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University of Calgary
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Graduate Studies The Vault: Electronic Theses and Dissertations
2017
Geometry Reconstruction and Finite Element
Modelling of Porcine Knee Joint
Zheng, Xiaoyue
Zheng, X. (2017). Geometry Reconstruction and Finite Element Modelling of Porcine Knee Joint
(Unpublished master's thesis). University of Calgary, Calgary, AB. doi:10.11575/PRISM/26291
http://hdl.handle.net/11023/4197
master thesis
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UNIVERSITY OF CALGARY
Geometry Reconstruction and Finite Element Modelling of Porcine Knee Joint
by
Xiaoyue Zheng
A THESIS
SUBMITTED TO THE FACULTY OF GRADUATE STUDIES
IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE
DEGREE OF MASTER OF SCIENCE
GRADUATE PROGRAM IN MECHANICAL AND MANUFACTURING ENGINEERING
CALGARY, ALBERTA
SEPTEMBER, 2017
© Xiaoyue Zheng 2017
ii
Abstract
Arthritis is a leading cause of disability in North America. It is believed to be associated
with the abnormal contact mechanics of articular cartilage. Contact analyses are widely used to
determine the mechanical interplays among the different tissues in the joint. Animal joints are
often used to validate a computational model and understand human joint mechanics.
The objectives of this research are to construct the geometry of the porcine stifle joint using
a combined CT and automated indentation mapping method, and build a finite element model in
ABAQUS to determine the joint mechanics.
The porcine knee joint model were reconstructed using MATLAB and Rhinoceros. A knee
compression was simulated with ABAQUS, which considered fluid pressure and flow in articular
cartilages and menisci. The reaction predicted by the model generally agrees with the
measurements from laboratory tests, which partially validates the modelling methodology.
Keywords: porcine knee joint; geometry reconstruction; CT images; automated indentation
mapping; finite element analysis
iii
Acknowledgements
I wish to thank my supervisor Dr. LePing Li for his patience and guidance throughout this
research study.
I would like to express my gratitude to Marcel Rodriguez, Di Zhu and Baaba Sekyiwaa
Otoo for their willingness to share knowledge and always provide constructive suggestions for
model enhancement.
I would like to acknowledge the use of Dr. Walter Herzog’s laboratory in the Faculty of
Kinesiology for sample treatment and loading tests on MTS device, of Dr. Steven Boyd’s
laboratory at the Cumming School of Medicine for the high-resolution micro CT scanner, and of
Dr. Martin Garon’s laboratory at Biomomentum Inc. in Laval for automated indentation mapping
tests.
I want to thank Ke Gu who compiled the subroutines for material modelling in the
simulation and Marcel Rodriguez for providing mechanical testing data.
Last, but not least, I wish especially to thank my parents for their support, encouragements
and patience during my graduate studies.
iv
Table of Contents
Abstract ............................................................................................................................... ii Acknowledgements ............................................................................................................ iii Table of Contents ............................................................................................................... iv List of Tables ..................................................................................................................... vi List of Figures and Illustrations ........................................................................................ vii List of Symbols, Abbreviations and Nomenclature ........................................................... xi
CHAPTER ONE: INTRODUCTION ..................................................................................1 1.1 Anatomy of the knee joint .........................................................................................1 1.2 Osteoarthritis ..............................................................................................................1 1.3 Motivation ..................................................................................................................2 1.4 Research objectives ....................................................................................................3
CHAPTER TWO: LITERATURE REVIEW ......................................................................4 2.1 Anatomy of human and porcine knee joints ..............................................................4 2.2 Experimental studies on articular cartilage ................................................................9 2.3 Constitutive modelling of soft tissues ......................................................................12 2.4 Geometrical acquisition of joint surfaces ................................................................13 2.5 Geometry reconstruction of knee joint ....................................................................15 2.6 Meshing generation ..................................................................................................17 2.7 Finite Element modelling.........................................................................................17
CHAPTER THREE: METHODS OF MODEL CONSTRUCTION .................................20 3.1 Data acquisition of joint components ......................................................................20
3.1.1 CT scan for bones ............................................................................................20 3.1.2 Automated indentation mapping and cartilage thickness determination .........21 3.1.3 Meniscus thickness measurement ....................................................................25
3.2 Segmentation and geometry generation ...................................................................27 3.2.1 Geometrical construction of the bones ............................................................27 3.2.2 Cartilage surface generation ............................................................................31 3.2.3 Meniscus surface generation ...........................................................................34 3.2.4 Ligament reconstruction ..................................................................................37
3.3 Finite element model of porcine knee joint .............................................................38
CHAPTER FOUR: MATERIAL MODELLING AND GEOMETRY MESHING ..........41 4.1 Split-line pattern of articular cartilage and meniscus ..............................................41 4.2 UMAT formulation in the numerical model ............................................................43 4.3 Implementation of material properties with ABAQUS ...........................................44 4.4 Geometry remeshing ................................................................................................47
4.4.1 Femoral cartilage meshing ..............................................................................49 4.4.2 Tibial cartilage meshing ..................................................................................50 4.4.3 Meniscus meshing ...........................................................................................50 4.4.4 Bones meshing .................................................................................................51 4.4.5 Ligaments meshing ..........................................................................................52
v
CHAPTER FIVE: RESULTS ............................................................................................53 5.1 Load, boundary conditions and constraints .............................................................53
5.1.1 Loading and boundary conditions ...................................................................53 5.1.2 Interactions and constraints .............................................................................53
5.2 Numerical results .....................................................................................................54 5.2.1 Contact status ...................................................................................................56 5.2.2 Fiber orientation and principal directions ........................................................58 5.2.3 Reaction force ..................................................................................................60 5.2.4 Fluid pressure ..................................................................................................61 5.2.5 Contact pressure ..............................................................................................64
CHAPTER SIX: DISCUSSION, CONCLUSION AND FUTURE WORK .....................67 6.1 Summary ..................................................................................................................67 6.2 Comparison with experimental data ........................................................................68 6.3 Significance .............................................................................................................70 6.4 Limitations and future directions .............................................................................70
REFERENCES ..................................................................................................................72
vi
List of Tables
Table 1-1 Advantages of pig model ................................................................................................ 2
Table 2-1 Direct measurement and the tibial index of MM and LM in human and pig knees (MM=medial meniscus; LM=lateral meniscus). Adapted from Proffen et al. (2012) ............ 9
Table 3-1 CT scanning parameters ............................................................................................... 21
Table 3-2 The thickness measurements of lateral meniscus ......................................................... 26
Table 3-3 The thickness measurements of medial meniscus ........................................................ 27
Table 3-4 Morphological operations on binary images ................................................................ 29
Table 3-5 Interpolation methods of the griddata function ............................................................ 34
Table 4-1 Average instantaneous modulus 𝑬𝑬 at different regions ................................................ 46
Table 4-2 Corresponding equilibrium modulus 𝑬𝑬𝑬𝑬 at different regions ..................................... 47
Table 4-3 Material properties implemented in the USER MATERIAL ....................................... 47
Table 5-1 Summary of five loading protocols with boundary and constraint conditions in ABAQUS modelling ............................................................................................................. 54
Table 6-1 Maximum reaction force at the compressive displacement of 200 μm ........................ 69
vii
List of Figures and Illustrations
Figure 2-1 Tibial plateau width (orange dash line) defined in the frontal plane (only femur and tibia are shown in the knee joint) ..................................................................................... 5
Figure 2-2 Direct measurements of range of motion for: a) human knee; b) pig knee with respect to the neutral 0º axis. Adapted from Proffen et al. (2012) .......................................... 6
Figure 2-3 Anterior aspect of a: a) human left knee; b) pig left knee. Lateral on the right. Redrew using Proffen et al. (2012) as a reference .................................................................. 7
Figure 2-4 Posterior aspect of a: a) human left knee; b) pig left knee. Lateral on the left. Redrew using Proffen et al. (2012) as a reference .................................................................. 8
Figure 2-5 Tibial attachments and menisci shape of a: a) human left knee; b) pig left knee. Interior view, Lateral on the right. Redrew using Proffen et al. (2012) as a reference ........... 8
Figure 2-6 Loading protocols and cartilage responses in the: a) stress relaxation test; b) creep test. Redrew using Jin (2014) as a reference ......................................................................... 11
Figure 2-7 Statistical femoral attachment sites of the cruciate ligaments and menisci in a right pig knee. From left to right are the lateral aspect of the medial femoral condyle, posterior aspect of femoral condyles and medial aspect of the lateral femoral condyle. Redrew using from Fuss et al. (1989) as a reference ............................................................ 16
Figure 2-8 Tunnel positions in the ACL reconstruction on the femur of a right porcine knee. The green circle represents the location of high AM and the yellow circle stands for the middle position of ACL. Redrew using Kato et al. (2010) as a reference ............................ 17
Figure 2-9 The modelling procedures of knee joint ...................................................................... 19
Figure 3-1 The equipment and specimen for bone scan: a) XtremeCT scanner; b) The porcine knee joint (kept in a plastic bag while scanning) .................................................................. 21
Figure 3-2 Uniform extraction of the articular surface (Red lines show the cuts) ....................... 22
Figure 3-3 Automated mapping system for cartilage (https://pbs.twimg.com/media/Co2qcKFWcAA-DvD.jpg:large) ......................................... 23
Figure 3-4 Mapping tips: a) spherical indenter; b) needle probe (http://biomomentum.com/wp-content/themes/biomomentum/library/images/zoho/Accessories/BMMT_MA680-1.jpg and http://www.farlamedical.com/images/product_images/popup_images/2455_0.jpg) ..... 24
Figure 3-5 Load vs. position curve in automated thickness mapping ........................................... 25
Figure 3-6 The gridding positions of menisci ............................................................................... 26
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Figure 3-7 Automated thresholding algorithm: a) Segmentation procedures; b) Processed image samples corresponding to each procedure .................................................................. 28
Figure 3-8 Reconstructed femur with: a) isosurfaces (composed of a loose upper part and a compact lower part); b) smoothed triangular surfaces .......................................................... 30
Figure 3-9 Reconstructed tibia with: a) isosurfaces; b) smoothed triangular surfaces ................. 31
Figure 3-10 Femoral cartilage surfaces: a) interpolated with ‘natural’ method and plotted in MATLAB; b) trimmed in Rhinoceros .................................................................................. 32
Figure 3-11 Reconstruction of femoral cartilages in SolidWorks: a) Side surfaces creation with Boundary Surface tool (yellow region represents boundary surface); b) Solid formation with Kit Surface tool ............................................................................................ 33
Figure 3-12 Reconstruction of tibial cartilages in SolidWorks: a) Side surfaces creation with Boundary Surface tool (yellow region represents boundary surface); b) Solid formation with Kit Surface tool ............................................................................................................. 33
Figure 3-13 Inferior surfaces construction of menisci: a) Rough sketch; b) Final profile (the menisci faces are in gray, the tibial cartilages are in green and the tibia is in pink. Black lines are outlines and loft lines) ............................................................................................ 35
