JOURNAL OF PHYSIOLOGY AND PHARMACOLOGY 2009, 60, Suppl 8, 69-72
www.jpp.krakow.pl
H. NAGERL1, J. WALTERS2, K.H. FROSCH3, C. DUMONT3, D. KUBEIN-MEESENBURG1,
J. FANGHANEL4, M.M. WACHOWSKI3
KNEE MOTION ANALYSIS OF THE NON-LOADED AND LOADED KNEE:
A RE-LOOK AT ROLLING AND SLIDING
Department of Orthodontics, Georg-August-University, Gottingen, Germany; 2Department of Orthopaedic Surgery,
Groote Schuur Hospital, University of Cape Town, Cape Town, South Africa; 3Department of Trauma Surgery,
Plastic and Reconstructive Surgery, Georg-August-University, Gottingen, Germany; 4Department of Oral Anatomy,
Ernst-Moritz-Arndt University, Greifswald, Germany
1
Many studies of knee motion have been reported in the literature over more than 100 years. Of particular interest to the
analysis presented here is the work of the Freeman group, who elegantly measured tibio-femoral kinematics in studies
made on cadavers and the knees of living individuals using MRI, anatomical dissection and RSA. We examined and reevaluated the data collected by Freeman’s group and suggest that their conclusion should be considered to be incorrect,
since their methods of evaluation were oversimplified from the mathematical and physical perspectives. By applying
appropriate methods, however, it is possible to show that the same data yield important insights into physiological knee
kinematics and reveal that the rolling-sliding relationship depends on the degree of flexion and on joint load in the
medial and lateral compartment, as well. In the initial range of flexion, a considerable amount of rolling was found to
occur. Based on this analysis, it is possible to gain useful insights of value for the design of total knee replacements.
K e y w o r d s : natural knee kinematics, rolling-sliding relationship, flexion, shear stresses, knee muscles
INTRODUCTION
Reports were made even a century ago that rolling motion
predominates from full extension sliding motion and then sliding
motion as flexion goes beyond 30° (1-4). In contrast, in a series of
papers authored by the Freeman group (5-7) in vivo and in vitro
data relating to the relative movements of the tibia and femur of the
non-loaded and loaded knee, obtained from MRI-measurements
and from dissections, were presented in which Pinskerova et al. (6)
specifically state that “there is no rolling at the contact area”.
We therefore wished to re-evaluate the data of the Freeman
group, applying more appropriate mathematical methods and
formulae, in order to assess knee kinematics under different
loads. To facilitate clarity and understanding of the problem, we
define the pertinent mathematical and physical principles
embodied in ‘rolling’ and ‘sliding’ motion and the derivations of
the formulae used are given, thereby enabling us to present a
mathematical and graphical re-evaluation of the published data.
[1]
ρ=
u 2 ds 2 dt ds 2
=
=
u1 ds1 dt ds1
Derivation of formulae
In words: the rolling-sliding ratio ρ is equal to the quotient
of the instantaneous speeds u2 and u1 with which the centre of
contact K of the touching articular surfaces moves along these
surfaces (Fig. 1). u2 or u1 = the instantaneous speed of the
contact K on the articular surface of the tibia or femur
respectively. s2 or s1 = position of the centre of contact K on the
tibial or femoral surface respectively; ds2 or ds1 = the
corresponding incremental alteration in position; dt = the
increment in time.
Conclusion: In order to make valid statements about the ratio
of rolling to sliding, the migrations of the contact centre K must
be related on both articular surfaces.
If the femur is rotating clockwise in relation to the tibia
solely around the femoral centre of curvature M1 by the angular
increment dα1, then the centre of contact K is displaced
anticlockwise by the increment:
[2]
ds1 = - dα1 • R1.
For analysis of rolling and sliding, the quasi-plane
movement used by the Freeman group was adopted. In
evaluating the corresponding data, one has to bear in mind the
physical definition of the rolling-sliding ratio ρ:
In this case, contact K on the tibial surface remains
stationary. This is type I of sliding which represents racing.
