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Using multibody dynamics to design total knee
replacement implants
1
John L. Williams,
2
Said T. Gomaa
1
Biomedical Engineering, University of Memphis, Memphis, Tennessee, USA
2
DePuy, a Johnson & Johnson Company, Warsaw, Indiana, USA
Abstract Computational mechanics methods, such as finite element analysis or
multibody dynamics, are usually employed toward the end of the design phase of a
total joint implant system. It is of greater benefit, however, to utilize these meth-
ods early in the design process as a benchmarking tool to compare competitive
products, as a screening tool to eliminate poor design concepts, and as a means to
virtually test selected designs to determine if they meet the functional require-
ments prior to cadaver testing in an experimental knee rig. The use of a purpose-
written commercial multibody dynamics program has provided computational ad-
vantages for this purpose in an industrial setting, saving an unprecedented amount
of time required for addressing design questions, prior to prototype manufacturing
and testing. Such methods can be successfully employed to deal with challenging
and clinically motivated design questions. This paper illustrates the use of compu-
tational mechanics as an enabling technology to discover design-related factors
that contribute to unsatisfactory functional performance in some patients. As an il-
lustrative example, it is demonstrated that the sagittal design of the femoral com-
ponent of a total knee replacement is responsible for the observed phenomenon of
paradoxical anterior motion in knee bending activities, and that minor design mod-
ifications can reduce or eliminate and even reverse paradoxical anterior displace-
ment in deep knee bending.
Published in: Adam Wittek, Karol Miller, Poul M.F. Nielsen (eds.).,
Computational Biomechanics for Medicine: Models, Algorithms and
Implementation, DOI 10.1007/978-1-4614-6351-1_14, Springer
Science+Business Media New York 2013, pp. 157-168.
Introduction
Paradoxical anterior motion has been clearly demonstrated to occur in total
knee replacement (TKR) subjects by in vivo fluoroscopy. The term ‘paradoxical’
describes the observed anterior sliding of the femur on the tibia while the subject
bends the knee whereas the femur is expected to roll back as it does in a healthy
knee. It has been reported to occur to varying degrees in most total knee replace-
ment devices. Paradoxical anterior motion has been observed in both posterior
cruciate-retaining and posterior cruciate substituting TKR designs [1]. During
weight-bearing knee flexion posterior ‘rollback’ of the femur occurs between 0
and 30 degrees of flexion, but beyond 30 degrees the medial side of a TKR femo-
ral component typically moves anteriorly with knee flexion while the lateral side
stays in a relatively fixed posterior position relative to the tibial component. Some
of the negative consequences of this paradoxical movement are: limited maximum
flexion, difficulty getting out of a chair, ascending and especially descending
stairs, accelerated wear of the tibial insert and the feeling of instability described
as “walking on ice” [2]. Although the phenomenon has been observed in most sin-
gle-leg knee bending fluoroscopic studies for some time, the exact cause of this
undesirable kinematic outcome has only recently been discovered [3-5]. In this
paper we demonstrate that paradoxical anterior sliding is caused by a common de-
sign aspect of TKR femoral components. We compare multibody dynamics simu-
lations of a double leg squat of several posterior cruciate retaining total knee im-
plant designs using measurements of the tibio-femoral ‘contact positions’ similar
to those published in fluoroscopic studies, in order to examine the nature and ex-
tent of paradoxical anterior motion in relation to the sagittal curvature of the femo-
ral and tibial component contact surfaces.
Materials and Methods
Multibody Dynamics Model A commercial multibody dynamics virtual knee
simulator (LifeMOD™/KneeSIM, LifeModeler, Inc., San Clemente, CA, USA)
was used to simulate a double-leg deep knee bend in a manner similar to the ‘Pur-
due Knee Simulator. Details of the KneeSIM program and many model parame-
ters have been described elsewhere [6]. The model included tibio-femoral and pa-
tello-femoral contact, ligaments (MCL, LCL and PCL, capsular tissues, and
quadriceps and hamstring muscles (Fig. 1).
Fig.1. Model of an implant-
ed left knee in a squat simu-
lation. The quadriceps and
hamstrings muscles and the
posterior cruciate ligament
and wrapping patellar tendon
and ligament are shown. The
capsular tissues and the col-
lateral ligaments are not
shown, but are present in the
model.
The MCL, LCL and PCL, and capsular stiffness properties were modeled
with point-to-point elements (multiple elements for the capsule, two elements for
the MCL and single elements for the PCL and LCL). Ligament lengths were cal-
culated using the distance between the attachments. Ligament force was calculated
from the relation: F = k L - cv, where k is the stiffness, e is the strain, L is the
original ‘free length’, c is the damping, and v is the velocity. The ligament stiff-
ness (slope of the force-strain curve) values - 1900 N (PCL), 3800 N (MCL), 3800
N (LCL), and 8900 N (capsule) - were based on comparisons to literature values
and on comparisons of model simulations to implanted cadaver knee experiments.