Figure 3-14 Superior surfaces construction of menisci (the menisci faces are in gray, the femoral cartilages are in yellow and the tibial cartilages are in green. Red lines represent the measured thicknesses of menisci. Black lines are outlines and loft lines) ...................... 36
Figure 3-15 Reconstructed lateral and medial menisci from different viewing angles ................ 36
Figure 3-16 Reconstructed four main ligaments: a) LCL; b) MCL; c) ACL; d) PCL. Each ligament has six surfaces ...................................................................................................... 38
Figure 3-17 A geometrical model of porcine knee joint with femur, tibia, femoral and tibial cartilages, menisci, ACL, PCL, MCL and LCL (pink lines are the center axes of femur and tibia in right view) .......................................................................................................... 39
Figure 3-18 Detailed drawings of contact pairs in the pig knee joint: a) Femur-femoral cartilage, femoral cartilage-meniscus, meniscus-tibial cartilage and tibial cartilage-tibia pairs in the lateral; b) Femur-femoral cartilage, femoral cartilage-meniscus, meniscus-tibial cartilage and tibial cartilage-tibia pairs in the medial; c) Lateral meniscus-femur and medial meniscus-tibia pairs; d) Meniscus-tibia pairs in the lateral and medial ............. 40
Figure 4-1 Fiber split lines of femoral cartilages for porcine stifle joint assigned in ABAQUS (Gu, 2010) ............................................................................................................................. 42
Figure 4-2 Fiber split lines in pig menisci from: a) the surface layer; b) the bulk tissue in the middle zone. Sectional view. Redrew using Aspden et al. (1985) as a reference................. 43
ix
Figure 4-3 Divided regions with the same average instantaneous modulus in the simulation: a) Femoral cartilages; b) Tibial cartilages............................................................................. 46
Figure 4-4 A typical structured piece in: a) shaded view; b) wireframe ...................................... 48
Figure 4-5 Lateral tibial cartilage partitioned by the datum planes .............................................. 49
Figure 4-6 FE meshing on femoral cartilages ............................................................................... 50
Figure 4-7 FE elements of tibial cartilages ................................................................................... 50
Figure 4-8 FE meshing of menisci ................................................................................................ 51
Figure 4-9 FE meshing of bones: a) Femur; b) Tibia ................................................................... 51
Figure 4-10 FE meshes of LCL .................................................................................................... 52
Figure 5-1 Schematic diagram of node penetration from the master surface into the slave surface resulting from rough meshes (indicated by the arrow). Redrew using ABAQUS Documentation 2014 as a reference ...................................................................................... 55
Figure 5-2 Operations to tight the contact surfaces: a) Interference Fit; b) Slave Node/Surface Adjustment. Redrew using ABAQUS Documentation 2014 as a reference ......................... 56
Figure 5-3 Contact status at 2 s (right before relaxation) on: a) the bottom surfaces of femoral cartilages; b) the top surfaces of tibial cartilages; c) the top surfaces of menisci; d) the bottom surfaces of menisci. Interior view, lateral on the right (CSTATUS = contact status) .................................................................................................................................... 57
Figure 5-4 Contact status at 1800 s (right after relaxation) on the: a) the bottom surfaces of femoral cartilages; b) the top surfaces of tibial cartilages; c) the top surfaces of menisci; d) the bottom surfaces of menisci. Interior view, lateral on the right (CSTATUS = contact status) ........................................................................................................................ 58
Figure 5-5 Material orientation plotted on the deformed shape of the: a) femoral cartilages; b) menisci. Interior view, lateral on the right (1-axis represents the fiber orientation)............. 59
Figure 5-6 Maximum principal stress tensor plotted at 2 s in the: a) femoral cartilages; b) menisci. Interior view, lateral on the right (Max. Principal = Maximum principal stress) .. 60
Figure 5-7 Reaction force of the tibia in the relaxation simulation .............................................. 61
Figure 5-8 Maximum pore pressure in soft tissues as a function of time ..................................... 62
Figure 5-9 Pore pressure distribution at 2 s on: a) the bottom surfaces of femoral cartilages; b) the top surfaces of tibial cartilages; c) the top surfaces of menisci; d) the bottom surfaces of menisci. Interior view, lateral on the right (POR = pore pressure) .................... 63
x
Figure 5-10 Pore pressure distribution at 1800 s on: a) the bottom surfaces of femoral cartilages; b) the top surfaces of tibial cartilages; c) the top surfaces of menisci; d) the bottom surfaces of menisci. Interior view, lateral on the right (POR = pore pressure) ........ 64
Figure 5-11 Contact pressure distribution at 2 s on: a) the bottom surfaces of femoral cartilages; b) the top surfaces of tibial cartilages; c) the top surfaces of menisci; d) the bottom surfaces of menisci. Interior view, lateral on the right (CPRESS = contact pressure) ................................................................................................................................ 65
Figure 5-12 Contact pressure distribution at 1800 s on: a) the bottom surfaces of femoral cartilages; b) the top surfaces of tibial cartilage; c) the top surfaces of menisci; d) the bottom surfaces of menisci. Interior view, lateral on the right (CPRESS = contact pressure) ................................................................................................................................ 66
Figure 6-1 Reaction force vs. compression at varied rates from experiment and simulation ....... 69
xi
List of Symbols, Abbreviations and Nomenclature
Symbol Definition ACL AL
Anterior cruciate ligament Anterior lateral
ALM AM
Anterior lateral meniscus Anterior medial
AMM DOF FE FEA FEM
Anterior medial meniscus Degree of freedom Finite element Finite element analysis Finite element method
LCL LM MCL MM OA PBS PCL PL
Lateral collateral ligament Lateral meniscus Medial collateral ligament Medial meniscus Osteoarthritis Phosphate Buffered Saline Posterior cruciate ligament Posterior lateral
PLM PM
Posterior lateral meniscus Posterior lateral
PMM Posterior medial meniscus 𝐸𝐸𝑚𝑚
𝑣𝑣𝑚𝑚
𝐸𝐸𝑥𝑥𝑓𝑓
𝐸𝐸𝑥𝑥0, 𝐸𝐸𝑥𝑥𝜀𝜀
𝜀𝜀𝑥𝑥
𝜎𝜎𝑥𝑥𝑓𝑓
Young’s modulus of non-fibrillar matrix Poisson’s ratio Fibrillar modulus in x direction As in 𝐸𝐸𝑥𝑥
𝑓𝑓 = 𝐸𝐸𝑥𝑥0 + 𝐸𝐸𝑥𝑥𝜀𝜀𝜀𝜀𝑥𝑥, 𝐸𝐸𝑥𝑥0 and 𝐸𝐸𝑥𝑥𝜀𝜀 are constants
Tensile strain in x direction Tensile stress in x direction
1
CHAPTER ONE: Introduction
1.1 Anatomy of the knee joint
The knee joint is the largest joint in the human body. It consists of femur, tibia, fibula and
patella. The ends of the bones are covered with articular cartilage layers, among which femoral
cartilages, tibial cartilages and patellar cartilages are of great interest. Between femur and tibia,
there are two wedge-shaped menisci: lateral meniscus and medial meniscus. Articular cartilages
and menisci help to protect and cushion the bones and joint as the legs perform mechanical
functions. Ligaments are long fibrous straps to connect the bones and provide the stability of the
joint. Anterior cruciate ligament and posterior cruciate ligament control the relative anterior-
posterior movements (rotation, sliding, etc.) of the femur in relation to the tibia. The medial
collateral ligament and lateral collateral ligament are located at the sides of the knee to restrain
valgus and varus angulation of the knee, respectively. Tendons and muscles are not modeled in
the present study.
1.2 Osteoarthritis
Osteoarthritis is commonly characterized by degeneration, wear and tear of cartilage, with
the knee being one of the most commonly affected joints. When knee osteoarthritis advances,
cartilage in the joint gradually wears away. Cartilage loss narrows the joint space and results in
more bone rubbing. Subsequently, pain, stiffness and swelling occur in the joint and even worse,
bone spurs come into being over time.
Although knee osteoarthritis is known to initiate from cartilage, the exact mechanism is
not fully understood. While age is a major risk factor for knee osteoarthritis, mechanical factors
are believed to play a vital role in developing osteoarthritis of the knee.
2
1.3 Motivation
The contact mechanics of articular cartilage is the key to better understanding the
mechanism of knee osteoarthritis. A thorough understanding of joint mechanics is in urgent need
for OA prevention and treatment. Progress has been continually made with constitutive models
describing the mechanical behaviors of articular cartilage, and these are successfully implemented
on cartilage disc samples or fresh cadaver human joint specimens. Our research group has
published a 3D human knee FE model with user-defined subroutines. However, this model is not
well validated against experimental data because large numbers of fresh cadaver human knee joints
are unavailable. Therefore, it is indispensable to use animal explants (Gregory et al., 2012).
Because of the availability and similarity in size to human knee, pig knee is a good substitute for
mechanical testing and simulation (Aspden et al., 1985; Gregory et al., 2012; Proffen et al., 2012),
as illustrated in Table 1-1. Therefore, the porcine knee is utilized to investigate the mechanics and
mechanobiology of the joint in this study and other studies in our research group.
Table 1-1 Advantages of pig model
Feasibilities of pig knee joint
Anatomically Proximity in size and morphology of the intra-articular structures of human knee joint
Pathologically Prone to spontaneous OA
Economically Low cost
Availability Easily obtained from the slaughter house nearby
3
1.4 Research objectives
The objectives of this study are to:
1) develop an alternative method for the construction of knee joint geometry;
2) build a finite element model of a pig knee joint to determine the joint mechanics.
Recently, magnetic resonance imaging (MRI) and computed tomography (CT)
technologies are widely used to measure the anatomical structures of the joint. MRI scanning has
sufficient contrast for all joint components (especially for soft tissues), but the cost is relatively
high and the waiting time is long. Although CT scan can only identify the geometries of bones, a
unique automated indentation mapping can be combined to capture the thickness of the articular
cartilages and menisci.
The described method for geometrical reconstruction is relatively straightforward. Meshes
can be then generated on the reconstructed model for finite element analysis, in which contact
status, stress and strain distribution, fluid pressure, etc. are calculated. These results can be
validated with mechanical tests of porcine joints performed by a team member. The validated finite
element model can also be used to predict the mechanics of human knee joints with confidence.
Furthermore, the mechanics of porcine joints may be able to aid another study in our group where
the links between mechanical factors and cartilage metabolism are investigated.
4
CHAPTER TWO: Literature review
This chapter investigates the anatomic similarities and differences between the human and
pig knee (stifle) joints. Typical standardized mechanical testing (stress relaxation and creep) and
constitutive models for soft tissues in the last decades are then introduced. Finally, techniques for
knee geometry reconstruction and the subsequent procedures of FE modelling are presented.
2.1 Anatomy of human and porcine knee joints
A knee joint consists of the femur, femoral cartilages, tibia, tibial cartilages, menisci and
ligaments in either species. Differences do exist in the intra-articular structures of the joint such as
menisci, ligaments and cartilages. Efforts have been made on the determination of anatomical
differences in the knees between species, for better serving the medical and clinical studies. It was
found that the meniscus of all the mammals has the same semilunar shape, despite of their walking
style or the size of the meniscus (Parson, 1899). The pig meniscus was identified as a practicable
and economically feasible model for meniscal biomechanical study (Aspden et al., 1985). The
anatomy and function of the cruciate ligaments of the domestic pig (Sus scrofa domestica) were
investigated in comparison with the human cruciate ligaments (Fuss, 1991). Specific to ACL, a
further functional comparison was conducted between species and it was concluded that the pig
was the preferred experimental animal model most closely resembles human beings (Xerogeanes
et al., 1998). Besides, the thickness of articular cartilage in pig knee joint was measured with an
average thickness of 1.5 mm, in comparison with 2.2 - 2.5 mm in humans (Hembry et al., 2001).
Proffen et al. (2012) first systematically and quantitatively compared the anatomy of the
joint tissue in the knee between human and pig. Statistical differences in motion range and tissue
size were found. To eliminate the impact on the size of different knees, all the measurements were
5
normalized by the width of the tibial plateau (highlighted in Fig. 2-1). The ratio between the width
of the tibial plateau and the direct measurement was referred to as the tibial index.
Figure 2-1 Tibial plateau width (orange dash line) defined in the frontal plane (only femur
and tibia are shown in the knee joint)
The major difference between human and porcine knees (and all animal knees) was the
extension angle. The human knee could be brought to almost full extension but the porcine knee
had a comparatively larger extension angle (~42º), as shown in Fig. 2-2. The difference in flexion
angle was unapparent between human and pig knees.
6
a) b)
Figure 2-2 Direct measurements of range of motion for: a) human knee; b) pig knee with
respect to the neutral 0º axis. Adapted from Proffen et al. (2012)
The menisci have two components (medial and lateral meniscus) with four horns attached
on the tibia. In the human knee, the medial meniscus is a bit narrower but significantly longer
compared with the pig medial meniscus (Table 2-1). Relative to the tibial plateau width, the tibial
index of the human medial meniscus in width was significantly smaller than the porcine one but
only slightly larger in length (Table 2-1). In both species, the bony insertion of AMM was the most
anterior structure in the tibial plateau right above the tibial tuberosity (Fig. 2-3 and Fig. 2-5). The
PMM was inserted on the most anteromedial facets of the posterior intercondylar area (Fig. 2-5).
The human lateral meniscus was a little narrower but much longer than the pig lateral meniscus
(Table 2-1). Neither the tibial index of the lateral meniscus in width nor in length was significantly
different between the species (Table 2-1). For human lateral meniscus, it only occupied a small
portion of the lateral tibial plateau and the ALM was attached to the most medial edge of the lateral
intercondylar eminence (Fig. 2-5). In pig knee, the lateral meniscus split the ACL bundles into
two parts (Fig. 2-5). The PLM connected the menisco-femoral ligament on the lateral back wall of
Flexion
Extension
7
the medial femoral condyle more inferiorly in the human specimen than in the pig one (Fig. 2-4
and Fig. 2-5).