But, if the femur is only rotating anticlockwise around the
tibial centre of curvature M2 by the angular increment dα2,
MATERIALS AND METHODS
70
contact K remains stationary on the femoral surface. On the tibial
surface, however, contact K is also displaced anticlockwise by:
[3]
ds2 = + dα2 • R2.1
This is type II of sliding which represents sledging. In
general, both rotation increments occur simultaneously, and
accumulate with respect to the total increment of flexion dα:
[4]
dα = dα1 + dα2
Inserting [2] and [3] into [4] and the result is
[5]
dα = −
ds1 ds 2
+
R1
R2
Combining [5] and [1], and by carrying out some algebraic
conversions and the result is
[6]
 dα
R 
ρ (α ) = −
⋅ R1 + 1 
R2 
 ds2
−1
or
[7]
ρ (α ) = −
ds 2 1
,
⋅
dα R1
Fig. 1. In the sagittal section of the medial compartment of the
knee the convexly curved femur (F) contacts the concave “tibia
plateau” (T) at site K. M1 and M2 are the centres of the circles of
curvature of the articular surfaces at contact K. R1 and R2 are the
corresponding radii of curvature. u1 = speed of migration of
contact K on femur F. u2 = speed of migration of contact K on the
“tibia plateau” T. When the tibial plateau is convexly curved, the
sign of term R2 is altered, since an anticlockwise rotating femur
around the tibial centre M2 makes the tibial contact K move in the
opposite direction in relation of the concave shape of the tibia.
if R 2 >> R1 .
The function ρ=ρ(α) means: a) the rolling-sliding ratio is
instantaneously defined, b) it depends on the angle of flexion,
and c) it alters in the course of flexion.
Note: If ρ(α)=1, the rolling would be pure. If ρ(α)<1, there
would be additional racing and, if ρ(α)>1 additional sledging of
the femur would occur in relation to the tibial articulating
surface. ρ(α)=0 pure racing. ρ(α)=infinite: pure sledging.
Procedures used for re-analysis of the Freeman group data (5-7)
1. The femoral radii of curvature and the position s2(α) of the
contact centre K on the “tibia plateau” as function of the flexion
angle α were taken from the Freeman data.
2. The derivative function ds2/dα was estimated from the
corresponding s2(α) data.
RESULTS
In Fig. 5, tracings of sagittal MRI sections through the medial
and lateral compartment were shown for -5°, 20° and 110° flexion.
Since each tracing was provided with a scale, the femoral radii R1,2
and s2, the corresponding tibial position of the contact K could be
easily determined. Using Iwaki’s tracings, the following
parameters were plotted: our determinations of the respective
positions s2 of contact K and the corresponding graphs s2(α) as
incorporated in Fig. 2. Though, for each graph, only three pairs of
values were available the data reveal a clear migration of the tibial
contact K posteriorly in both compartments. Maximal distances
covered were 14 mm medially and 24 mm laterally. By taking into
account the corresponding difference quotients and the respective
radii of curvature EF and FF (Fig. 2) the mean rolling-sliding
relationship
ρEF between -5° and 20° and
ρFF between 20° and
110° flexion was calculated by means of the following formulae:
[8]
ρ EF = −
s 2 ( −5°) − s 2 ( 20°) 1
s ( 20 °) − s 2 (110 °) 1
.
, ρ FF = − 2
⋅
⋅
FF
− 5° − 20°
20 ° − 110 °
EF
Fig. 2. Migration of contact K in flexion Iwaki et al. A) along the
medial “tibial plateau” and B) along the lateral “tibial plateau” with
our readings for the position s2 of contact K as well as the graphs
of the corresponding position-flexional angle characteristics s2(α).
Readings for the femoral radii of curvature. Medial compartment:
EF=33mm, FF=22mm. Lateral compartment: EF=31mm,
FF=20mm. Calculated using the radii from Iwaki’s paper.
