arXiv: 1611.07297v1 [] 26 Nov 2016

High-Performance and Distributed Computing in a
Probabilistic Finite Element Comparison Study of the
Human Lower Leg Model with Total Knee
Replacement
Corneliu Arsene
School of Engineering Sciences
University of Southampton, SO17BJ
Southampton, United Kingdom
CorneliuArsene@gmail.com
than 20 years [3-5]. The majority of the studies have focussed
on the implanted proximal femur but subsequently, there has
been considerable interest in modelling of Total Knee
Replacement (TKR). Modelling the behaviour of TKR is
challenging, as the stresses generated within the prosthesis and
the supporting bone are a function of the kinematics, and the
kinematics in turn are a function of the implant design, relative
position of the components (with respect to each other and
with respect to the bones) and the balance of the soft tissues.
Clinical studies [6,7] have shown that the kinematics of TKR
are highly variable and a potential cause of this is variability in
implant positioning.
For computer studies, in order to assess the likely
performance of an implant, all of the input variability needs to
be incorporated into a suitable computational model [8-10].
Laz et al. [11,12] have pioneered the application of
probabilistic FE analyses to assess the performance of TKRs.
They used the Advance Mean Value (AMV) method together
with a 1000 run Monte Carlo Simulation Technique (MCST)
for assessing the impact of experimental variability in the
Stanmore knee wear simulator on predicted TKR mechanics
by determining the envelope of performance of joint
kinematics and contact parameters. Combination of the FE
solvers with probabilistic methods (i.e. probabilistic FE
analyses) have been applied to reliability of other orthopaedic
components such as hip replacements [13-16] and for a longer
time in the assessment of structural reliability [17-19]. This
combination results in a more rigorous validation of the
computation model and a more realistic comparison of the
predicted results with results obtained by other means such as
experimental results [11,12].
Providing a more complicated setup than the isolated TKR
mode, the passive flexion FE lower limb model with TKR is
studied in this paper. A brief comparison is made also with
previous studies [1,2] involving an active flexion FE human
lower limb model with TKR. The probabilistic FE studies

Abstract—Reliability theory is used to assess the sensitivity of a
passive flexion and active flexion of the human lower leg Finite
Element (FE) models with Total Knee Replacement (TKR) to the
variability in the input parameters of the respective FE models.
The sensitivity of the active flexion simulating the stair ascent of
the human lower leg FE model with TKR was presented before in
[1,2] whereas now in this paper a comparison is made with the
passive flexion of the human lower leg FE model with TKR. First,
with the Monte Carlo Simulation Technique (MCST), a number
of randomly generated input data of the FE model(s) are
obtained based on the normal standard deviations of the
respective input parameters. Then a series of FE simulations are
done and the output kinematics and peak contact pressures are
obtained for the respective FE models (passive flexion and/or
active flexion models). Seven output performance measures are
reported for the passive flexion model and one more parameter
was reported for the active flexion FE model (patello-femoral
peak contact pressure) in [1]. A sensitivity study will be
performed based on the Response Surface Method (RSM) to
identify the key parameters that influence the kinematics and
peak contact pressures of the passive flexion FE model. Another
two MCST and RSM-based probabilistic FE analyses will be
performed based on a reduced list of 19 key input parameters. In
total 4 probabilistic FE analyses will be performed: 2
probabilistic FE analyses (MCST and RSM) based on an
extended set of 78 input variables and another 2 probabilistic FE
analyses (MCST and RSM) based on a reduced set of 19 input
variables. Due to the likely computation cost in order to make
hundreds of FE simulations with MCST, a high-performance and
distributed computing system will be used for the passive flexion
FE model the same as it was used for the active flexion FE model
in [1].
Keywords—high performance and distributed computing;
probabilistic finite element study; total knee replacement; human
lower leg model; passive flexion; active flexion.

I. INTRODUCTION
This Finite Element (FE) studies have been used for
orthopaedic problems in computational biomechanics for more

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presented in this paper will benefit especially from a powerful
high-performance and distributed computing platform.