For the cruciate ligaments (ACL and PCL), the femoral ACL insertion site is located at the
posteromedial edge of the lateral condyle in both species, but the tibial insertion sites varied. For
the human knee, the ACL attachment on the tibia was adjacent to the tibial ALM insertion, as
demonstrated in Fig. 2-3 and Fig. 2-5. In comparison, a pig knee had the tibial ACL insertion of
the AM and PL bundles separated by the tibial ALM attachment (Fig. 2-5). In human and pig
knees, PCL attachment to the femur and tibia had the same location (Fig. 2-3, Fig 2-4 and Fig 2-
5).
a) b)
Figure 2-3 Anterior aspect of a: a) human left knee; b) pig left knee. Lateral on the right.
Redrew using Proffen et al. (2012) as a reference
8
a) b)
Figure 2-4 Posterior aspect of a: a) human left knee; b) pig left knee. Lateral on the left.
Redrew using Proffen et al. (2012) as a reference
a) b)
Figure 2-5 Tibial attachments and menisci shape of a: a) human left knee; b) pig left knee.
Interior view, Lateral on the right. Redrew using Proffen et al. (2012) as a reference
9
Table 2-1 Direct measurement and the tibial index of MM and LM in human and pig knees
(MM=medial meniscus; LM=lateral meniscus). Adapted from Proffen et al. (2012)
Species
MM LM
Width Length Width Length
Human
Direct measurement/mm 9.50 39.80 9.83 33.28
Tibial index/mm 0.14 0.57 0.14 0.48
Pig
Direct measurement/mm 10.44 25.32 10.26 25.60
Tibial index/mm 0.20 0.49 0.20 0.50
In conclusion, comparative studies on the size and morphology show that the pig knee joint
has a considerably similar structure to the human knee joint, except for the limited extension which
is less of an important factor in compressive testing. Overall, the porcine knee joint is the optimal
animal model for simulating the mechanics of the human knee.
2.2 Experimental studies on articular cartilage
Articular cartilage consists of proteoglycan, collagen fiber and interstitial fluid (Mow et
al., 1989). From the cartilage surface to the subchondral bone, articular cartilage is often
subdivided into three zones: superficial, middle and deep zones (Kempson et al., 1973; Muehleman
et al., 2004; Schenk et al., 1986). The depth of the three zones from the articular surface to the
subchondral bone is 10-20%, 40-60% and 20-50% of the total thickness of the tissue, respectively.
The collagen fibers are distributed tangentially to the articular surface in the superficial zone but
perpendicular to the cartilage-bone interface in the deep zone. The collagen fibers in the middle
zone are randomly oriented (Choi and Gold, 2011; Mow, 2005)
10
Articular cartilage is highly inhomogeneous and anisotropic and its mechanical properties
vary throughout the tissue depth. To perform experimental observations and evaluate the
mechanical properties of cartilage, three different standardized compression tests are frequently
used: confined compression, unconfined compression and indentation test (Buschmann et al.,
1997; Suh and Mow, 1990; Suh et al. 1990).
Relaxation and creep loading protocols are often used in the standardized experiments. In
stress relaxation, a ramp displacement is applied to the cartilage specimen and the position is held
for a period of time, which is shown in Fig. 2-6a. At the beginning, the stress increases with the
compressive displacement before the fluid pressure reaches the maximum. As the compression
remains constant, the reaction force starts to decrease till the fluid pressure ceases. This behavior
can be explained by the fluid exudation and intrinsic viscoelasticity in the tissue (Mow et al., 1980).
In the creep protocol, a given force is applied to the specimen and kept constant. The resulting
displacement continues going up before reaching its asymptote at equilibrium, also on account of
the fluid exudation and viscoelasticity (Mow et al., 1980).
11
a)
b)
Figure 2-6 Loading protocols and cartilage responses in the: a) stress relaxation test; b) creep
test. Redrew using Jin (2014) as a reference
Articular cartilage plays a vital role in joint mechanics owing to its unique structure. This
study mainly focuses on the compression of the knee. Therefore, cartilage and meniscus are very
important to be taken into account. Other soft tissues such as ligament and tendon are not
introduced in detail because they make less contribution to the compressive stiffness in the joint.
12
2.3 Constitutive modelling of soft tissues
The simplest constitutive models of the articular cartilage are single-phase elastic models,
which only consider the solid phase in the tissue. These models are commonly assumed as a linear
elastic solid matrix (Bahrani and Gardner, 1980; Hayes et al., 1972; Kempson et al., 1971), and
are inadequate to explain the effects due to the fluid-flow. Therefore, they only have the capability
of capturing cartilage behavior in two extreme cases: at the instant of loading before flow comes
into being; at equilibrium when the fluid-flow has ended (Goldsmith et al., 1996). The responses
are elastic in both cases. Researchers then provided amendments accounting for the viscoelasticity
in the cartilage (Coletti et al., 1972; Hayes and Mockros, 1971; Parsons and Black, 1977). In short,
the single-phase models are unsuitable to describe the fluid-flow that causes the time-dependent
behavior of the cartilage.
Biphasic models are the second generation of constitutive models for cartilage, and
consider both solid and fluid phases, which enables them to describe the effect of the fluid flow in
the tissue. These models represent hydrated tissues (collagen fibers and proteoglycan matrix) as
comprised of a solid phase and an interstitial fluid phase (Mow et al., 1980). The solid phase is
treated as a linearly elastic, homogenous, isotropic and permeable matrix, and the fluid is basically
assumed to be inviscid and no shear stress is considered. The early linear biphasic model
considered the permeability as a constant and has been widely employed for modelling the
mechanical behavior of cartilage (Mow et al., 1980). Later, a nonlinear biphasic model was
proposed to consider the permeability as a function of volumetric strain of the solid matrix (Lai et
al., 1980; Mow and Roth, 1981). Large deformation biphasic model was first formulated with a
hyperelastic solid matrix (Holmes and Mow, 1990; Kwan et al., 1990). Biphasic models have
limitations in modelling the mechanical response of articular cartilage with the short-term and
13
time-dependent response at a high compressive strain-rate. One probable cause is that the fluid
pressure is fairly high with respect to the compressive stress in the tissue (Brown and Singerman,
1986; Miller, 1998).
The fibril-reinforced poroelastic models have made a progress describing the cartilage
response at high strain-rate compressions (Li et al., 1999; Soulhat et al., 1999). In these models,
the solid phase is further divided into two parts: the porous non-fibrillar matrix representing the
proteoglycans; the fibrillar network, which represents the collagen fibrils reinforcing the non-
fibrillar solid matrix. The tensile loads in the cartilage are carried by both of the components but
the compressive ones are only supported by the non-fibrillar matrix, due to the fact that the
collagen fibers cannot resist compressive loading. Therefore, the fibrils are usually described by
the tensile modulus, and the non-fibrillar matrix is modelled as a linearly elastic material with
Young’s modulus and Poisson’s ratio. The early fibril-reinforced poroelastic model was first
proposed where the non-fibrillar matrix was assumed to be comprised of linear elastic
proteoglycans reinforced by linear elastic collagen fibrils (Soulhat et al., 1999). Li and his
coworkers (1999, 2001 and 2002) amended the fibril-reinforced poroelastic model and considered
the nonlinear permeability, nonlinear fibrillar modulus and finite deformation. It was also proposed
that pore fluid pressurization dominated the transient mechanical response of the tissue in
compression (Li et al., 2004b). In conclusion, the fibril-reinforced poroelastic models are able to
account for the fluid pressurization and accurately define the strain- and time-dependent
mechanical response in the cartilage.
2.4 Geometrical acquisition of joint surfaces
Many methods have been used to quantify knee joint geometries. Mechanical techniques
include the production of plastic moldings (Seedhom et al., 1972), the production of a silicone
14
rubber mold used to make a plaster casting (Scherrer, 1977; Scherrer and Hillberry, 1979), and the
use of a mechanical measuring pin attached to a dial gauge (Wismans et al., 1980). Other methods
such as slicing (Meachim et al., 1977; Shiba et al., 1988), radiographs (Armstrong and Gardner,
1977; Hall and Wyshak, 1980) and ultrasound (Modest et al., 1989; Rushfeldt et al., 1981) have
also been commonly used. Optical techniques have been used such as successively photographing
specimen slices (Mcleod et al., 1976), close-range photogrammetry (Ghosh, 1983) and analytical
stereophotogrammetry (Ateshian et al., 1988; Huiskes et al., 1985; Soslowsky et al., 1989). CT
and MRI technologies have also been used to image the anatomical structures of the joint (Belsole
et al., 1988; Feldkamp et al., 1989; Garg and Walker, 1990; Moon et al., 1983).
CT imaging is superior to conventional radiography because it offers a sectioning
evaluation and high-contrast resolution of osseous changes (Chan et al., 1991). Micro CT has been
successfully implemented in rabbit and canine models to accurately image subchondral bone
changes and thereby follow disease progression of osteoarthritis (Batiste et al., 2004; Boyd et al.,
2000 and 2002). A limitation of μCT is its inferior contrast for soft tissue due to the low X-ray
attenuation, so cartilage is unable to be directly imaged by this modality (Piscaer et al., 2008;
Renders et al., 2014). μCT-arthrography provides the possibility of indirect visualization of
cartilage morphology by injection of a radio-opaque contrast agent into the joint cavity ex vivo
(Roemer et al., 2005).
MRI scan has the capability of direct multi-planar imaging and provides a higher contrast
for soft tissues than CT scan (Chan et al., 1991). MRI imaging is considered as a non-invasive tool
for accessing pathologic variations in the articular cartilage, meniscus, and ligament of the knee
(Braunstein et al., 1990; Disler et al., 2000). Researchers have demonstrated that MRI was
sensitive to the changes of cartilage integrity in the rabbit (Batiste et al., 2004; Calvo et al., 2001;
15
Laurent et al., 2003; Wachsmuth et al., 2003), guinea pig (Tessier et al., 2003), and rat (Kapadia
et al., 1998; Loeuille et al., 1997; Spandonis et al., 2004). MRI imaging is widely used to identify
the geometry of soft tissue in vivo but the cost is relatively high and the waiting time is long.
2.5 Geometry reconstruction of knee joint
Image processing software packages, such as Mimics (Materialise, Leuven Belgium),
Simpleware (Exeter, UK) and Rhinoceros 3D (Seattle, WA, USA), are available to reconstruct the
3D geometry of knee joint from 2D images with different tools including segmentation, model
construction and model refinement (Kazemi Miraki, 2013). First of all, segmentation of 2D images
is to precisely identify the tissues and their boundaries. This process can be performed
automatically, semi-automatically or manually. It is time-consuming when the collection data set
is large. A built-in tool called 3D LiveWire in Mimics is commonly used to segment the tissues
from MRI/CT slices (Bowers et al., 2008; Gougoutas et al., 2004; Steines et al., 2000). After using
the thresholding tool and performing some manual edits, a 3D surface model can be obtained from
the segmented mask; or marching cubes algorithm (Ferrant et al., 2001; Viceconti et al., 1999;
Wang et al., 2005) and NURBS algorithm (Lee et al., 2002; Wu et al., 2007) are other good choices
for the primary surface generation. The coarse facets generated from the segmentation procedure
have many bumps and invalid areas superficially. To deal with these issues, the smoothing tools
like open-sources or programs such as MeshLab (Cignoni et al., 2008) are utilized to acquire the
optimal surface condition.
Bones (femur and tibia) and soft tissues (femoral cartilages, tibial cartilages and menisci)
can be reconstructed with the approaches mentioned above. However, for ligament reconstruction,
only a smooth surface with accurate insertion sites to the bones is required. In most of the finite
element models, they were modelled as springs (Bendjaballah et al., 1995; Blankevoort & Huiskes,
16
1991; Li et al., 1999). Proffen and his group (2012) located the tibial attachments of menisci and
ligaments. To confirm the locations of femoral attachments, Fuss et al. (1989) performed a novel
dissection technique on 20 fresh knee joints from 20 pigs (sus scrofa domestica) and showed the
insertion sites of ACL, PCL and PLM on the femur, as shown in Fig. 2-7. Bowman et al. (2009)
compared the single-bundle and the double-bundle (AL and PM) reconstructions of PCL for
providing surgical technique support. Iriuchishima et al. (2009) explored how intercondylar roof
impingement pressure changed with different tunnel reconstructed positions (AM-AM and PL-
High AM) in the single-bundle ACL model with the concept of AM and PL bundles. Kato et al.
(2010) further defined three more different tunnel positions (PL-PL, MID-MID and DB) for single-
bundle ACL reconstruction with AM and PL bundle admitted and figured out which method
provided the best stability and more closely restored normal knee kinematics.