An initial pre-tension of 44 N was applied to each of the ligaments.
Flexion/extension at the hip and ankle joints, and abduction/adduction, var-
us/valgus and axial rotation at the ankle joint were unconstrained while a constant
vertical load of 463 N (equal to half of the body weight of a 95 kg subject) was
applied at the hip. A proportional/derivative (PD) feedback control system was
applied to the quadriceps and hamstrings muscle forces to maintain a user-input
knee flexion angle history. The PD controller compared the instantaneous knee
flexion angle during the simulation to the desired (input) knee flexion angle. The
error was then multiplied by a Pgain resulting in a quadriceps/hamstrings muscle
tension force. The derivative gain (Dgain) was a fixed percentage of the Pgain,
and was applied as a damping force to the error rate. The force applied to the ham-
strings was proportional to the product of the error and quadriceps force. The sys-
tems were subjected to one 9-second cycle of knee bending up to 120 degrees of
flexion. The anterior-posterior (AP) positions of the lowest points on the femoral
lateral and medial condyles closest to the tibial tray (the same measure used in
fluoroscopy studies) were recorded relative to the dwell points of the inserts.
Model validation, in the sense of evidence that the model is suitable for the tasks
at hand, has been addressed by several studies [6], including comparisons to the
ASTM F1223-89 laxity test and comparisons to in vitro knee cadaver testing and
in vivo fluoroscopic data [7].
Commercial Designs Reverse-engineered models of commercially available
fixed-bearing cruciate-retaining total knee implant systems (NexGen®, Zimmer,
Warsaw, IN, USA; Biomet Vanguard™, Biomet, Warsaw, IN, USA) were im-
ported into the model without modification to the design of the tibial insert or
femoral components. These were among many such simulations performed on a
wide variety of commercial designs to determine if multibody dynamics could
predict the fluoroscopically observed paradoxical anterior motion.
Influence of Femoral Contact Geometry on Kinematics In order to isolate the
influence of the femoral condyle geometry from tibial insert geometry, simula-
tions were also performed after replacing the commercial tibial insert with a per-
fectly flat tibial insert. For this purpose, we report simulation results of reverse-
engineered models of fixed-bearing cruciate-retaining total knee implants: The an-
terior-posterior positions of the condylar lowest points (CLP) on the femoral con-
dyles closest to the tibial tray (the same measure used in fluoroscopy studies) were
recorded relative to the equilibrium position in full extension. For the flat inserts
the CLP coincided with the ‘contact points’ or centers of pressure (COP) of the
calculated tibio-femoral contact areas.
Theoretical Designs Several theoretical TKR models were designed to further
explore the relationship between design and kinematics. The anterior geometry
(trochlear and patello-femoral surfaces) and patella of the theoretical designs were
identical in all of these theoretical models. The tibio-femoral contact surfaces of
the femoral components were designed to have multi-radial curvatures in the para-
sagittal planes (polycentric designs) (Fig. 2).
Fig. 2. Lateral view of a para-sagittal cross-
section of a generic femoral component
showing a radius of curvature in the posterior
condylar region decreasing from R
1
to R
2
. The
angle where the change in curvature occurs
was measured from the vertical axis.
Two cases are presented: In the first the para-sagittal radius decreased from
35 mm at full extension to 25 mm at 30 degrees of flexion, and in the second the
radius increased from 25 mm to 30 mm at 30 degrees of flexion. Simulations with
these two femoral designs were performed using both a generic perfectly flat tibial
insert and an insert with curvature in the para-sagittal and para-coronal planes in
order to isolate the influence of the femoral condyle geometry from that of the tib-
ial insert geometry. Both inserts had the same antero-posterior dimensions. In ad-
dition to the CLP, we also report translations of the calculated centers of pressure
(COP) of the medial and lateral tibio-femoral contact patches. More extensive
simulations using curved tibial inserts and numerous variations in the femoral sag-
ittal radius of curvature change can be found in a recent patent application [3].
Results
Commercial Designs In the original commercial designs analyzed in this fash-
ion we noted that the onset of paradoxical anterior motion occurred at angles that
corresponded approximately with the flexion angles at which the femoral para-
sagittal radius abruptly decreased (Table 1, Fig. 3 and 4).
Fig. 3. Antero-posterior (+ANT)
translation (mm) of the lowest
condyle points (CLP) of a
Zimmer NexGen TKR, as a
function of knee flexion angle
during a lunge simulation (me-
dial condyle: dashed line; lateral
condyle: solid line).
Fig. 4. Antero-posterior (+ANT)
translation (mm) of the lowest
condyle points (CLP) of a Biomet
Vanguard TKR, as a function of
knee flexion angle during a lunge
simulation.