For each of the three components, the rotations will be applied
to a node called centre of gravity which will be attached to the
respective components. A penalty-based method was
employed to define contact [20]. Contact is modelled between
the femoral and tibial components (tibio-femoral contact) and
the patellar and femoral components (patello-femoral contact).
Friction is modelled between the contacting components and
is given a value of 0.04 [20, 21]. The friction parameter will
be varied in the probabilistic FE study. The final part included
in the FE models is the soft tissues. These are currently
modelled as simple non-linear tension-only bars, which can be
seen in Fig.1. The soft tissues are created by modelling a
nonlinear tension-only bar element between 2 nodes, one on
each rigid body. A single bar is used to represent the Medial
and Lateral Collateral Ligaments (MCL and LCL) and medial
and lateral patellar retinaculum, whilst three bars in parallel
model the quadriceps tendon and the patellar ligament. The
positions of the insertion nodes for the MCL and LCL will be
varied in the probabilistic FE study for 3 degrees of freedom
(x, y, z). The material used for ligaments has zero resistance in
compression and behaves in a nonlinear elastic-plastic fashion
in tension. Additionally, these elements are unable to transmit
bending and torsion moments. The MCL, LCL and patellar
retinaculum are modelled with an initial pretension (pre-strain)
in order to replicate their properties in vivo. The material
properties (stiffness, pre-strain) of the collateral ligaments will
be varied in the probabilistic FE study. The stiffness and two
initial rotations of the quadriceps around the medial lateral
axis and around the anterior posterior axis will be varied as
well.

II. ACTIVE/PASSIVE FLEXION FINITE ELEMENT MODELS WITH
TOTAL KNEE REPLACEMENT
The passive flexion lower limb model consists of three main
modelling parts bones, TKR components and soft tissues
(Fig.1a). Preliminary loading and boundary conditions are
applied through the hip and ankle joints, as well as a
quadriceps force for the passive flexion, ensuring that the knee
is kinematically unconstrained. The model includes the pelvis,
femur, tibia and fibula. These bones are modelled using shell
elements and defined as a rigid body. In the stochastic study,
the positions of the center of hip and the centre of ankle are
modified for three degrees of freedom (x, y and z).
The mean values and the standard deviations of the normal
distributions of the 77 input variables of the passive flexion
FE model from which the sampling process is done are shown
in [1] for the active flexion. To the respective list is added in
the passive flexion model the quadriceps load variable which
forms 78 input variables in total. The variability of quadriceps
load used in the probabilistic FE analyses in this paper is 10%.
Kinematic joints are modelled at the hip and the ankle to
replicate the soft tissue constraints that are in place in vivo.
Each joint is fixed in the translational degrees of freedom but
free in the rotational degrees of freedom. The stiffness and the
friction of the two kinematic joints will be varied in the
stochastic study. Figure 1 shows the implanted TKR which is
a PFC Sigma (DePuy) posterior cruciate sacrificing device.
The position of the femoral, tibial and patellar components
that are modelled using rigid shell elements, as described
above, will be varied for all 6 degrees of freedom (3
translations and 3 rotations) in the probabilistic FE studies.

a) Passive flexion FE model
b) Active flexion FE model
Fig. 1. a) Passive flexion FE model of the human lower limb includes bones with TKR component, the soft tissues modelled as
tension bars, patellar ligament, quadriceps tendon, MCL and LCL ligaments, the patellar retinaculum; b) Active flexion FE model
of the human lower limb where forces are applied (i.e. FE model and probabilistic FE analyses presented in detail in [1]).

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Also the friction of the patellar ligament and the quadriceps
tendon will be varied. Forces or displacements can be applied
to the model to generate motion. In this paper, a passive
flexion cycle is modelled while in [1] the active flexion was
presented in detailed. Passive flexion is a clinically relevant
movement, as during TKR surgery the surgeon will passively
flex the limb in order to check the implanted components. The
flexion cycle lasts for 1000 ms. Passive flexion is achieved by
applying two functions to the lower limb: firstly, a small
quadriceps load of 50N is applied that always acts along the
mechanical axis of the femur, regardless of the orientation of
the femur in the global coordinate system. The FE model has
a settling phase of 100 ms during which the quadriceps load is
not applied. The quadriceps load will be varied in the
probabilistic FE study. Secondly, motion is achieved by
applying a 100N force to the ankle that acts in the direction of
the hip. This results in a flexion of the knee joint of
approximately 135o. The force is applied after 500 ms.

