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Three-Dimensional
Dynamic Anatomical
Modeling of the
1
Human Knee Joint
1.1
Biomechanical Systems • Physical Knee Models •
Phenomenological Mathematical Knee Models • Anatomically
Based Mathematical Knee Models
1.2
Tibio-Femoral Joint: Model Formulation
Kinematic Analysis • Contact and Geometric Compatibility
Conditions • Ligamentous Forces • Contact Forces • Equations
of Motion
1.3
DAE Solvers • Load Vector and Stiffness Matrix
Mohamed Samir Hefzy
1.4
1.5
Varus-Valgus Rotation • Tibial Rotation • Femoral and Tibial
Contact Pathways • Velocity of the Tibia • Magnitude of the
Tibio-Femoral Contact Forces • Ligamentous Forces
University of Toledo
Eihab Muhammed Abdel-
Rahman
1.6
1.7
Virginia Polytechnic Institute
Three-dimensional dynamic anatomical modeling of the human musculo-skeletal joints is a versatile tool
for the study of the internal forces in these joints and their behavior under different loading conditions
following ligamentous injuries and different reconstruction procedures. This chapter describes the three-
dimensional dynamic response of the tibio-femoral joint when subjected to sudden external pulsing loads
utilizing an anatomical dynamic knee model. The model consists of two body segments in contact (the
femur and the tibia) executing a general three-dimensional dynamic motion within the constraints of
the ligamentous structures. Each of the articular surfaces at the tibio-femoral joint is represented by a
separate mathematical function. The joint ligaments are modeled as nonlinear elastic springs. The six-
degrees-of-freedom joint motions are characterized using six kinematic parameters and ligamentous
forces are expressed in terms of these six parameters. In this formulation, all the coordinates of the
ligamentous attachment points are dependent variables which allow one to easily introduce more liga-
ments and/or split each ligament into several fiber bundles. Model equations consist of nonlinear second
order ordinary differential equations coupled with nonlinear algebraic constraints. An algorithm was
developed to solve this differential-algebraic equation (DAE) system by employing a DAE solver, namely,
the Differential Algebraic System Solver (DASSL) developed at Lawrence Livermore National Laboratory.
© 2001 by CRC Press LLC
 Model calculations show that as the knee was flexed from 15 to 90°, it underwent internal tibial
rotation. However, in the first 15° of knee flexion, this trend was reversed: the tibia rotated internally
as the knee was extended from 15° to full extension. This indicates that the screw-home mechanism
that calls for external rotation in the final stages of knee extension was not predicted by this model.
This finding is important since it is in agreement with the emerging thinking about the need to re-
evaluate this mechanism.
It was also found that increasing the pulse amplitude and duration of the applied load caused a decrease
in the magnitude of the tibio-femoral contact force at a given flexion angle. These results suggest that
increasing load level caused a decrease in joint stiffness. On the other hand, increasing pulse amplitude
did not change the load sharing relations between the different ligamentous structures. This was expected
since the forces in a ligament depend on its length which is a function of the relative position of the tibia
with respect to the femur.
Reciprocal load patterns were found in the anterior and posterior fibers of both anterior and posterior
cruciate ligaments, (ACL) and (PCL), respectively. The anterior fibers of the ACL were slack at full
extension and tightened progressively as the knee was flexed, reaching a maximum at 90° of knee flexion.
The posterior fibers of the ACL were most taut at full extension; this tension decreased until it vanished
around 75° of knee flexion. The forces in the anterior fibers of the PCL increased from zero at full
extension to a maximum around 60° of knee flexion, and then decreased to 90° of knee flexion. On the
other hand, the posterior fibers of the PCL were found to carry lower loads over a small range of motion;
these forces were maximum at full extension and reached zero around 10° of knee flexion. These results
suggest that regaining stability of an ACL deficient knee would require the reconstruction of both the
anterior and posterior fibers of the ACL. On the other hand, these data suggest that it might be sufficient
to reconstruct the anterior fibers of the PCL to regain stability of a PCL deficient knee.