Table 1. Magnitude and angle of discrete femoral radius reduction as determined
from reverse-engineered components
Knee System
Medial
Lateral
NexGen®
35% at 37 deg.
50% at 16 deg.
Vanguard™
35% at 20 deg. and
20% at 90 deg.
35% at 20 deg. and
20% at 90 deg.
Scorpio®
6% at 75 deg.
6% at 75 deg.
Influence of Femoral Contact Geometry on Kinematics Posterior translation
of the medial and lateral tibio-femoral ‘contacts’ of NexGen®, Vanguard™ and
Scorpio® either ceased or slowed at flexion angles between 16 and 75 degrees
(Figs. 5-7). Simulations using flat tibial inserts indicate paradoxical anterior mo-
tion started at about 40 degrees for NexGen® (Fig. 5), 90 degrees for Vanguard™
(Fig. 6) and almost imperceptibly at about 70 degrees for Scorpio® (Fig. 7).
Fig. 5. Antero-posterior (+ANT)
motion (mm) of the condyle low
points (CLP) relative to the tibi-
al tray during flexion from 0 to
120 degrees for a design with a
polycentric sagittal femoral
curve. The medial and lateral
condyles in this design have dif-
ferent sagittal curvatures and the
radius decrease occurs at knee
flexion angles shown by arrows
(see Table 1).
Fig. 6. Antero-posterior (+ANT)
motion (mm) of the condyle low
points (CLP) relative to the tibial
tray during flexion from 0 to 120
degrees for a design with a polycen-
tric sagittal femoral curve. The me-
dial and lateral condyles in this de-
sign have the same sagittal
geometry in the tibio-femoral con-
tact regions and the radius decrease
occurs at knee flexion angles indi-
cated by arrows (angle for R
2
change was estimated by adding the
angles for the two radii of curva-
ture, R
1
and R
2
(see Fig. 2 and Ta-
ble 1).
Fig. 7. Antero-posterior (+ANT)
motion (mm) of the condyle low
points (CLP) relative to the tibial
tray during flexion from 0 to 120
degrees for a design with a polycen-
tric sagittal though nearly constant
femoral curve. The medial and lat-
eral condyles in this design have the
same sagittal geometry in the tibio-
femoral contact regions and the
slight radius decrease observed in
the reversed-engineering compo-
nent occurs at knee flexion angle
marked by the arrow (see Table 1).
Theoretical Designs For the theoretical design with a femoral radius reduction
on a flat insert: The tibio-femoral lowest points (CLP) coincided with the centers
of pressure (COP) and both translated posteriorly from 0 to 120 degrees, but the
rate of ‘rollback’ slowed at 30 degrees of flexion (Fig. 8).
Fig. 8. Theoretical femoral de-
sign with a 10 mm reduction in
sagittal radius at approximately
30 degrees of knee flexion. An-
tero-posterior (AP, +ANT) trans-
lations (mm) of the condyle low-
est points (CLP) and centers of
pressure (COP) relative to the
tibial tray with a flat insert.
For the design with a femoral radius reduction on a curved insert: The tibio-
femoral lowest points (CLP) translated posteriorly from 0 to 30 degrees, after
which the medial side translated anteriorly while the lateral side rolled back at a
slower rate (Fig. 9). The centers of pressure (COP) showed similar ‘rollback’ be-
havior to that of the CLP. Due to the sagittal curvature of the insert, the COP
translated posteriorly faster than the CLP. Since contact was made between the in-
sert and the reduced radius at an earlier flexion angle than for a flat insert, ‘roll-
back’ of the COP was correspondingly slowed earlier in the flexion cycle. For the
design with a femoral radius increase on a flat insert: Both CLP and COP translat-
ed posteriorly continuously and a slight acceleration of ‘rollback’ could be seen at
30 degrees (Fig. 10).
Fig. 9. Theoretical femoral de-
sign with a 10 mm reduction in
sagittal radius at approximately
30 degrees of knee flexion. An-
tero-posterior (AP, +ANT)
translations (mm) of the condyle
lowest points (CLP) and centers
of pressure (COP), and supero-
inferior (SI, +SUP) translations
(mm) of the COP relative to the
tibial tray with a curved insert.
Fig. 10. Theoretical femoral de-
sign with a 5 mm increase in
sagittal radius at approximately
30 degrees of knee flexion. An-
tero-posterior (AP, +ANT)
translations (mm) of the condyle
lowest points (CLP) and centers
of pressure (COP) relative to the
tibial tray with a flat insert.
For the design with a femoral radius increase from 25 mm to 30 mm at 30 de-
grees on a curved insert: CLP ‘rollback’ accelerated after 30 degrees, especially
on the lateral side, while COP ‘rollback’ preceded CLP rollback due to the addi-
tional curvature of the insert (Fig. 11). At approximately 105 degrees the lateral
condyle of the femur with the increasing radius rode up onto the posterior lip of
the curved insert and reached the edge of the flat insert. At that point the COP of
the lateral side remained on the rim, while the CLP continued past the physical
boundary of the inserts (Figs. 10 & 11).