Trials could be random, but a better result is achieved by
distributing the trials regularly across the sample space.
Secondly, with the least squares method we can represent the
results as a system of equations and arrange thus:



T

b X X



1

T

(1)

X y

where b denotes the coefficients of the RSE.
Third, the RSE (b) together with a large number of
Gaussian distributed samples (e.g. 1000) for each input
variable forming matrix X1 will generate the response vector
of interest y1:

y1  b X 1

(2)

Similar statistical measures can be applied to the response
vector y1 (e.g. mean, range, standard deviation, etc). This
method works best when the true output is well represented by
the mathematical function, for example relatively linear,
smooth and monotonic models can easily be fitted [24,25]. It
is expected that the passive flexion lower limb model will be
relatively linear when a reduced set of input variables will be
used.
Finally a sensitivity study is performed as described in [1]
where a sensitivity matrix is calculated for each output
measure, which sensitivity matrix has the number of rows
equal to the number of input variables and the number of
columns equal to the number of samples used to calculate Eq.
1. The matrix X1 of each output kinematics and peak contact
pressure variable is divided by the standard deviations σ of the
input variables:

III. PROBABILISTIC FINITE ELEMENT STUDIES OF THE
ACTIVE/PASSIVE FLEXION FINITE ELEMENT MODELS WITH
TOTAL KNEE REPLACEMENT
There will be two reliability techniques used in the
probabilistic FE study of the passive flexion FE lower limb
model. The effect of the full 78 input variables, from which 77
input variables are presented in [1], have on the kinematics
and contact pressure will be investigated. Two probabilistic
methods will be used: the MCST and the Response Surface
Method (RSM). Following a sensitivity study, a reduced set of
input variables will be derived, which will represent the key
parameters for the passive flexion lower limb model. The
reduced set will be investigated in a second probabilistic FE
study. The aim is first to find out how well the reduced set can
approximate the numerical results obtained with the full set of
input variables as well as whether there are any changes in the
envelopes of performance for the results obtained with the
MCST and the same reduced set.
The first probabilistic method used is MCST. For the
MCST method, each input variable assumes a normal
probability distribution from which a number of random
samples are generated. A set of samples are used in the FE
simulation to obtain output values. The sampling process and
the FE simulations are repeated until a wide sample of output
results are obtained.
The second probabilistic method used in this paper is the
RSM [22]. The RSM fits a mathematical function of the input
variables (called the Response Surface Equation, RSE) to
approximate the output parameter, across the full range of the
sample space. Typically, this will be a low-order polynomial,
and regression techniques are used to select the term
coefficients. The method comprises three steps: first, a
response vector y (i.e. AP translation, IE rotation, Peak
Contact Pressure) is obtained from a probabilistic FE
simulation based on a sample of input conditions, denoted by
vector X (the number of point will be smaller than that needed
for the MCST method; in this case no more than 100 trials).

Ab

X

1

(3)



where A is the sensitivity matrix with the number of rows
equal to the number of input variables and the number of
columns equal to the number of Monte Carlo samples (e.g. 50,
100) used in equation (2).
The sensitivity of an output variable sji (e.g. tibial anteriorposterior translation) with respect to an input variable i is
calculated as the mean of the absolute values of the input
variable row i of the sensitivity matrix Aj corresponding to the
respective output variable j.
s ji  mean ( abs ( A j (i, :)))

(4)

where sji is the sensitivity of j output variable with respect to
the input variable i, j is an index with values from 1 to 7 for
each output variable, i is an index with values from 1 to 78 for
the probabilistic FE analysis with RSM based on extended set
of input variables or with values from 1 to 19 for the
probabilistic FE analysis with RSM based on the reduced set
of input variables, Aj is the j-th matrix which corresponds to
the j output variable.

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IV. HIGH-PERFORMANCE AND DISTRIBUTED COMPUTING FOR
COMPUTATIONAL BIOMECHANICS

passive flexion FE simulation, took about 50 minutes to
complete, 800 points were obtained in 11 days by using in
parallel the 2 PCs described above. Obviously further
reduction in computational time can be achieved by adding
more PCs in parallel which is not regarded as a problem at the
present cost of a PC. For example, adding one more PC of
type (a) in parallel would reduce the computational time to
approximately 6 days and with more PCs added in parallel
(i.e. 3 or 4) the computational time would be drastically
reduced to even few hours. A further reduction in time would
be achieved by modifying the characteristics of the passive FE
model in order to run far less than 50 minutes as for example
in under 10 minutes.