1.1 Background
Biomechanical Systems
Biomechanics thus provides the means to study and analyze the behaviors of the different
biological systems as well as their components. Models have emerged as necessary and effective tools to
be employed in the analysis of these biomechanical systems.
In general, employing models in system analysis requires two prerequisites: a clear objective identifying
the aims of the study, and an explicit specification of the assumptions to be made. A system mainly
depends upon what it is being used to determine, e.g., joint stiffness or individual ligament lengths and
forces. The system is thus identified according to the aim of the study. Assumptions are then introduced
in order to simplify the system and construct the model. These assumptions also depend on the aim of
the study. For example, if the intent is to determine the failure modes of a tendon, it is not reasonable
to model the action of a muscle as a single force applied to the muscle’s attachement point. On the other
hand, this assumption is appropriate if it is desired to determine the effect of a tendon transfer on gait.
After the system has been defined and simplified, the modeling process continues by identifying system
variables and parameters. The parameters of a system characterize its components while the variables
describe its response. The variables of a system are also referred to as the quantities being determined.
Modeling activities include the development of physical models and/or mathematical models. The
mechanical responses of physical models are determined by conducting experimental studies on fabri-
cated structures to simulate some aspect of the real system. Mathematical models satisfy some physical
laws and consist of a set of mathematical relations between the system variables and parameters along
with a solution method. These relations satisfy the boundary and initial conditions, and the geometric
constraints.
Major problems can be encountered when solving the mathematical relations forming a mathematical
model. They include indeterminacy, nonlinearity, stability, and convergence. Several procedures have
© 2001 by CRC Press LLC
Biomechanics is the study of the structure and function of biological systems by the means of the methods
of mechanics.
63
  presented in the
literature effectively allow solution of most nonlinear systems of equations, yet reaching a stable and
convergent solution never may be achieved in some situations. However, with the recent advances in
computers, mathematical models have proved to be effective tools for understanding behaviors of various
components of the human musculo-skeletal system.
Mathematical models are practical and appealing because:
16
1. For ethical reasons, it is necessary to test hypotheses on the functioning of the different components
of the musculo-skeletal system using mathematical simulation before undertaking experimental
studies.
115
experimental procedures.
3. The complex anatomy of the joints means it is prohibitively complicated to instrument them or
study the isolated behaviors of their various components.
4. Due to the lack of noninvasive techniques to conduct
in vivo
and/or
in vitro
in vivo
experiments, most experimental
work is done
in vitro
.
This chapter focuses on the human knee joint which is one of the largest and most complex joints
forming the musculoskeletal system. From a mechanical point of view, the knee can be considered as a
biomechanical system that comprises two joints: the tibio-femoral and the patello-femoral joints. The
behavior of this complicated system largely depends on the characteristics of its different components.
As indicated above, models can be physical or mathematical. Review of the literature reveals that few
physical models have been constructed to study the knee joint. Since this book is concerned with
techniques developed to study different biomechanical systems, the few physical knee models will be
discussed briefly in this background section.
Physical Knee Models
Physical models have been developed to determine the contact behavior at the articular surfaces and/or
to simulate joint kinematics. In order to analyze the stresses in the contact region of the tibio-femoral
joint, photoelasticity techniques have been employed in which epoxy resin was used to construct models
of the femur and tibia.
31,87
100,101
This construct consists of two crossed rods that are hinged at one end
and have a length ratio equal to that of the normal anterior and posterior cruciate ligaments. The free
ends of these two crossed rods are connected by a coupler that represents the tibial plateau. This simple
apparatus was used to demonstrate the shift of the contact points along the tibio-femoral articular surfaces
that occur during knee flexion.
Another model, the Burmester curve, has been used to idealize the collateral ligaments.
76,88,89
This curve
is comprised of two third order curves: the vertex cubic and the pivot cubic. The construct combining
the crossed four-bar linkage and the Burmester curve has been used extensively to gain an insight into
knee function since the cruciate and collateral ligaments form the foundation of knee kinematics.
However, this model is limited because it is two-dimensional and does not bring tibial rotations into the
picture. A three-dimensional model proposed by Huson allows for this additional rotational degree-of-
freedom.