Figure 2-7 Statistical femoral attachment sites of the cruciate ligaments and menisci in a
right pig knee. From left to right are the lateral aspect of the medial femoral condyle,
posterior aspect of femoral condyles and medial aspect of the lateral femoral condyle.
Redrew using from Fuss et al. (1989) as a reference
17
Figure 2-8 Tunnel positions in the ACL reconstruction on the femur of a right porcine
knee. The green circle represents the location of high AM and the yellow circle stands for
the middle position of ACL. Redrew using Kato et al. (2010) as a reference
2.6 Meshing generation
FE meshing can be generated using the built-in functions of the image processing tools,
such as MATLAB (Mathworks, Natick, MA, USA), or in FE programs, such as ABAQUS
(Simulia, Providence, USA), or in specialized meshing programs, such as MeshLab and
HyperMesh (Altair, Troy, MI, USA). On the other hand, automatic or free meshing normally yields
triangular elements which have limited ability to generate specific meshes such as meshes with
hexahedral elements. Nevertheless, the commercial packages are able to achieve it. Specifically,
when contact interaction and fluid pressurization are both considered, convergence is relatively
faster with hexahedral elements (ABAQUS Documentation 2014).
2.7 Finite Element modelling
FE technique is one of the effective numerical methods to predict the biomechanics of
articular cartilage. Chand and his colleagues (1976) proposed a 2D FE model of the knee joint in
18
NASTRAN (MSC Software Corporation, Santa Ana, CA, USA) to attain the relation between
force and deformation. FEAP (University of California, Berkeley, USA), another FE package, was
then utilized by Huber-Betzer et al. (1990) for the development of a plane-strain knee model, in
order to investigate the contact mechanics in the joint. van der Voet et al. (1992 and 1993) used
the poroelastic elements in ABAQUS to model 2D joint contact problems. Li and Soulhat (1999)
started to model the fibrillar network in ABAQUS with discrete spring elements or continuum
elements. So far, computational modelling of articular cartilage had been mostly performed with
the FE package ABAQUS.
Fig. 2-9 shows a general work flow how the knee joint is reconstructed and analyzed by
FEM. First of all, anatomically accurate geometries are obtained from MRI or CT slices. Then, all
the geometrical images are imported to a data processing software package like Mimics to get a
smoothed joint model. After generating structured meshes in ABAQUS CAE, FEA can be running
with a user-defined material subroutine (UMAT) to get joint mechanics.
19
Figure 2-9 The modelling procedures of knee joint
20
CHAPTER THREE: Methods of model construction
This chapter describes the various technologies that were employed for geometrical
acquisition and reconstruction. A micro CT (µCT) was used to obtain femur and tibia geometries
of the porcine knee joint. An automated indentation mapping tester was adapted to gain the surface
data of articular cartilages. Menisci thicknesses were measured by needle probe testing. Surface
generation was achieved in MATLAB for all the joint components. Coarse facets were trimmed
and assembled in Rhinoceros 3D for further FEA use. At last, ligaments were constructed at the
attachment sites where they insert into the bones.
3.1 Data acquisition of joint components
There are a few preliminary preparations before collecting the geometrical information.
Firstly, a porcine knee (stifle) joint with intact joint capsule was obtained within 24 hours after
slaughter. Secondly, an electric saw was used to cut the excess bones in order to make sure the
knee joint could fit the cylindrical cast for the CT and did not exceed the maximum scan size.
Thirdly, the joint was kept hydrated in a sealed plastic bag with PBS solution and stored in a
refrigerator at 4°C for 15 hours before testing.
3.1.1 CT scan for bones
After muscles and tendons were removed (still with complete joint capsule), the porcine
knee joint was scanned with a high-resolution of 60.7 µm CT scanner in Dr. Steve Boyd’s lab at
the University of Calgary. The XtremeCT scanner (Fig. 3-1a) is a clinical scanner located in the
Foothills Hospital. The slice thickness for the scanned pig joint was 0.607 mm and the CT images
were saved as DICOM format. Other main parameters for the CT scanner in data acquisition
process are shown in Table 3-1.
21
Table 3-1 CT scanning parameters
Specifications Parameters
Scan field of view 139.85 mm
Display field of view 102.58 mm
Image matrix 1690 × 1633
Data collection number 3401
a) b)
Figure 3-1 The equipment and specimen for bone scan: a) XtremeCT scanner; b) The
porcine knee joint (kept in a plastic bag while scanning)
3.1.2 Automated indentation mapping and cartilage thickness determination
Once completing the CT scan (3 hours later), the joint capsule was opened up carefully.
Femoral cartilage and tibial cartilage were cut off by an electric saw for thickness mapping tests.
The condyles were removed in a plane with at least 5 mm of bone left on all sides (Fig. 3-2), which
ensured the integrity and uniformity of the testing samples.
22
Figure 3-2 Uniform extraction of the articular surface (Red lines show the cuts)
Automated indentation mapping test of the cartilage was conducted with a Mach-1™
V500css tester (Biomomentum, Laval, Canada), as shown in Fig. 3-3. Cartilage sample was fixed
in a testing chamber and kept hydrated with PBS. A camera-registration system was equipped
alongside for image acquisition. A position grid was superimposed on the cartilage sample for a
mechanically-controlled surface mapping (illustrated in the detailed drawing of Fig. 3-3).
Mechanical properties were first mapped ex vivo (1 measurement per site) using the automated
indentation mapping and then mapped for thickness using the automated thickness mapping.
23
Figure 3-3 Automated mapping system for cartilage
(https://pbs.twimg.com/media/Co2qcKFWcAA-DvD.jpg:large)
The steps of automated indentation mapping are described in the following. In order to be
optimal with the species (pig cartilage), the radius of the spherical indenter used was 0.5 mm (Fig.
3-4a) and it was installed under a tri-axial load cell (70N range and 3.5mN resolution on the vertical
axis, 50N range and 2.5mN resolution on the horizontal axes). At each projected position, the
spherical indenter measured the contact coordinates (x, y, z) of the grid point itself and four
neighboring locations (Δx = Δy = ± 1 mm), so that the surface orientation could be calculated. The
perpendicular indentation was performed by using three displacement components simultaneously
to provide perpendicular displacement based on the measured surface orientation. Finally, the
normal force was calculated using multiaxial load cell components (Fx, Fy and Fz) (Sim et al.,
2014).
24
a) b)
Figure 3-4 Mapping tips: a) spherical indenter; b) needle probe (http://biomomentum.com/wp-
content/themes/biomomentum/library/images/zoho/Accessories/BMMT_MA680-1.jpg and
http://www.farlamedical.com/images/product_images/popup_images/2455_0.jpg)
After finishing the indentation test, the spherical indenter was replaced by a needle probe
for automated thickness mapping. The optimized needle tip corresponding to the porcine cartilage
is a 26G 3/8” PrecisionGlide intradermal bevel needle (BD, Franklin Lakes, NJ, USA) (Fig. 3-4b).
A vertical needle penetration test was performed at each predefined grid and led to a force curve
as a function of position, which was demonstrated in Fig. 3-5. There was a small force increase as
the needle reached the cartilage surface. The load kept rising steadily along the cartilage thickness
but a sharp growth in force occurred at the subchondral bone. Therefore, cartilage thickness could
be determined by the vertical distance multiplying the cosine of the surface angle, as illustrated in
Fig. 3-5 (Sim et al., 2014).
25
Figure 3-5 Load vs. position curve in automated thickness mapping
Combining automated indentation mapping with automated thickness mapping, the
instantaneous modulus can be obtained by fitting the load-displacement curve to an elastic model
in the indentation (Sim et al., 2014). The elastic modulus will be used later in Chapter 4 for
implementation of material property in the FE model.
3.1.3 Meniscus thickness measurement
Thicknesses of the menisci were manually measured at a various predefined positions (Fig.
3-6). At first, a uniform position grid was imposed on the menisci sample. Then, a needle probe
vertically penetrated the menisci till the bottom. A mark was put on the needle where it was just
exposed on the menisci surface. Thickness could be denoted by the distance between the needle
tip and the mark site. The thicknesses of lateral and medial menisci at each gridding point were
listed in Table 3-2 and Table 3-3.
26
Figure 3-6 The gridding positions of menisci
Table 3-2 The thickness measurements of lateral meniscus
Position # 1 2 3 4 5 6 7 8 9 10
Thickness/mm 4.0 6.0 11.0 10.0 10.0 10.0 10.0 7.0 7.0 4.0
Position # 11 12 13 14 15 16 17 18 19 20
Thickness/mm 5.0 4.5 2.0 1.5 2.0 6.0 4.5 4.0 5.0 11.0
27
Table 3-3 The thickness measurements of medial meniscus
Position # 1 2 3 4 5 6 7 8 9 10
Thickness/mm 10.0 6.5 8.5 9.0 11.5 7.0 3.5 2.0 2.0 3.5
Position # 11 12 13 14 15 16 17
Thickness/mm 4.5 8.5 12.0 8.5 3.5 11.5 5.0
3.2 Segmentation and geometry generation
3.2.1 Geometrical construction of the bones
In order to distinguish the bone tissues from the non-bone tissues, the image processing
toolbox in MATLAB was utilized to identify and obtain the boundaries of bones from the high-
resolution CT slices. As mentioned before, CT images were stored in the standard DCM format,
which MATLAB was unable to modify directly. Thus, Sante DICOM Viewer Free (Santesoft,
Athens, Greece) software was utilized to view DICOM files and transform them into JPG format
in grayscale mode. Grayscale images were then segmented by an automated thresholding
algorithm compiled in MATLAB (Fig. 3-7). Transforming grayscale images to binary images, all
the biological tissues (muscles, tendons or ligaments) visible in CT slices were removed except
for bones. For example, the femur had maximum contrast in CT scanning so that it had a
sufficiently high brightness value (a pixel depth of 127) in grayscale images. MATLAB could
filter out the darker pixels with a grayscale value lower than 127. As a consequence, only the femur
was kept in pure white (a pixel depth of 1) with a pure dark background (a pixel depth of 0) in
binary images.
28
However, there still existed many noisy points around the bones. The ‘clean’ mode of built-
in morphological operation in MATLAB was applied on binary images to eliminate the isolated
white pixels (noises). Table 3-4 summarizes a couple of morphological operations commonly used
in binary image processing. Although the morphological-operated images were good enough to
generate the primary surface of the bones, the ‘remove’ operation was further applied to acquire
the boundary of the cross-section for the sake of fast reconstruction. All the operated slices were
then saved in a 3D array called volume data 𝑉𝑉 for surface generation at next step.
a)
b)
Figure 3-7 Automated thresholding algorithm: a) Segmentation procedures; b) Processed
image samples corresponding to each procedure
Grayscale
image
Binary
image
Morphological-
operated image
29
Table 3-4 Morphological operations on binary images
Operation Description
'clean' Removes isolated pixels (individual 1s that are surrounded by 0s)
'fill' Fills isolated interior pixels (individual 0s that are surrounded by 1s)
'remove' Removes interior pixels and leaves only the boundary pixels on, by setting a pixel to 0 if all its 4-connected neighbors are 1
'thicken' Adds pixels to the exterior of objects and thickens the boundaries, until doing so would result in previously unconnected objects being 8-connected
'thin' Removes pixels so that an object without holes shrinks to a minimally connected stroke, and an object with holes shrinks to a connected ring halfway between each hole and the outer boundary
There were two major steps to generate the bone surface. First, the isosurface function in
MATLAB was used to make up a coarse surface with triangle meshes (Fig. 3-8a). Second, the
Poisson algorithm in MeshLab was used to get a smoothed surface (Fig. 3-8b).
An isosurface is a surface that represents points of a constant value (isovalue) within a
volume of space. The isosurface function in MATLAB connects the points that had the same color
value much the way contour lines connect the points of equal elevation. The isosurface function
computes from the 3D volume data 𝑉𝑉 at a specific value and returns a 2D array of faces and a 2D
array of vertices, which constitute the reconstructed object. Here, the given isosurface value was
255 representing the bone boundary in comparison with the black background, which was 0. To
generate the isosurface for the femur without femoral condyles, 64 CT images out of 960 were
uniformly selected out. Note that the femoral condyles were built separately with more slices (15
CT images chosen out of 90) because of their small dimensions; otherwise, inadequate information
30
would be obtained and holes would be formed on the apex of condyles for lack of details. The
same procedures applied to the tibia and 70 CT images out of 770 were used for reconstruction
(Fig. 3-9).