The optimization/distributed computing software PamOpt
(ESI, Paris) will be used together with the FE software
PamCrash [26,27] to implement the MCST and the RSM.
PamOpt is able to call PamCrash processes in parallel on
several Personal Computers (PCs) assuming that
PamCrash/PamOpt software was installed before on the
respective PCs. Fig.2 (courtesy of ESI, Paris) shows how
PamOpt makes simultaneous calls on remote platforms. This
way it is possible to drastically reduce [1] the computational
times required to obtain the envelope of performance for the
kinematics and the contact pressure of the active/passive
flexion FE lower limb models. This forms a high-performance
and distributed computing platform for computational
biomechanics. In order to implement the distributed
computing framework formed by PamOpt/PamCrash software
the Cygwin software [23] had to be installed and configured
so that to be able to run the native Linux application PamOpt
on Windows. There were used 2 PCs with the following
specifications: a) Intel(R) Core(TM)2 CPU 6600 @ 2.40 GHz
2.40 GHz and b) Intel(R) Pentium(R) 4 CPU 3.20 GHz 3.19
GHz. The MCST points were distributed on the two different
PCs and the computational time was reduced. While one

V. NUMERICAL RESULTS AND CONCLUSIONS
The kinematics of the tibio-femoral contact joint and the
patello-femoral contact joint were reported in Grood and
Suntay joint coordinate system.

Fig.2. PamOpt makes simultaneous calls on remote platforms of the FE software PamCrash (courtesy of ESI, Paris).
The following 7 output performance metrics were considered
for the passive flexion: tibio-femoral flexion angle, tibiofemoral peak contact pressure, tibial anterior-posterior
translation, tibial internal-external rotation, patello-femoral
flexion angle, patellar medial-lateral displacement, patellar tilt.
During the MCST, it was looked to the 5% and 95% percentile
(i.e. the values which are below, respectively above the
respective percentiles were taken out), and the mean value of
the peak value of the tibio-femoral flexion angle. It was
observed that after 400 FE simulations the mean and the
percentile values became constant (Fig. 3).
Fig.3 Peak flexion angle function of the number of
simulations for the probabilistic FE analysis based on MCST
with the full set of 78 input variables.

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However, there are used 800 FE simulation points after which
the probabilistic FE analysis based on the MCST was stopped.
Furthermore the RSM with 800 FE simulation points obtained
a second set of envelopes which compared very well (i.e not
shown in the paper) with the performance envelopes obtained
with MCST with 800 FE simulations. With the RSM were
also calculated the sensitivity coefficients. The most important
19 key parameters which influence the output kinematics and
the peak contact pressure of the passive flexion model were
calculated as described in [1] for the active model and in the
previous section. The key parameters were calculated by
summing their ranking positions from the seven lists of 78
input variables. After the 19th key parameter a significant
dropped was noticed in the total scores of the key parameters.
The 19 key input parameters are friction of the pelvis-hip
joint on the r-rotational direction (anterior-posterior:
coordinates 0.71 (x) and 34.56 (y)), anterior-posterior position
of the patellar component, quadriceps load, quadriceps initial

rotation around the medial lateral direction, quadriceps initial
rotation around the anterior-posterior direction, varus-valgus
position of the femoral component, varus-valgus position of
the tibial component, inferior-superior position of the femoral
component, stiffness of the LCL ligament, friction of the leg
ankle on the r-rotational direction (anterior-posterior:
coordinates 0.71 (x) and 34.56 (y)), inferior-superior position
of the tibial component, tilt of the tibial component, friction of
the pelvis-hip joint on the r-rotational direction (anteriorposterior: coordinates 0.289 (x) and y (2.41)), internal-external
position of the patellar component, x coordinate of the
insertion point node 9400 (i.e. specific to our FE model) of
LCL, medial-lateral position of the tibial component, anteriorposterior position of the femoral component, stiffness of the
MCL ligament, x coordinate of the insertion node 9403 of
MCL.

a) Tibio-femoral flexion angle (degrees)

b) Tibial anterior-posterior translation (mm)

c) Tibial internal-external rotation (degrees)

d) Patello-femoral flexion angle (degrees)