90
76
Phenomenological Mathematical Knee Models
Several mathematical formulations have been proposed to model the response of the knee joint which
constitutes a biomechanical system. Three survey papers appeared in the last decade to review
© 2001 by CRC Press LLC
been developed to overcome some of these difficulties. For example, eliminating indeterminacy requires
using linear or nonlinear optimization techniques. Different numerical algorithms
2. It is more economical to use mathematical modeling to simulate and predict joint response under
different loading conditions than costly
Kinematic physical knee models were also proposed to demonstrate the
complex tibio-femoral motions that can be described as a combination of rolling and gliding.
The most common physical model that has been developed to illustrate the tibio-femoral motions is
the crossed four-bar linkage.
mathematical knee models which can be classified into two types:
phenomenological
and
anatomically
based
models.
66,69,70
The phenomenological models are gross models, describing the overall response of the knee without
considering its real substructures. In a sense, these models are not real knee models since a model’s
effectiveness in the prediction of
in vivo
simple hinge
models, which consider the knee a hinge joint connecting the femur and tibia, and
rheological
models, which consider the knee a viscoelastic joint.
Simple Hinge Models
This type of knee model is typically incorporated into global body models. Such whole-body models
represent body segments as rigid links connected at the joints which actively control their positions.
Some of these models are used to calculate the contact forces in the joints and the muscle load sharing
during specific body motions such as walking,
38,73,86,112,114
running,
25
and lifting and lowering tasks.
39,44
These models provide no details about the geometry and material properties of the articular surfaces
and ligaments. Equations of motion are written at the joint and an optimization technique is used to
solve the system of equations for the unknown muscle and contact forces. Other simple hinge models
were developed to predict impulsive reaction forces and moments in the knee joint under the impact of
a kick to the leg in the sagittal plane.
83,121,122
Rheological Models
Masses, springs, and dampers are used to represent
the velocity-dependent dissipative properties of the muscles, tendons, and soft tissue at the knee joint.
These models do not represent the behavior of the individual components of the knee; they use exper-
imental data to determine the overall properties of the knee. While phenomenological models are of
limited use, their dynamic nature makes them of interest.
97
or a Kelvin body idealization.
37,108
Anatomically Based Mathematical Knee Models
Anatomically based models are developed to study the behaviors of the various structural components
forming the knee joint. These models require accurate description of the geometry and material properties
of knee components. The degree of sophistication and complexity of these models varies as rigid or
deformable bodies are employed. The analysis conducted in most of the knee models employs a system
of rigid bodies that provides a first order approximation of the behaviors of the contacting surfaces.
Deformable bodies have been introduced to allow for a better description of this contact problem.
Employing rigid or deformable bodies to describe the three-dimensional surface motions of the
tibia and/or the patella with respect to the femur using a mathematical model requires the development
of a three-dimensional mathematical representation of the articular surfaces. Methods include describ-
ing the articular surfaces using a combination of geometric primitives such as spheres, cones, and
cylinders,
4-7,116,125,136,137
and describing the articular surfaces utilizing the piecewise continuous
parametric bicubic Coons patches.
21,23,75
12,14,67,68
The B-spline least squares surface fitting method is also used
to create such geometric models.
13
models.
Kinematic models describe and establish relations between motion parameters of the knee joint. They
do not, however, relate these motion parameters to the loading conditions. Since the knee is a highly
compliant structure, the relations between motion parameters are heavily dependent on loading condi-
tions making each of these models valid only under a specific loading condition. Kinetic models try to
remedy this problem by relating the knee’s motion parameters to its loading condition.
66
further classified anatomically based models into
kinematic
and
kinetic
© 2001 by CRC Press LLC
response depends on the proper simulation of the knee’s
articulating surfaces and ligamentous structures. Phenomenological models are further classified into
In these models, the thigh and the leg were considered as a
double pendulum and the impulse load was expressed as a function of the initial and final velocities of
the leg.