Poisson Surface Reconstruction operation in MeshLab was used to remesh the bony models
and get smoothed triangular surfaces. Several parameters are available to enhance the remeshing
result, including Depth, SolverDivide, IsoDivide, SamlePerNode, Scale and Offset. Especially the
parameters Depth and SamplesPerNode have a great influence on the generated mesh. The higher
the value of depth is, the more detailed the results are. A high SamplesPerNode parameter provides
a smoothing with loss of vertices, while a low value keeps the detail level high. In this study, femur
and tibia were managed by default options and values in Poisson Surface Reconstruction operation.
a) b)
Figure 3-8 Reconstructed femur with: a) isosurfaces (composed of a loose upper part and a
compact lower part); b) smoothed triangular surfaces
31
a) b)
Figure 3-9 Reconstructed tibia with: a) isosurfaces; b) smoothed triangular surfaces
3.2.2 Cartilage surface generation
For cartilage surface reconstruction, the griddata function in MATLAB was utilized to
interpolate the scattered surface coordinate data (Fig. 3-10a), which was collected from the
automated indentation mapping. The syntax of griddata specifies the triangulation interpolation
method to compute the surface value. The triangulation method uses Renka's algorithm (Renka,
1984) to perform a Delaunay triangulation (Okabe et al., 1992) for the scattered points nearby, and
identify a neighborhood to be used in the interpolation. Table 3-5 lists different interpolation
operations generally utilized for 2D/3D scattered data. Here, ‘natural’ mode was chosen because
it worked best among available trials.
Since the fitting surfaces had irregular shapes that were not suitable for FE meshing in the
subsequent step, all the cartilage surfaces were trimmed properly with a structured contour in
Rhinoceros (Fig. 3-10b). Side faces of articular cartilages were then built by Boundary Surface
32
command and connected with the rest to form a solid model by Knit Surface command in
SolidWorks, as shown in Fig. 3-10. The Boundary Surface feature created surfaces between
boundaries using two types of guidelines. First, the Direction 1 lines were set, which were actually
the edges of the two surfaces where the boundary surface was required. Then Direction 2 of
guidelines were defined, along which the curvature combs were generated (Fig. 3-11a and Fig. 3-
12a). Having selected all the cartilage surfaces, an enclosed volume was created with Knit Surface
tool under default Gap Control. A solid model could also be produced if the Create Solid option
was checked.
a) b)
Figure 3-10 Femoral cartilage surfaces: a) interpolated with ‘natural’ method and plotted in
MATLAB; b) trimmed in Rhinoceros
33
a) b)
Figure 3-11 Reconstruction of femoral cartilages in SolidWorks: a) Side surfaces creation
with Boundary Surface tool (yellow region represents boundary surface); b) Solid formation
with Kit Surface tool
a) b)
Figure 3-12 Reconstruction of tibial cartilages in SolidWorks: a) Side surfaces creation with
Boundary Surface tool (yellow region represents boundary surface); b) Solid formation with
Kit Surface tool
34
Table 3-5 Interpolation methods of the griddata function
Operation Description
'linear' Triangulation-based linear interpolation (default)
'nearest' Triangulation-based nearest neighbor interpolation
'natural' Triangulation-based natural neighbor interpolation. This method is an efficient tradeoff between linear and cubic
'cubic' Triangulation-based cubic interpolation supporting 2D interpolation only
'v4' Biharmonic spline interpolation supporting 2D interpolation only
3.2.3 Meniscus surface generation
As mentioned, only the thicknesses of menisci at grid positions were obtained. Portions of
tibial articular cartilages were determined to be the datum surfaces for thickness construction (Fig.
3-13a). Referring to the realistic anatomical features in the pig knee joint, the outlines of menisci
inferior surfaces were modified based on the datum surfaces, and then lofted into smoothed faces
in Rhinoceros (Fig. 3-13b). According to the reconstructed inferior surfaces, the measured
thicknesses were precisely indicated by vectors (red lines in Fig. 3-14). The other end of the vectors
roughly located the superior surfaces of menisci, which were also constrained by the femoral
cartilages in space (Fig. 3-14). In compliance with these rules, the profiles of superior menisci
were represented by splines. Superior surfaces of menisci were then generated by lofting these
splines (Fig. 3-14). Similar to cartilage side surface reconstruction, structured menisci side surfaces
were constructed in SolidWorks with Boundary Surface tool and combined to make a solid with
Kit Surface tool (Fig. 3-15).
35
a) b)
Figure 3-13 Inferior surfaces construction of menisci: a) Rough sketch; b) Final profile (the
menisci faces are in gray, the tibial cartilages are in green and the tibia is in pink. Black lines
are outlines and loft lines)
36
Figure 3-14 Superior surfaces construction of menisci (the menisci faces are in gray, the
femoral cartilages are in yellow and the tibial cartilages are in green. Red lines represent the
measured thicknesses of menisci. Black lines are outlines and loft lines)
a) b)
Figure 3-15 Reconstructed lateral and medial menisci from different viewing angles
37
3.2.4 Ligament reconstruction
After getting the bony models and reconstructing soft tissues, ligaments were created as
structured hexahedrons in Rhinoceros (Fig. 3-16). The insertion sites of all the ligaments (ACL,
PCL, MCL and LCL) were manually located on the 3D model in accordance with the statistical
results of their attachment distributions from published experimental data (Fuss, 1991; Kato et al.,
2010; Proffen et al., 2012), which were specified in details in Chapter 2. Hexahedral ligaments
were then imported to SolidWorks for a solid generation.
38
a) b)
c) d)
Figure 3-16 Reconstructed four main ligaments: a) LCL; b) MCL; c) ACL; d) PCL. Each
ligament has six surfaces
3.3 Finite element model of porcine knee joint
Finally, all the components were assembled in Rhinoceros with the correct orientation.
Thus, a complete 3D solid model of a porcine knee joint was obtained and ready to be meshing for
FEA (Fig. 3-17). Angle tool in Rhino showed the angle between the femur and tibial was 39.54°.
Contact state in the reconstructed geometry is illustrated in Fig. 3-18.
39
Figure 3-17 A geometrical model of porcine knee joint with femur, tibia, femoral and tibial
cartilages, menisci, ACL, PCL, MCL and LCL (pink lines are the center axes of femur and
tibia in right view)
40
a) b)
c) d)
Figure 3-18 Detailed drawings of contact pairs in the pig knee joint: a) Femur-femoral
cartilage, femoral cartilage-meniscus, meniscus-tibial cartilage and tibial cartilage-tibia
pairs in the lateral; b) Femur-femoral cartilage, femoral cartilage-meniscus, meniscus-tibial
cartilage and tibial cartilage-tibia pairs in the medial; c) Lateral meniscus-femur and medial
meniscus-tibia pairs; d) Meniscus-tibia pairs in the lateral and medial
41
CHAPTER FOUR: Material modelling and geometry meshing
In this study, femoral cartilages, tibial cartilages and menisci were modeled as fibril-
reinforced fluid-saturated composites. The non-fibrillar matrix represented the solid phase
excluding collagen fibers, which was described as linearly elastic by Young’s modulus 𝐸𝐸𝑚𝑚 and
Poisson’s ratio 𝑣𝑣𝑚𝑚. The fibrillar matrix referring to the collagen network was considered to be
nonlinearly viscoelastic. However, the zonal difference of cartilage was ignored in FE modelling
for simplification, which means the fibers at a certain region were oriented in a specific direction
across the tissue thickness. Material properties and fiber orientation were incorporated in a user-
defined material subroutine for numerical implementation. Finally, the meshing techniques for
each joint tissue were presented.
4.1 Split-line pattern of articular cartilage and meniscus
The alignment of collagen fibers in femoral cartilages for the human knee joint has been
approximated from the experimental results by Below et al. (2002). A dissecting needle dipped in
India ink was inserted into the superficial layer of femoral cartilages. The resulting direction of
ink diffusion identified the preferential orientation of the collagen fibers at each needle insertion
point (Below et al., 2002). It has been found that the collagen content and network architecture in
articular cartilages of pig stifle joint were analogous to humans (Rieppo et al., 2009). Due to the
absence of relevant literature for the fiber orientation in the porcine knee joint, the split-line pattern
from human femoral cartilages was utilized in this study (Fig. 4-1).
42
Figure 4-1 Fiber split lines of femoral cartilages for porcine stifle joint assigned in
ABAQUS (Gu, 2010)
Aspden et al. (1985) made an observation to explore the fiber orientation in pig meniscus.
Fig. 4-2a indicates the split-line on the inferior surfaces of pig menisci in a shallow manner. At the
surface, the collagen fibrils were aligned in the radial direction (Aspden et al., 1985; Fithian et al.,
1990). If the split lines were sufficiently deep, the reorientation could be seen from their surface
direction to a different one in the bulk tissue, as shown in Fig. 4-2b. The split-line pattern was
more diffuse and the fibers tended to be oriented circumferentially in the bulk tissue (Aspden et
al., 1985; Fithian et al., 1990). The latter pattern was adopted for entire menisci in this study.
The split-pattern of tibial cartilage was not considered in the current FE modelling. Fiber
orientations in the medial and lateral tibial cartilages were both arranged to point to the medial
direction.
43
a) b)
Figure 4-2 Fiber split lines in pig menisci from: a) the surface layer; b) the bulk tissue in the
middle zone. Sectional view. Redrew using Aspden et al. (1985) as a reference
In ABAQUS CAE, the command ORIENTATION defines the material orientation (x was
the fiber orientation) for soft tissues by specifying the local coordinate system oxyz. Under the
global Cartesian coordinate system OXYZ, the local frame of reference can be oriented in any
direction, thus creating anisotropic properties for cartilages (Li et al., 2009).
4.2 UMAT formulation in the numerical model
The fibril-reinforced model can be implemented numerically in a user-defined material
subroutine with ABAQUS modelling. The Jacobian matrix of the material 𝐷𝐷 = 𝜕𝜕∆𝜎𝜎/𝜕𝜕∆𝜀𝜀 was
introduced in the subroutine calculations, which describes the material behavior at each time
increment. However, the total stress 𝜎𝜎 in the fibril-reinforced solid included the normal stress in
the fibrillar matrix 𝜎𝜎𝑓𝑓𝑓𝑓𝑓𝑓 and the stress in the linear elastic non-fibrillar matrix 𝜎𝜎𝑛𝑛𝑛𝑛𝑛𝑛𝑓𝑓𝑓𝑓𝑓𝑓. The tensile
stress in the fibirillar matrix could be determined by quasi-linear viscoelasticity integral (Fung,
2013; Suh and Bai, 1998; Weiss and Puso, 1998). Li and Herzog (2004b) further deduced a
numerical form of the fibrillar stress for the sake of iterative steps. The stress in the non-fibrillar
solid matrix could be easily approximated from the Hooke’s law for isotropic behavior. Only small
44
deformation theory was considered in this research. The Young’s modulus 𝐸𝐸𝑥𝑥𝑓𝑓 was simplified as a
linear function of the tensile strain 𝜀𝜀𝑥𝑥, for example, at the instantaneous loading, the effective
fibrillar modulus was
𝐸𝐸𝑥𝑥𝑓𝑓(𝜀𝜀𝑥𝑥) = 𝐸𝐸𝑥𝑥0 + 𝐸𝐸𝑥𝑥𝜀𝜀𝜀𝜀𝑥𝑥, (4-1)
where 𝐸𝐸𝑥𝑥0 and 𝐸𝐸𝑥𝑥𝜀𝜀 were the constants independent of strain but dependent on the direction.
On the other hand, the time-dependent response arising from the fluid-flow should also be
combined in the numerical procedure with anisotropic permeabilities specified in the local
Cartesian coordinate system oxyz at an element level. The permeability perpendicular to the fiber
orientation was considered to be smaller compared to the permeability parallel to the fiber direction
(Federico and Herzog, 2008).