5

e) Patellar medial-lateral displacement (mm)

f) Patellar tilt (degrees)

g) Tibio-femoral peak contact pressure (MPa)
Fig. 4. Comparison between RSM model-predicted envelopes with 50 FE simulation points (dashdot line), 100 FE simulation
points (dashed line), and MCST (solid line) with 800 points for the passive flexion FE model with the reduced set of 19 input
variables.
It is important to underline for this paper that a comparison
of this list with the similar list of 22 input key parameters
obtained for the active flexion FE model [1,2] reveals that the
quadriceps load, the quadriceps initial rotations and the
friction of the leg-ankle and the pelvis-hip joints play a more
important role in the passive flexion than in the active flexion
motion. These results confirm similar findings from the
literature regarding the relationship between the quadriceps
tendon and the passive flexion movement of the human lower
leg [28, 29].
Finally, two probabilistic FE studies were implemented
with the reduced set of 19 input variables: a MCST with 800
FE simulation points (i.e. mean/percentile values became
constant at 400 FE points) was run and the RSM with 50 and
100 FE simulations were used to calculate the envelopes
(Fig.4). The envelopes of performance obtained with the 800
MCST FE simulations and the reduced set of 19 input
variables compared very well with the envelope of
performance obtained with the 800 MCST FE simulations
with the full set of 78 variables and the envelopes of

performance obtained with the RSM-50 and RSM-100 for the
passive flexion FE model with the reduced set of 19 input
variables. For example, the differences between the MCST
with the full set of 78 input variables and 800 FE simulation
points and the MCST with the reduced set of 19 input
variables and 800 FE simulation points were 1.07 degrees for
tibio-femoral flexion angle, 0.81 mm for tibial anteriorposterior translation, 2.44 degrees for tibial internal-external
rotation, 2.46 degrees for patella-femoral flexion angle, 0.89
mm for patella medial-lateral displacement and 1.48 degrees
for patellar tilt.
The differences between the MCST with the reduced set
of 19 input variables and the RSM-50 with the reduced set of
19 input variables were of 5.82 degrees for tibio-femoral
flexion angle, 1.24 mm for tibial anterior-posterior translation,
1.06 degrees for tibial internal-external rotation, 3.59 degrees
for patella-femoral flexion angle, 1.17 mm for patella mediallateral displacement, 1.11 degrees for patellar tilt, 1.6 MPa for
tibio-femoral peak contact pressure. The differences between
the MCST with the reduced set of 19 input variables and the

6

RSM-100 with the reduced set of 19 input variables were of
6.34 degrees for tibio-femoral flexion angle, 1.05 mm for
tibial anterior-posterior translation, 1.44 degrees for tibial
internal-external rotation, 3.81 degrees for patella-femoral
flexion angle, 1.39 mm for patella medial-lateral
displacement, 1.16 degrees for patellar tilt, 1.55 MPa for tibiofemoral peak contact pressure.
This suggests that a smaller number of FE simulations can
be used in order to obtain the envelopes of performance based
on the RSM-50/100 method and the reduced set of 19 input
variables. However, the probabilistic FE analyses based on the
MCST (i.e. with either the full set or the reduced set of input
variables) represents the gold standard that it should be
employed in every probabilistic FE analysis and in addition
other probabilistic methods can be employed for exploratory
studies.
In conclusion there were realized four probabilistic FE
analyses: two probabilistic FE analyses based on MCST and
RSM and based on the full set of 78 input variables and
another two probabilistic FE analyses based on MCST and
RSM and based on the reduced set of 19 input variables.
Further work will involve the development of a real-life
decision support system for orthopaedics which to include the
present results and the ones shown in [1,2] which were based
also on a similar distributed computing paradigm.

[8]

ACKNOWLEDGMENT

[18]

[9]

[10]

[11]

[12]

[13]

[14]
[15]
[16]

[17]

The author would like to thank to European Union
Framework Programme 6 (FP6) for financing this work under
the project DESSOS (Decision Support Software for
Orthopedics). The author would like to thank also to the
following people for their support in this work Prof Mark
Taylor, Dr Elena Samsonova, Dr. Magdalena Janosova, Dr
Lucy Knight, Mrs Muriel Beaugonin, Mr. Pierre Guyon, Dr.
Michael Strickland and Prof David Barrett.

[19]

[20]
[21]

[22]

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