the medialaspectmay exist during knee to this date (e.g.Reithmeierand Plitz, 1990;Yamaguchi
flexion. Current resultsfor the hydrostatic stresspatterns and Zajac, 1989). Finally, Hirokawa (1991)presented
computedin both specimensmadeit unclear, however, a three-dimensionalgeneralizationof van Eijden’stwowhether such tensile stresspatterns occur: in the first dimensionalmodel.
specimenthey appearedover the lateral sideasopposed
In contrast to rigid body models,continuousdeformto the secondspecimenwhere sometensilestresswas able modelsare characterizedby an infinite number of
found in the posterior medialaspect.Most of the tensile degreesof freedom.Hence,equilibrium conditions take
stresswas concentrated near the PT insertion area, the form of a systemof non-linear partial differential
whereasthere was no suchconcentration at the quad- equation (NPDE) instead of a finite set of equations
ricepsinsertion,underscoringthe importanceof the bone characteristicof rigid body systems.However,whenconshapein stressdistribution (the PT insertsin a region fronted with theseNPDE.over suchcomplex shapesas
wherethe sagittalradiusof curvature of the cortical shell thosefound in humanbones,analytical solutionsdo not
is smallerthan the one near the quadricepsinsertion, exist. TheseNPDE are thus approximated (to a fixed
thus favoring stressconcentrations).
degreeof accuracy)by a set of non-linear equationsby
The highestshear stressvalues were observedalong discretizing the geometricaldomain under stiidy using
the anterior surfaceof the cancellousbone,which in case the finite element method (FEM) (e.g. Cumier, 1994;
of homogeneous
and isotropic materialagreeswith pre- Hughes,1987;Zienkiewicz and Taylor, 1991).A general
vious results(Hayeset al., 1982).High shearstresses
were descriptionof this method,its principles,its possibilities
alsofound beneaththe patellar tendon insertionsite. In and limitations in orthopedic biomechanicshas been
termsof tissuedifferentiation, this could further explain publishedby Huiskesand Chao (1983).As for rigid body
how the tendon insertsin a reinforcedregion,by assum- models,only a few mathematicalstudieshave beendeing that ossificationof cartilage is acceleratedin those voted to patellar stressanalysis(e.g.Hayes et al., 1982;
regions exposed to higher shear (deviatoric) stresses Minns et al., 1979;Minns and Braiden, 1981).
(Wong and Carter, 1990).
Besidesthe rigid or deformableapproach in joint
At this point it is appropriateto mention the essential biomechanics,a still challengingproblem arisesfrom the
featuresof the computermodel presentlyusedto assess contact betweenthe componentsconstituting the joint
the globalpatello-femoralbiomechanics.
Currently avail- (i.e. they are free to separatebut cannot penetrateeach
able mathematicalmodelsin joint biomechanicsusually other). The finite element method has provided the
deal either with rigid body systems,in which forcesand groundfor a numberof efficientsolutionsto this problem
momentsare related to rigid body motionsthrough the by the implementationof contact elements
(e.g.Alart and
laws of classicalmechanics(i.e. Newton Laws) or with Cumier, 1991;Chan andTuba, 1971;Hugheset al., 1976).
deformable bodies, in which stresses
are relatedto strains The effectiveness
of suchelementsin joint biomechanics
through constitutive lawsin addition to the former rela- was illustrated by, for example,Chan and Rim (1976),
tionships.On the one hand, rigid body systemsprovide Rapperport et al., (1987), Huber-Betzer et al., (1990),
only a coarseapproximation of the joint’s interface be- Weinanset al., (1990)and Rubin et al., (1993).However,
havior while deformablebody systemsare$xed and are thesecontact elementswerecharacterizedby a node-onfurther submittedto fully prescribedexternal loads rep- node geometryand werethereforerestrictedto small slips
resentingonly an approximation of the unknown joint betweenthe contactingbodies.Thus,moving joints, where
forces.
relative motion becomeslarge,wereprecludedwith these
Most modelsdealing with joint kinematicsare two- elements.
dimensional(resp.three-dimensional)
rigid body models,
In the presentmathematicalmodelof the patello-femwhereeachcomponentof the systemhasthree (resp.six) oral joint all the foregoing limitations were overcome.
degreesof freedom.Sometimes,
thesemodelsalsoinclude First, the geometricalnon-linearcontinuum formulation
a few additional degreesof freedomtaking into account adopted to describedeformablebodies kinematicsalthe joint componentsflexibility and are governed by lowedone to considerlargedisplacements
of the patella,
a constitutive law (e.g.ligaments,articular contact). All including finite rotations, from which the pseudo-rigid
thesemodelsend up with a set of non-linear equations body kinematicscould be extracted.Secondly,by assum-
The biomechanics of the human patella during passive knee flexion
ing the patella as a moving deformablebody slidingon
the femur, stressesoccurring in the patella could be
evaluatedduring its motion and not only at a prescribed
fixed position.
Finally an essentialfeature of this model was to considerthe forcesacting on the joint aspart of the problem
unknowns.For each successive
patellar position occurring during knee flexion, the corresponding
system of
forces, including quadricepsforces (imposed),patellar
tendon forcesand contact forces,representeda loading
systemconsistent with the computedpatellar tracking,
and thereforeprovided more realisticloadingconditions
acting on the patella than applying
fully imposed forces.
1277
Ahmed, A. M., Chan, K. H., Shi, S. and Lanzo, V. (1989)
Correlation of patellar tracking motion with the articular
surface topography. Proc. 35th. Annual Meeting ORS, p. 202.
Ahmed, A. M., Yu, A. and Burke, D. L. (1983) In t&o measure
ment of static pressure distribution in synovial joints-part II:
retropatellar surface. J. biomech. Engng 105, 226236.