4.3 Implementation of material properties with ABAQUS
As mentioned in Chapter 3, the instantaneous modulus 𝐸𝐸 of the non-fibrillar matrix in
articular cartilages could be determined from mechanical tests, when the indentation data were
curve fit to the elastic model of Hayes et al (1972) with the measured thickness. Fig. 4-3 illustrates
the distribution of the grid positions on articular cartilages and the divided regions with the same
modulus. Table 4-1 shows the average instantaneous modulus 𝐸𝐸 at each region in femoral
cartilages (①~⑧) and tibial cartilages (⑨~⑬). Sim et al. (2014) observed a strong correlation
between instantaneous modulus 𝐸𝐸 measured in indentation test and equilibrium modulus 𝐸𝐸𝑚𝑚 in
confined compression. The ratio between 𝐸𝐸 and 𝐸𝐸𝑚𝑚 is 4.8 with a coefficient of determination value
of 0.4480, which means that about 45% of the total variation in y can be explained by the linear
relationship between x and y. Table 4-2 demonstrates the corresponding equilibrium modulus 𝐸𝐸𝑚𝑚
at different regions. The tensile properties of the fiber network in articular cartilages were
45
approximated from the methodology in the previous models created by Li et al. (2003) and Woo
et al. (1976). With respect to the menisci, Young’s modulus 𝐸𝐸𝑚𝑚 and Poisson’s ratio 𝑣𝑣𝑚𝑚 of the non-
fibrillar matrix were taken as 0.5 MPa and 0.36, respectively. The modulus of the fibrillar matrix
in menisci were set as 28 and 5 MPa in the circumferential and radial directions respectively
(Shirazi et al., 2008). The hydraulic permeability of soft tissues was orthotropic in different
directions (Li et al., 2009 and Proctor et al., 1989). Table 4-3 lists all the material properties of
articular cartilages and menisci in the ABAQUS modelling. The coefficient of friction 0.087 on
the articular surfaces was taken from a reference of the average value of a pendulum technique test
in the intact guinea pig tibiofemoral joint (Teeple et al. 2007), which was slightly greater than the
ones found in the previous studies for cadaver finger, knee, and ankle joints (0.005~0.024)
(Charnley & Dintenfass, 1959). The higher coefficients of friction could be explained by the
restraint from capsular and ligamentous attachments (Teeple et al. 2007). Femur and tibia were
considered as rigid bodies during the analysis.
46
a) b)
Figure 4-3 Divided regions with the same average instantaneous modulus in the simulation:
a) Femoral cartilages; b) Tibial cartilages
Table 4-1 Average instantaneous modulus 𝑬𝑬 at different regions
Region ① ② ③ ④ ⑤ ⑥ ⑦ ⑧
𝑬𝑬/MPa 1.714 2.123 2.623 2.353 2.212 2.414 2.359 2.159
Region ⑨ ⑩ ⑪ ⑫ ⑬
𝑬𝑬/MPa 2.879 3.994 0.002 2.166 3.480
47
Table 4-2 Corresponding equilibrium modulus 𝑬𝑬𝑬𝑬 at different regions
Region ① ② ③ ④ ⑤ ⑥ ⑦ ⑧
𝑬𝑬𝑬𝑬/MPa 0.357 0.442 0.546 0.490 0.461 0.503 0.491 0.450
Region ⑨ ⑩ ⑪ ⑫ ⑬
𝑬𝑬𝑬𝑬/MPa 0.600 0.832 0.0004 0.451 0.725
Table 4-3 Material properties implemented in the USER MATERIAL
Tissue
Fibrillar Moduli
/MPa
Non-fibrillar
Porous Matrix
Permeability
/(10-3mm4/Ns)
𝐸𝐸𝑥𝑥𝑓𝑓 𝐸𝐸𝑦𝑦
𝑓𝑓 𝐸𝐸𝑧𝑧𝑓𝑓 𝐸𝐸𝑚𝑚/MPa 𝑣𝑣𝑚𝑚 x y z
Femoral
Cartilage
3+2400𝜀𝜀𝑥𝑥
0.9+1200𝜀𝜀𝑦𝑦
0.9+1200𝜀𝜀𝑦𝑦
0.357~0.600 0.36
2
1
1 (region-dependent)
Tibial
Cartilage
2+1600𝜀𝜀𝑥𝑥
2+1600𝜀𝜀𝑦𝑦
2+1600𝜀𝜀𝑧𝑧
0.0004~0.832 0.36
2
1
1 (region-dependent)
Meniscus 28 5 5 0.5 0.36 2 1 1
Bone Rigid
Coefficient of friction between all sliding surfaces: 0.087
4.4 Geometry remeshing
After importing to ABAQUS CAE, smoothed surfaces of all the joint tissues were meshed
part by part. In this study, soft tissues were discretized with pure hexahedral elements on account
48
of its relatively fast convergence, comparing with tetrahedral elements when contact interactions
and fluid pressurization were both considered (ABAQUS Documentation 2014).
To get the pure hexahedral meshes of soft tissues in ABAQUS, the solid was partitioned
into smaller structured and meshable pieces (Fig. 4-4) by the datum planes (Fig. 4-5). In general,
the shape of the small piece should be close to a cube. To ensure the entire mesh was uniform,
transitions between partitioned cubes were mild at edges (Fig. 4-5). The distortion of elements was
eased somehow because the meshes fit nicely with the actual geometry.
Structured, sweep and bottom-up meshing techniques were performed on the partitioned
cubes according to the local geometry.
a) b)
Figure 4-4 A typical structured piece in: a) shaded view; b) wireframe
49
Figure 4-5 Lateral tibial cartilage partitioned by the datum planes
4.4.1 Femoral cartilage meshing
The femoral cartilage was meshed using 8-node brick elements with linear displacement
and constant pore pressure (C3D8P) (Fig. 4-6). Since the cartilage was too thin, only 3 layers of
elements were generated as illustrated. The total number of elements was 2295 with average aspect
ratio 1.76. None of the elements were distorted which means that they had an angle on quadrilateral
faces no greater than 135° or no less than 45°.
50
Figure 4-6 FE meshing on femoral cartilages
4.4.2 Tibial cartilage meshing
Similar to femoral cartilages, tibial cartilages were meshed with C3D8P elements and also
have 3 layers of the thickness (Fig. 4-7). In total, the number of elements was 2757 with average
aspect ratio 2.17 and no distortion.
Figure 4-7 FE elements of tibial cartilages
4.4.3 Meniscus meshing
Since the menisci are wedge-shaped, the meshes were yielded along the peripheral
direction using C3D8P (Fig. 4-8). The number of the layers across the thickness was 4. In total,
the number of elements was 1808 with average aspect ratio 2.03. Only 2 elements were distorted
(~0.3%).
51
Figure 4-8 FE meshing of menisci
4.4.4 Bones meshing
All the bony components were modeled as discrete rigid bodies in ABAQUS because they
are three orders of magnitude stiffer than the soft tissues. In this study, a 3-node 3D rigid triangular
facet element (R3D3) was selected for bone discretization (Fig. 4-9) and the numbers of elements
were 4248 and 5394 for femur and tibia, respectively.
a) b)
Figure 4-9 FE meshing of bones: a) Femur; b) Tibia
52
4.4.5 Ligaments meshing
Four ligaments (ACL, PCL, MCL and LCL) were considered as fibril-reinforced solids in
the FE simulations, because they were in tension with little fluid pressure. Each one was meshed
with 8-node linear brick elements (C3D8) and in total, 735 elements were used without distortion.
The meshing of LCL is illustrated in Fig. 4-10 as an example.
Figure 4-10 FE meshes of LCL
53
CHAPTER FIVE: Results
This chapter introduces the loading conditions such as boundary conditions, constraints,
and contact interactions. Finally, principal stress direction, reaction force, contact status and fluid
pressure were analyzed.
5.1 Load, boundary conditions and constraints
5.1.1 Loading and boundary conditions
A 0.2 mm of displacement was applied to the femur along the vertical axis with a rate of
10 µm/s, 50 µm/s, 100 µm/s, 1000 µm/s and 2000 µm/s, respectively. Tibia was completely fixed
in all six DOF during analysis.
5.1.2 Interactions and constraints
To address the contact problem, the implicit FE technique in ABAQUS/Standard was
utilized. The nonlinear node-to-surface contact discretization was applied on the contact surfaces.
The hard contact option was chosen for pressure-overclosure behavior and the linear penalty
method was used for the contact constraint enforcement. The friction coefficient was 0.087, an
average value in the normal range of friction of pig knees (Teeple et al. 2007). Here, six contact
pairs were considered with three on the medial compartment and the other three on the lateral
compartment: femoral cartilages- menisci, tibial cartilages-menisci and femoral cartilages-tibial
cartilages. For each contact pair, meniscus interface was always defined as the master surface and
cartilage interface as the slave on account of their different stiffness. With the node-to-surface
discretization method, the contact conditions are established such that each slave node on one side
of a contact interface effectively interacts with a point of projection on the master surface on the
opposite side of the contact interface. Thus, each contact condition involves a single slave node
and a group of nearby master nodes from which values are interpolated to the projection point. The
54
slave nodes are constrained not to penetrate into the master surface; however, the nodes of the
master surface can, in principle, penetrate into the slave surface (ABAQUS Documentation 2014).
Table 5-1 Summary of five loading protocols with boundary and constraint conditions in
ABAQUS modelling
Compression rate 10 μm/s 50 μm/s 100 μm/s 1000 μm/s 2000 μm/s
Ramp displacement 𝑈𝑈3 = −0.2 mm (total knee compression)
Loading time 20 s 4 s 2 s 0.2 s 0.1 s
Relaxation time 1800 s
Boundary condition
1) Fixed the tibia 2) Fixed the horns of the menisci to tibia 3) Uniformly applied the displacement on the femur within
the total time
Interactions
1) Medial femoral cartilage and medial meniscus 2) Lateral femoral cartilage and lateral meniscus 3) Medial tibial cartilage and medial meniscus 4) Lateral tibial cartilage and lateral meniscus 5) Medial femoral cartilage and medial tibial cartilage 6) Lateral femoral cartilage and lateral tibial cartilage
Constraints 1) Articular cartilages tied on the bones 2) Attachments of the ligaments inserted to the bones
5.2 Numerical results
The numerical computation for contact analysis in the joint is extremely time-consuming.
The Lattice cluster (Compute Canada, WestGrid) was used to perform the large jobs. 19 tokens
were taken for a typical run on the platform of 24 CPUs and the wallclock time was about 1600 s
(~26 min). There were two effective measures to reduce the computational cost. First, nice meshes
55
were achieved for contact pairs in order to avoid uneven penetration of master surface into slave
surface (Fig. 5-1), which would increase the difficulty of calculating. However, the master surface
shouldn’t be too coarse in case the program produced wrong results.
a)
b)
Figure 5-1 Schematic diagram of node penetration from the master surface into the slave
surface resulting from rough meshes (indicated by the arrow). Redrew using ABAQUS
Documentation 2014 as a reference
Second, the incremental time for each step was allocated efficiently. For example, at the
beginning, while the stable contact was establishing, the increment of time was confined within a
small limit. After a quarter of the running time, the time increment was gradually raised so that the
total iteration was significantly cut down.
56
To assure contact convergence, the most challenging work was to eliminate the gap or
overclosure between contact areas. Here, two main arrangements were made: Interference Fit and
Slave Node/Surface Adjustment. As indicated in Fig. 5-2a, ABAQUS would try to reduce the
overclosure h between master and slave surfaces at the first analysis step if the Interference Fit
option was checked. In Slave Node/Surface Adjustment option, the tolerance a could be specified
so that the nodes on slave surface would be adjusted (Fig. 5-2b).
a)
b)
Figure 5-2 Operations to tight the contact surfaces: a) Interference Fit; b) Slave
Node/Surface Adjustment. Redrew using ABAQUS Documentation 2014 as a reference
Incorporated with the techniques mentioned above, the stress relaxation of the porcine knee
joint was simulated with small deformation modelling. Displacement, fluid pressure and stress
distribution were obtained. The following results are illustrated taking 100 μm/s as an example.
5.2.1 Contact status
Fig. 5-3 and Fig. 5-4 demonstrate the contact regions and statuses among contact pairs
before and after relaxation. ABAQUS assigns one of the three contact statuses to individual nodes
on the master and slave surfaces: Closed (Sticking) if any of the overlapping nodes are sticking;
57
Closed (Slipping) if none of the overlapping nodes are sticking and one or more of the overlapping
nodes are slipping; Open if all of the overlapping nodes are open (ABAQUS Documentation 2014).
Therefore, green and red parts are the contact regions. Almost all of the interactive nodes present
to be slipping in compression at 2 s. However, most of the overlapping nodes are sticking at 1800
s after relaxation and just a few are still in slipping, which means the system has approached its
equilibrium.
a) b)
c) d)
Figure 5-3 Contact status at 2 s (right before relaxation) on: a) the bottom surfaces of femoral
cartilages; b) the top surfaces of tibial cartilages; c) the top surfaces of menisci; d) the bottom
surfaces of menisci. Interior view, lateral on the right (CSTATUS = contact status)
58
a) b)
c) d)
Figure 5-4 Contact status at 1800 s (right after relaxation) on the: a) the bottom surfaces of
femoral cartilages; b) the top surfaces of tibial cartilages; c) the top surfaces of menisci; d)
the bottom surfaces of menisci. Interior view, lateral on the right (CSTATUS = contact
status)
5.2.2 Fiber orientation and principal directions
Material orientation was plotted on the deformed model, as shown in Fig. 5-5. It can be
observed that the maximum principal stress is tensile stress with its direction in accord with the
fiber orientation (Fig. 5-6). For femoral cartilages, it obeys the split-line pattern specified in
Chapter 4; for menisci, the maximum principal stress is aligned circumferentially. No matter in
femoral cartilages or menisci, the maximum principal stress was larger on the medial side.