These models use linear viscoelasticity theory to model the knee joint using a Maxwell fluid
approximation
describing each of the articular surfaces by a separate polynomial function of
the form y = y (x, z),
Hefzy and Grood
. Quasi-static models determine forces
and motion parameters of the knee joint through solution of the equilibrium equations, subject to
appropriate constraints, at a specific knee position. This procedure is repeated at other positions to cover
a range of knee motion. Quasi-static models are unable to predict the effects of dynamic inertial loads
which occur in many locomotor activities; as a result, dynamic models have been developed. Dynamic
models solve the differential equations of motion, subject to relevant constraints, to obtain the forces
and motion parameters of the knee joint under dynamic loading conditions. In a sense, quasi-static
models march on a space parameter, for example, flexion angle, while dynamic models march on time.
quasi-static
and
dynamic
Quasi-Static Anatomically Based Knee Models
Several three-dimensional anatomical quasi-static models are cited in the literature. Some of these models
are for the tibio-femoral joint, some for the patello-femoral joint, some include both tibio-femoral and
patello-femoral joints, and some include the menisci. The most comprehensive quasi-static models for
the tibio-femoral joint include those developed by Wismans et al.,
129,130
Andriacchi et al.,
9
and Blankevoort
The most comprehensive quasi-static three-dimensional models for the patello-femoral joint
include those developed by Heegard et al.,
20-23
64
Essinger et al.,
50
Hirokawa,
72
and Hefzy and Yang.
68
The
are the only
models that realistically include both tibio-femoral and patello-femoral joints. The latter model is the
only and most comprehensive quasi-static three-dimensional model of the knee joint available in the
literature.
118
Gill and O’Connor,
57
and Bendjaballah et al.
17
This model includes menisci, tibial, femoral and patellar cartilage layers, and ligamentous
structures. The bony parts were modeled as rigid bodies. The menisci were modeled as a composite of
a matrix reinforced by collagen fibers in both radial and circumferential directions. However, this com-
prehensive model is limited because it is valid only for one position of the knee joint: full extension.
This chapter is devoted to the dynamic modeling of the knee joint. Therefore, the previously cited
quasi-static models will not be further discussed. The reader is referred to the review papers on knee
models by Hefzy et al. for more details on these quasi-static models.
17
66,70
Dynamic Anatomically Based Knee Models
Most of the dynamic anatomical models of the knee available in the literature are two-dimensional,
considering only motions in the sagittal plane. These models are described by Moeinzadeh et al.,
93-99
Engin and Moeinzadeh,
47
Wongchaisuwat et al.,
131
Tumer et al.,
118-119
Abdel-Rahman and Hefzy,
1-3
and
Ling et al.
84
Four ligaments, the two cruciates and the two collaterals were modeled by a spring element
each. Ligamentous elements were assumed to carry a force only if their current lengths were longer than
their initial lengths, which were determined when the tibia was positioned at 54.79° of knee flexion. A
quadratic force elongation relationship was used to calculate the forces in the ligamentous elements. A
one contact point analysis was conducted where normals to the surfaces of the femur and the tibia, at
the point of contact, were considered colinear. The profiles of the femoral and tibial articular surfaces
were measured from X-rays using a two-dimensional sonic digitizing technique. A polynomial equation
was generated as an approximate mathematical representation of the profile of each surface. Results were
presented for a range of motion from 54.79° of knee flexion to full extension under rectangular and
exponential sinusoidal decaying forcing pulses passing through the tibial center of mass. No external
moments were considered in the numerical calculation.
Moeinzadeh et al.’s theoretical formulation included three differential equations describing planar
motion of the tibia with respect to the femur, and three algebraic equations describing the contact
condition and the geometric compatibility of the problem.
93,96
93-96
Using Newmark’s constant-average-accel-
the three differential equations of motion were transformed to three nonlinear algebraic
equations. Thus, the system was reduced to six nonlinear algebraic equations in six independent
unknowns: the
15
x
and
y
coordinates of the origin of the tibial coordinate system with respect to the femoral
© 2001 by CRC Press LLC
In turn kinetic models are classified as
et al.
models developed by Tumer and Engin,
Moeinzadeh et al.’s two-dimensional model of the tibio-femoral joint represented the femur and the
tibia by two rigid bodies with the femur fixed and the tibia undergoing planar motion in the sagittal
plane.
eration scheme,
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