Alart, P. and Curnier, A. (1991) A mixed formulation for friFtional contact problems prone to Newton like methods. Comput. Meth.
appl. Mech.
Engng
9, 353-375:
Andriacchi, T. P., Mikosz, R. P., Hampton, S. J. and Galante, J.
0. (1983) Model studies of the stiffness characteristics of the
human knee joint. J. Biomechanics 16, 2329.
Bandi, W. (1970) Chondromallacia Patellae und Femoro-Patellare Arthrose. Helu. Chir. Acta (suppl.) 11, l-70.
Blankevoort, L. and Huiskes, R. (1991) Ligament bone interaction in a three-dimensional model of the knee. J. biomech.
This wasespeciallyrelevant when consideringthe wide
Engng 113,263-269.
range of contact pressure variations occurring during Bonnel, F. (1988) Anatomie de I’appareil extenseur du genou. In:
knee flexion, and which could directly affect suhchondral
bonestresses.
It followsthat the hydrostatic pressureand
von Mises stress could he further related to theseloading
conditions, unlike the aforementionedstressstudies,in
L’appareil
extenseur du genou (Edited by Mansat, C., Bonnel,
F. and Jaeger, J. H.). Masson, Paris.
Brossmann, J., Muhle, C., Schroder, C., Melchert, U. H., Bull, C.
C., Spielmann, R. P. and Heller, M. (1993) Patellar tracking
patterns during active and passive knee extension: evaluation
with motion-triggered tine MR imaging. Radiology
187,
205-212.
Chan, R. and Rim, K (1976) Stresses in the human knee joint.
J. Biomechanics
9, 417-422.
Chan, S. H. and Tuba, I. S. (1971) A finite element method for
contact problems in solid bodies. Int. J. Mech. Sci. 13,
615-639.
Cumier, A. (1985) TACT: A Contact Analysis Program. In
which all the forces acting on the patella were prescribed.
This important feature followed from the adopted
method to include unilateral large slip contact in a nonlinear continuum model of the joint: the problem could
he stated as a constrained optimization
one, where the
total energy of the extensor system, including the patella
and the patellar tendon, had to be minimized under the
Unilateral
Problems
in Structural
Analvsis-2.
CISM
304.
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(Edited by Maceri, F. and Del Piero, G.).- Springer, Wien.
Methods
in Solid Mechanics.
the femur he non-negative. This minimum was then char- Curnier, A. (1994) Computational
Kluwer, Amsterdam.
acterized by means of an augmented Lagrangian
funcSilva, 0. L. and Bratt, J. F. (1970) Stress trajectories in the
tional. The resulting model highlighted the accuracy that Dapatella.
Acta orthop. stand. 40, 608-618.
could he obtained by combining a rigorous contact law Eckstein, F., Muller-Gerbl, M. and Putz, R. (1992) Distribution
and a precisegeometricmodel.
of subchondral bone density and cartilage thickness in the
human patella. J. Anat. 180, 425-433.
A natural extension to the present model would he to
consider the trabecular bone as inhomogeneous,
with Ellis, M. E., Seedhorn, B. B., Wright, V. and Dowson, D. (1980)
An evaluation of the ratio between the tensions along the
spatial dependent density (or equivalently spatial depenquadriceps tendon and the patellar ligament. Engng Med. 9,
dent stiffness) which according to Hayes et al. (1982)
189-194.
could lead to differences, from the homogeneous case, as Essinger, J., Leyvraz, P. F., Heegaard, J. H. and Robertson, D.
D. (1989) A mathematical model for the evaluation of the
large as 200% in the peak von Mises stresses. Such
behaviour
during flexion of condylar type knee prostheses.
features are already available in the present model
J. Biomechanics
22, 1229-1241.
(Rakotomanana
et al., 1992), but were neglected here for Ferguson, A. B., Brown, T. D., Fu, F. H. and Rutkowski, R.
simplicity (identification
of the associated material con(1979) Relief of patellofemoral contact stress by anterior displacement of the tibia1 tubercle. J. Bone Ji Surg. 61A,
stants remainsdifficult). The articular cartilagecould be
159-166.
further considered as a bi-phasic material, which would
Ficat, R. P. (1970) Pathologie @more-patellaire.
Masson et Cie,
require further development. It can be noticed that the
Paris.
augmented Lagrangian formalism introduced to enforce Ficat, R. P. and Hungerford, D. S. (1977) Disorders of the
exactly contact conditions could also he used to handle
Patello-femoral
Joint. Masson, Paris.
der
the cartilage fluid phase incompressibility
condition. Fi- Fick, R. (1904) Handbuch der Anatomic und Mechanick
Gelenke. G. Fisher, Jena.
nally, including bone remodeling in such a model could Freehafer,
A. A. (1962) A study of the function of the patella.
help to understandthe relationshipbetweenjoint motion
Clin. Orthop. 25, 162-167.
and underlying bone morphology and structure better. Fujikawa, K., Seedhom, B. B. and Wright, W. (1983) Biomechanics of the patello-femoral joint. Part I. Engng Med. 12,
3-11.
Acknowledgements-This
research project was sponsored in
S. A., Coale, E., Weiss, A.-P. C. and Grossnickle, M.
part by the Swiss National Fund for Scientific Research, Grant Goldstein,
(1986) Patellar surface strain. J. orthop. Res. 4, 372-377.
No. 32-30013.90. The authors thank Pierre Alart, Dominique
J., Hungerford, D. S. and Zindel, M. (1976) PatelPi;i;tti and Pascal Rubin for valuable contributions to this Goodfellow,
lofemoral joint mechanics and pathology. J. B&e it Surg.
58-B. 287-299.
Goym&, V. and Mueller, H. G. (1974) New Calculations of the
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