59
a)
b)
Figure 5-5 Material orientation plotted on the deformed shape of the: a) femoral cartilages;
b) menisci. Interior view, lateral on the right (1-axis represents the fiber orientation)
60
a)
b)
Figure 5-6 Maximum principal stress tensor plotted at 2 s in the: a) femoral cartilages; b)
menisci. Interior view, lateral on the right (Max. Principal = Maximum principal stress)
5.2.3 Reaction force
Fig. 5-7 shows the amplitude of reaction force on the tibial plateau at a typical rate of 100
μm/s. The reaction force reaches a maximum (25.9332 N) at 2 s and then goes down to equilibrium
(~900 s) during stress relaxation process.
61
Figure 5-7 Reaction force of the tibia in the relaxation simulation
5.2.4 Fluid pressure
Fig. 5-8 plots the maximum fluid pressure for articular cartilages and menisci with respect
to time in the stress relaxation simulation. When the ramp displacement is gradually loading within
2 s, the fluid starts to get trapped in the tissues and causes pore pressure. After relaxation, the fluid
pressure is decreasing because the water-flow has ceased. In general, soft tissues need to take at
least 900 s (~15 mins) to come to the equilibrium and reach an asymptote.
-5
0
5
10
15
20
25
30
-100 100 300 500 700 900
Rea
ctio
n fo
rce/
N
Time/s
100 μm/s
62
Figure 5-8 Maximum pore pressure in soft tissues as a function of time
Fig. 5-9 and Fig. 5-10 show the distribution of the fluid pressure in the articular cartilages,
which are represented by the output POR in ABAQUS. The maximum pore pressure on the contact
surfaces is 0.341 MPa for the femoral cartilages and 0.314 MPa for the tibial cartilages. In menisci,
maximum fluid pressures on the upper and lower surfaces are 0.266 and 0.208 MPa, respectively.
However, maximum pore pressure in the entire femoral cartilages reaches 0.602 MPa (0.314 MPa
of the whole tibial cartilage pieces). Similarly, pore pressure is much severer at the medial side in
the whole knee, which coincides with the changes of contact pressure in the joint. As time increases
to 1800 s, the fluid pressure decreases and gets more uniformly redistributed, whereas the soft
tissues are still pressurized by the trapped fluid and the medial components are under more pressure
than the lateral components (Fig. 5-10).
-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
-100 100 300 500 700 900
Max
imum
por
e pr
essu
re
Time/s
Femoral cartilage
Tibial cartilage
Menisci
63
a) b)
c) d)
Figure 5-9 Pore pressure distribution at 2 s on: a) the bottom surfaces of femoral cartilages;
b) the top surfaces of tibial cartilages; c) the top surfaces of menisci; d) the bottom surfaces
of menisci. Interior view, lateral on the right (POR = pore pressure)
64
a) b)
c) d)
Figure 5-10 Pore pressure distribution at 1800 s on: a) the bottom surfaces of femoral
cartilages; b) the top surfaces of tibial cartilages; c) the top surfaces of menisci; d) the bottom
surfaces of menisci. Interior view, lateral on the right (POR = pore pressure)
5.2.5 Contact pressure
Fig. 5-11 and Fig. 5-12 show the contact pressure (normal stress) distribution in the
cartilages and menisci. For femoral cartilages, the maximum contact pressure is 0.091 MPa found
in the medial condyle. In tibial cartilages, the largest contact pressure is also on the medial side
with a value of 0.049 MPa. In menisci, consequently, the medial surfaces withstand more pressure
than the lateral ones. On the other hand, the pressure is more concentrated within the interaction
65
between the femoral cartilages and menisci. Nevertheless, the contact pressures on the tibial
cartilages and the top surfaces of menisci are more uniformly distributed.
As time increases (Fig. 5-12), the maximum contact pressures on the tibial cartilages and
menisci increase to an equilibrium level (~0.04MPa). Pressures in the femoral cartilages decrease
to the same value. Also, pressure distribution on all the contact surfaces turns to be more uniform.
It is notable that the lateral condyles and menisci are more pressurized later comparing to the
beginning.
a) b)
c) d)
Figure 5-11 Contact pressure distribution at 2 s on: a) the bottom surfaces of femoral
cartilages; b) the top surfaces of tibial cartilages; c) the top surfaces of menisci; d) the bottom
surfaces of menisci. Interior view, lateral on the right (CPRESS = contact pressure)
66
a) b)
c) d)
Figure 5-12 Contact pressure distribution at 1800 s on: a) the bottom surfaces of femoral
cartilages; b) the top surfaces of tibial cartilage; c) the top surfaces of menisci; d) the bottom
surfaces of menisci. Interior view, lateral on the right (CPRESS = contact pressure)
67
CHAPTER SIX: Discussion, conclusion and future work
The objectives of this study were to develop a new combined CT-based and automated
indentation mapping method for porcine knee construction, and determine the joint mechanics
using ABAQUS subroutines. Displacement, reaction force, contact pressure and pore pressure
were obtained from the FE knee model.
6.1 Summary
In general, the thesis can be summarized as follows:
1) The geometry of each joint component was acquired and constructed separately. Femur
and tibia were scanned by a micro CT device and were reconstructed with MATLAB. For articular
cartilages, automated indentation mapping on a micromechanical tester was used to get the
thickness and compressive Young’s modulus. The thicknesses of menisci were obtained by needle
probe testing. The four main ligaments (ACL, PCL, LCL and MCL) were constructed in a narrow
and long hexahedral shape with their attachments on the bones precisely located. All the soft
tissues were then processed and exported as IGES format in Rhinoceros 3D or SolidWorks.
2) Bones and soft tissues were discretized with different FE elements and meshing
techniques in ABAQUS. Femur and tibia were considered as rigid shell parts and meshed with the
3-node 3D rigid triangular facets. 8-node hexahedrons with linear displacement and constant pore
pressure were used to discretize articular cartilages and menisci in three or four layers across the
thickness. The ligaments were meshed with 8-node linear brick elements.
3) Relaxation tests at five different loading rates were simulated with small deformation
theory. A ramp displacement was applied on the femur, the tibia was fixed in the six DOF. Tie
constraints and contact interactions were defined in ABAQUS to mimic the bone-cartilage
interfaces and the contact pairs between soft tissues. The fibril reinforcement and fluid
68
pressurization were both considered with a UMAT subroutine developed previously by our
research group.
4) In the relaxation simulation, displacement of the femur, reaction force on the tibia,
contact pressure and pore pressure of articular cartilages and menisci were investigated. Once
verified, they may have significance in studying the contact mechanics of the human knee.
6.2 Comparison with experimental data
The FE modelling can be validated by mechanical testing under same conditions. Fresh
porcine knee joints with intact capsule were tested on MTS in an independent study. The two ends
of the joint were fixed and the angle between the femur and tibia was kept under natural conditions
during the testing (~40°). After preconditioning was performed, an 800 µm ramp compression was
applied on femur at 6 rates: 10, 50, 100, 500, 1000 and 2000 µm/s, respectively, followed by a 20
min relaxation period. The displacement and reaction force were monitored. The tests were
conducted by a member of our research group, Marcel Rodriguez. As a matter of convenience,
three rates (10, 100 and 1000 μm/s) were considered for comparison with model prediction. Also,
only the front compression section ranging from 0 to 200 μm was selected as compared with the
simulation. The reaction force versus compressive displacement curves at different rates in testing
and modelling groups are shown in Fig. 6-1. The maximum reaction forces on the tibia at the
displacement of 200 μm for all cases are shown in Table 6-1.
69
Figure 6-1 Reaction force vs. compression at varied rates from experiment and simulation
Table 6-1 Maximum reaction force at the compressive displacement of 200 μm
Cases
Maximum reaction force/N
10 μm/s 100 μm/s 1000 μm/s
Mechanical testing 24.12 25.35 31.80
FE modelling 25.03 25.93 25.91
Fig. 6-1 and Table 6-1 indicate that the simulation results are within the range of the
experimental ones, but the FE modelling shows a weaker rate-dependency when the displacement
is relatively small. To a certain degree, the simulated reaction force agrees with the test data.
0
5
10
15
20
25
30
35
0 50 100 150 200
Rea
ctio
n fo
rce/
N
Compression/μm
10 μm/s (experiment)100 μm/s (experiment)1000 μm/s (experiment)10 μm/s (simulation)100 μm/s (simulation)1000 μm/s (simulation)
70
6.3 Significance
Soft tissues are prevalently detected via MRI scanning and processed with Mimics
software, both of which are expensive. This study proposes a combined CT scan and automated
indentation mapping method to collect the tissue geometries of the porcine knee joint, which is
beneficial in saving time and cost. Joint tissues were reconstructed and smoothed with customer-
defined codes in MATLAB. The FE mesh was nicely generated in ABAQUS with manual partition
and adjustment.
The FE analysis shows the contact stresses in the porcine knee joint under compression,
which may have similarities with the mechanical behavior of human knee joint. With the validation
of the mechanical tests on the pig knee, the FE modelling can be used to better understand the joint
mechanics of the human knee.
Another group member, Baaba Sekyiwaa Otoo has made a progress on the determination
of the site-specific changes in the gene expression level for porcine stifle knee at compression. The
present study may help to explain the influence of the mechanical loading on cartilage metabolism
through the analysis of contact pressure and fluid flow.
6.4 Limitations and future directions
First of all, for geometry construction, thicknesses of menisci were roughly obtained with
a needle probe and the accuracy may not be satisfactory. In future, the automated mapping
approach (Chapter 3) or MTS can be applied to meniscus thickness measurement. Menisci sample
along with tibia and tibial cartilages can be fixed in the chamber together. As the indenter or the
needle penetrates into the tissue, the vertical displacement can be detected for thickness
reconstruction thereafter.
71
In the present research, viscoelasticity in the knee was investigated using small deformation
theory. The current model can be further improved by applying large deformation theory. On the
other hand, the loading protocol utilized here was a preliminary study for the real physiological
condition, which is more realistic.
Moreover, the effect of meniscectomy in the pig knee joint may be explored using the
current FE model. Thus, the mechanical responses of intact and meniscectomized joints can be
determined, the results of which could be further used to compare with test data obtained from our
group.
In conclusion, FE relaxation analysis on porcine knee joint demonstrates that the reaction
force is consistent with mechanical tests. As joint mechanics of human knee may have similarities
with pig knee, the present study supports the fact that FE modeling of human knees may serve as
an effective methodology for the prediction of mechanical testing results.
72
References
• ABAQUS, V. (2014). 6.14 documentation. Dassault Systemes Simulia Corporation.
• Armstrong, C. G., & Gardner, D. L. (1977). Thickness and distribution of human femoral
head articular cartilage. Changes with age. Annals of the rheumatic diseases, 36(5), 407-
412.
• Aspden, R. M., Yarker, Y. E., & Hukins, D. W. (1985). Collagen orientations in the
meniscus of the knee joint. Journal of anatomy, 140(Pt 3), 371.
• Ateshian, G. A., Soslowsky, L. J., Froimson, M. I., Lai, W. M., & Mow, V. C. (1988).
Stereophotogrammetric determinations of patellar cartilage thickness and surface
geometry. ASME Adv. Bioengng, 8, 167-170.
• Bahrani, A. S., & Gardner, D. L. (1980). Changes in the deformational behavior of human
hip cartilage with age.
• Baka, N., Kaptein, B. L., de Bruijne, M., van Walsum, T., Giphart, J. E., Niessen, W. J., &
Lelieveldt, B. P. (2011). 2D–3D shape reconstruction of the distal femur from stereo X-ray
imaging using statistical shape models. Medical image analysis, 15(6), 840-850.
• Batiste, D. L., Kirkley, A., Laverty, S., Thain, L. M., Spouge, A. R., Gati, J. S., ... &
Holdsworth, D. W. (2004). High-resolution MRI and micro-CT in an ex vivo rabbit anterior
cruciate ligament transection model of osteoarthritis. Osteoarthritis and cartilage, 12(8),
614-626.
• Below, S., Arnoczky, S. P., Dodds, J., Kooima, C., & Walter, N. (2002). The split-line
pattern of the distal femur: a consideration in the orientation of autologous cartilage grafts.