Fig. 11. Theoretical femoral de-
sign with a 5 mm increase in
sagittal radius at approximately
30 degrees of knee flexion. An-
tero-posterior (AP, +ANT)
translations (mm) of the condyle
lowest points (CLP) and centers
of pressure (COP), and supero-
inferior (SI, +SUP) translations
(mm) of the COP relative to the
tibial tray with a curved insert.
Discussion and Conclusions
Commercial Designs NexGen® and Vanguard™ are multi-radius (polycentric)
designs, whereas Scorpio® is a single-radius design. Manufacturing, polishing
and/or reverse engineering tolerances may result in radius variations slightly dif-
ferent from original design intent. Despite differences in both the femoral and the
tibial component para-sagittal plane curvatures between these implants, the onset
of paradoxical anterior motion was approximately related to the flexion angle at
which the para-sagittal femoral radius of curvature at the contact points decreased
abruptly relative to the radius at the contact points in extension. These findings
suggest the principal cause of paradoxical anterior motion is a 30–50% discrete
decrease in the femoral condyle para-sagittal radius at flexion angles between 20–
40 deg. It should be noted that the flexion angle in the simulations can only be ap-
proximately synchronized with the angle at which the radius of curvature change
is observed in the implant component, due to the fact that R1 and R2 have differ-
ent centers of curvature and that the centers of curvature for polycentric designs
are located in positions that differ from the Grood & Suntay origin used in the
simulations.
Influence of Femoral and Tibial Contact Geometry on Kinematics
These deep knee bending simulations indicate that the primary cause of para-
doxical anterior motion in posterior cruciate retaining total knee replacements is
the discrete reduction in posterior condylar radius of curvature in the para-sagittal
plane of the femoral component. While the radius of curvature of the bearing sur-
face on the tibial side can influence the kinematics as well, the femoral curvature
appears to dominate the onset of paradoxical anterior motion. Designs with a near-
ly constant radius of curvature (single radius design) over the bearing surfaces of
the femoral condyles in the para-sagittal planes appear to be less susceptible, or
even immune, to paradoxical anterior motion.
The results for AP motion of the lowest condylar points are similar to what
has been reported for cruciate-retaining total knee implants in a single-leg deep
knee bend [1]: In a deep knee bend, PCL-retaining total knees pivot on the lateral
side, rather than the medial, in association with paradoxical anterior motion. Due
to the different sagittal curvatures of the inserts, the AP movements of the centers
of pressure (true contact points) in the medial and lateral compartments are differ-
ent from those of the lowest condylar points. We have provided results for the lat-
ter because they correspond with what has most frequently been reported in fluor-
oscopic studies. The results of these simulations suggest that inserts with less
sagittal plane conformity have greater posterior motion in early flexion, but also
have more undesired anterior motion past 30-40 degrees of flexion. Paradoxical
anterior sliding is likely to occur independently of the insert sagittal radius. The
results further suggest that paradoxical anterior sliding is associated with a reduc-
tion in the condylar radius in the sagittal plane, which causes slippage at the con-
tacts between the condyles and insert. Patients may achieve improved ‘rollback’
and deeper flexion with a lower conformity design, but may still suffer from a
sense of instability due to paradoxical anterior sliding.
Conclusions This paper focused on the application of multibody dynamics in
understanding a clinically motivated question and determining a design parameter
associated with the clinical challenge. In this case the designs were all posterior
cruciate retaining TKR designs. The same problem of paradoxical anterior sliding
occurs in posterior cruciate substituting designs and is related to the same design
factors. The remedy may therefore be similar, however, this has to be carefully
coordinated with the timing of the cam-post engagement so as not to engage the
cam and post at the wrong time or cause large transfer of load to the cam mecha-
nism.
One study has reported that patients with TKR designs based on a single radi-
us design do not suffer from paradoxical anterior motion of the femur [8] lending
support to conclusions drawn from these simulation results.
Although the insert design can moderate the severity of paradoxical anterior
motion, these results for theoretical designs point to the reduction in femoral para-
sagittal radius as the trigger for paradoxical anterior motion and suggest a simple
design modification to alleviate or eliminate it, or even to potentially enhance
rollback in deep flexion. Further examples of how even a 1 mm increase in the
femoral sagittal radius of curvature can enhance femoral rollback have been pub-
lished elsewhere [3].
Finally, it should be recognized that, although paradoxical anterior motion
can be dealt with by making small changes to the radial curvature of the femoral
component, it is not at all clear how such changes affect knee kinematics in load-
bearing activities other than deep knee bending.
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