71
Table 1. Rolling-sliding ratios. Details in the text.
Range of
flexion
ρ EF
0° - 20°
Medial compartment
Load
Non-load Cadaver
0.96
0.80
0.63
ρ FF
45° - 90°
0.10
0.09
The calculations yield
ρEF = 49% on the medial side and
ρEF
= 76% on the lateral plus
ρFF = 20% on the medial side and
ρFF
= 43% on the lateral side, respectively.
Pinskerova et al. (7) showed the data for migration of the
articular contacts on the tibia plateau were presented in a tabular
form: in Fig. 3, we present the same information in the form of
ρEF
a graph. The corresponding mean rolling-sliding ratios2
and
ρFF, calculated from these graphs, are listed in Table 1.
DISCUSSION
Beyond doubt the sliding-rolling relationships, determined
by calculation, only represent approximated values.
Nevertheless, despite this shortcoming, there is no material
deficit. Hence, the in vivo MRI-data of the Freeman group are of
great value. In particular, they substantiate the predominanting
0.08
Lateral compartment
Load
Non-load Cadaver
1.24
0.17
0.37
0.20
0.43
0,43
rolling motions of the articular surfaces near full extension under
a load. This initial range of motion is functionally, as well as
clinically, critical for the natural knee and the functional
prerequisites for joint replacement design (8), because it
corresponds to the stance phase during which the knee is loaded
during gait. The data (5-7) beautifully demonstrate how nature
solves the problem of friction kinematically when the joint
surfaces are under load. At peak loads in the stance phase of
normal gait, nature capitalises on the smaller friction generated
by rolling. Since, in the stance phase, three reversals of motion
occur, as is demonstrated by the well-known graphs of flexion in
gait cycles (9), the predominance of rolling has an additional
advantage: the articulating surfaces are less exposed to
disadvantageous shear stresses. At the points of reversal of
motion - when the motion is momentarily at rest - racing under
load would produce static friction. With subsequent motion, the
contacting surfaces are subjected to shear stress until this static
friction connection ceases. With rolling motion, the contacting
surfaces, and hence the articular cartilage, are only subjected to
compressive loads. The AEQUOS total knee replacement (10)
shows a similar rolling-sliding characteristic to that of the
natural knee.
Our re-analysis of Pinskerova data reveals that the rollingsliding relationship substantially depends on the amount of load,
especially in the lateral compartment. Since muscles are able to
alter their forces without altering their lengths – as shown in
several papers (11-18) for knee muscles – our results
demonstrate that the rolling-sliding pattern can also be adjusted
by the muscular system acting over the knee.
CLINICAL RELEVANCE
The re-analysis of the data presented here has revealed their
particular clinical significance: the natural knee possesses a
special gear mechanism which dictates that, initially, the
articular surfaces roll back both medially and laterally when
under a load (Fig. 3). During the stance phase, when the load is
at a maximum, the posterior migration of the load on the tibial
plateau minimizes the shear stress at the point of contact, thereby
protecting the articular cartilage. During the swing phase, when
compressive and shear loads are at a minimum, sliding motion
predominates. Therefore, knee prostheses which claim to
possess natural kinematics should mimic this specific ‘roll back
and sliding’ pattern of motion, as has been put into practice by
the AEQUOS total knee replacement (8, 10).
Conflict of interests: None declared.
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Fig. 3. Position of the contact-flexion angle characteristics s2(α)
after Pinskerova et al. (7). A) Medial compartment: the initial
slope –ds2/dα was maximal for knee under load. B) Lateral
compartment: the initial slope –ds2/dα was maximal for knee
under load.
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R e c e i v e d : October 22, 2009
A c c e p t e d : December 18, 2009
Author’s address: Dr. med. Martin Wachowski, Department
of Trauma Surgery, Plastic and Reconstructive Surgery, GeorgAugust-University, Robert-Koch-Strasse 40, 37075 Gottingen,
Germany; Phone: +49551396114; Fax: +49551398981; E-mail:
[email protected]