Arthroscopy: The Journal of Arthroscopic & Related Surgery, 18(6), 613-617.
73
• Belsole, R. J., Hilbelink, D. R., Llewellyn, J. A., Stenzler, S., Greene, T. L., & Dale, M.
(1988). Mathematical analysis of computed carpal models. Journal of orthopaedic
research, 6(1), 116-122.
• Bendjaballah, M. Z., Shirazi-Adl, A., & Zukor, D. J. (1995). Biomechanics of the human
knee joint in compression: reconstruction, mesh generation and finite element analysis. The
knee, 2(2), 69-79.
• Blankevoort, L., & Huiskes, R. (1991). Ligament-bone interaction in a three-dimensional
model of the knee. Journal of biomechanical engineering, 113(3), 263-269.
• Bowers, M. E., Trinh, N., Tung, G. A., Crisco, J. J., Kimia, B. B., & Fleming, B. C. (2008).
Quantitative MR imaging using “LiveWire” to measure tibiofemoral articular cartilage
thickness. Osteoarthritis and cartilage, 16(10), 1167-1173.
• Bowman, K. F., & Sekiya, J. K. (2009). Anatomy and biomechanics of the posterior
cruciate ligament and other ligaments of the knee. Operative Techniques in Sports
Medicine, 17(3), 126-134.
• Boyd, S. K., Müller, R., & Zernicke, R. F. (2002). Mechanical and architectural bone
adaptation in early stage experimental osteoarthritis. Journal of bone and mineral research,
17(4), 687-694.
• Boyd, S. K., Müller, R., Matyas, J. R., Wohl, G. R., & Zernicke, R. F. (2000). Early
morphometric and anisotropic change in periarticular cancellous bone in a model of
experimental knee osteoarthritis quantified using microcomputed tomography. Clinical
Biomechanics, 15(8), 624-631.
74
• Braunstein, E. M., Brandt, K. D., & Albrecht, M. (1990). MRI demonstration of
hypertrophic articular cartilage repair in osteoarthritis. Skeletal radiology, 19(5), 335-339.
• Brown, T. D., & Singerman, R. J. (1986). Experimental determination of the linear biphasic
constitutive coefficients of human fetal proximal femoral chondroepiphysis. Journal of
biomechanics, 19(8), 597-605.
• Buschmann, M. D., Soulhat, J., Shirazi-Adl, A., Jurvelin, J. S., & Hunziker, E. B. (1997).
Confined compression of articular cartilage: linearity in ramp and sinusoidal tests and the
importance of interdigitation and incomplete confinement. Journal of biomechanics, 31(2),
171-178.
• Calvo, E., Palacios, I., Delgado, E., Ruiz-Cabello, J., Hernandez, P., Sanchez-Pernaute, O.,
... & Herrero-Beaumont, G. (2001). High-resolution MRI detects cartilage swelling at the
early stages of experimental osteoarthritis. Osteoarthritis and Cartilage, 9(5), 463-472.
• Chan, W. P., Lang, P., Stevens, M. P., Sack, K., Majumdar, S., Stoller, D. W., ... & Genant,
H. K. (1991). Osteoarthritis of the knee: comparison of radiography, CT, and MR imaging
to assess extent and severity. AJR. American journal of roentgenology, 157(4), 799-806.
• Charnley, J., & Dintenfass, L. (1959). The lubrication of animal joints, Symp. on
Biomechanics Inst. Mech. Engrs., London, 12.
• Choi, J. A., & Gold, G. E. (2011). MR imaging of articular cartilage physiology. Magnetic
resonance imaging clinics of North America, 19(2), 249-282.
• Cignoni, P., Callieri, M., Corsini, M., Dellepiane, M., Ganovelli, F., & Ranzuglia, G.
(2008, July). Meshlab: an open-source mesh processing tool. In Eurographics Italian
Chapter Conference (Vol. 2008, pp. 129-136).
75
• Coletti, J. M., Akeson, W. H., & Woo, S. L. (1972). A comparison of the physical behavior
of normal articular cartilage and the arthroplasty surface. J Bone Joint Surg Am, 54(1),
147-160.
• der Voet, V., & Frank, A. (1992). Finite element modelling of load transfer through
articular cartilage.
• der Voet, V., & Frank, A., Shrive, N. G., & Schachar, N. S. (1993). Numerical modelling
of articular cartilage in synovial joints—poroelasticity and boundary conditions. Computer
Methods in Biomechanics and Biomedical Engineering. IBJ Publishers, Swansea, 200-209.
• Disler, D. G., Recht, M. P., & McCauley, T. R. (2000). MR imaging of articular cartilage.
Skeletal radiology, 29(7), 367-377.
• Dunne, F., & Petrinic, N. (2005). Introduction to Computational Plasticity.
• Federico, S., & Herzog, W. (2008). On the anisotropy and inhomogeneity of permeability
in articular cartilage. Biomechanics and modeling in mechanobiology, 7(5), 367-378.
• Feldkamp, L. A., Goldstein, S. A., Parfitt, M. A., Jesion, G., & Kleerekoper, M. (1989).
The direct examination of three-dimensional bone architecture in vitro by computed
tomography. Journal of bone and mineral research, 4(1), 3-11.
• Ferrant, M., Nabavi, A., Macq, B., Jolesz, F. A., Kikinis, R., & Warfield, S. K. (2001).
Registration of 3-D intraoperative MR images of the brain using a finite-element
biomechanical model. IEEE transactions on medical imaging, 20(12), 1384-1397.
• Fithian, D. C., Kelly, M. A., & Mow, V. C. (1990). Material properties and structure-
function relationships in the menisci. Clinical orthopaedics and related research, 252, 19-
31.
76
• Fung, Y. C. (2013). Biomechanics: mechanical properties of living tissues. Springer
Science & Business Media.
• Fuss, F. K. (1989). Anatomy of the cruciate ligaments and their function in extension and
flexion of the human knee joint. American Journal of Anatomy, 184(2), 165-176.
• Fuss, F. K. (1991). Anatomy and function of the cruciate ligaments of the domestic pig
(Sus scrofa domestica): a comparison with human cruciates. Journal of anatomy, 178, 11.
• Garg, A., & Walker, P. S. (1990). Prediction of total knee motion using a three-dimensional
computer-graphics model. Journal of Biomechanics, 23(1), 45-58.
• Ghosh, S. K. (1983). A close-range photogrammetric system for 3-D measurements and
perspective diagramming in biomechanics. Journal of biomechanics, 16(8), 667-674.
• Goldsmith, A. A. J., Hayes, A., & Clift, S. E. (1996). Application of finite elements to the
stress analysis of articular cartilage. Medical engineering & physics, 18(2), 89-98.
• Gougoutas, A. J., Wheaton, A. J., Borthakur, A., Shapiro, E. M., Kneeland, J. B., Udupa,
J. K., & Reddy, R. (2004). Cartilage volume quantification via live wire segmentation.
Academic radiology, 11(12), 1389-1395.
• Gregory, M. H., Capito, N., Kuroki, K., Stoker, A. M., Cook, J. L., & Sherman, S. L.
(2012). A review of translational animal models for knee osteoarthritis. Arthritis, 2012.
• Gu, K. (2010). An Anatomically-accurate Finite Element Knee Model Accounting for
Fluid Pressure in Articular Cartilage.
• Hall, F. M., & Wyshak, G. (1980). Thickness of articular cartilage in the normal knee. The
Journal of Bone & Joint Surgery, 62(3), 408-413.
77
• Hayes, W. C., & Mockros, L. F. (1971). Viscoelastic properties of human articular
cartilage. Journal of applied physiology, 31(4), 562-568.
• Hayes, W. C., Keer, L. M., Herrmann, G., & Mockros, L. F. (1972). A mathematical
analysis for indentation tests of articular cartilage. Journal of biomechanics, 5(5), 541-551.
• Hembry, R. M., Dyce, J., Driesang, I., Hunziker, E. B., Fosang, A. J., Tyler, J. A., &
Murphy, G. (2001). Immunolocalization of matrix metalloproteinases in partial-thickness
defects in pig articular cartilage: a preliminary report. JBJS, 83(6), 826-838.
• Hipp, J. A., Goel, P., Newman, P. S., & Chan, E. F. Accuracy and Reproducibility of
Radiographic Knee Joint Space Narrowing Measurements Referenced to the Mid-Coronal
Plane.
• Holmes, M. H., & Mow, V. C. (1990). The nonlinear characteristics of soft gels and
hydrated connective tissues in ultrafiltration. Journal of biomechanics, 23(11), 1145-1156.
• Huiskes, R., Kremers, J., De Lange, A., Woltring, H. J., Selvik, G., & Van Rens, T. J.
(1985). Analytical stereophotogrammetric determination of three-dimensional knee-joint
geometry. Journal of biomechanics, 18(8), 559-570.
• Iriuchishima, T., Tajima, G., Ingham, S. J., Shen, W., Horaguchi, T., Saito, A., Smolinski,
P. & Fu, F. H. (2009). Intercondylar roof impingement pressure after anterior cruciate
ligament reconstruction in a porcine model. Knee Surgery, Sports Traumatology,
Arthroscopy, 17(6), 590-594.
• Jin, Z. (Ed.). (2014). Computational modelling of biomechanics and biotribology in the
musculoskeletal system: biomaterials and tissues. Elsevier.
78
• Kapadia, R. D., Stroup, G. B., Badger, A. M., Koller, B., Levin, J. M., Coatney, R. W., ...
& Gowen, M. (1998). Applications of micro-CT and MR microscopy to study pre‐clinical
models of osteoporosis and osteoarthritis. Technology and Health Care, 6(5, 6), 361-372.
• Karol Miller (1998). Modelling soft tissue using biphasic theory—a word of caution.
COMPUTER METHODS IN BIOMECHANICS AND BIO MEDICAL ENGINEERING,
1(3), 261-263.
• Kato, Y., Ingham, S. J., Kramer, S., Smolinski, P., Saito, A., & Fu, F. H. (2010). Effect of
tunnel position for anatomic single-bundle ACL reconstruction on knee biomechanics in a
porcine model. Knee Surgery, Sports Traumatology, Arthroscopy, 18(1), 2-10.
• Kazemi Miraki, M. (2013). Finite Element Study of the Healthy and Meniscectomized
Knee Joints Considering Fibril-Reinforced Poromechanical Behaviour for Cartilages and
Menisci (Doctoral dissertation, University of Calgary).
• Kempson, G. E., Freeman, M. A. R., & Swanson, S. A. V. (1971). The determination of a
creep modulus for articular cartilage from indentation tests on the human femoral head.
Journal of biomechanics, 4(4), 239-250.
• Kempson, G. E., Muir, H., Pollard, C., & Tuke, M. (1973). The tensile properties of the
cartilage of human femoral condyles related to the content of collagen and
glycosaminoglycans. Biochimica et Biophysica Acta (BBA)-General Subjects, 297(2), 456-
472.
• Kwan, M. K., Lai, W. M., & Mow, V. C. (1990). A finite deformation theory for cartilage
and other soft hydrated connective tissues—I. Equilibrium results. Journal of
biomechanics, 23(2), 145-155.
79
• Lai, W. M., & Mow, V. C. (1980). Drag-induced compression of articular cartilage during
a permeation experiment. Biorheology, 17(1-2), 111.
• Laurent, D., Wasvary, J., O'Byrne, E., & Rudin, M. (2003). In vivo qualitative assessments
of articular cartilage in the rabbit knee with high-resolution MRI at 3 T. Magnetic
resonance in medicine, 50(3), 541-549.
• Lee, K. K., Tian-Xia, Q., & Teo, E. C. (2002). 3-D finite element modeling of lumbar spine
(L2/L3) using digitizer. International Journal of Information Technology, 8(2), 1-9.
• Li, G., Gil, J., Kanamori, A., & Woo, S. Y. (1999). A validated three-dimensional
computational model of a human knee joint. Journal of biomechanical engineering, 121(6),
657-662.
• Li, L. P., & Herzog, W. (2004a). Strain-rate dependence of cartilage stiffness in unconfined
compression: the role of fibril reinforcement versus tissue volume change in fluid
pressurization. Journal of biomechanics, 37(3), 375-382.
• Li, L. P., & Herzog, W. (2004b). The role of viscoelasticity of collagen fibers in articular
cartilage: theory and numerical formulation. Biorheology, 41(3-4), 181-194.
• Li, L. P., Buschmann, M. D., & Shirazi-Adl, A. (2001). The asymmetry of transient
response in compression versus release for cartilage in unconfined compression. Journal
of biomechanical engineering, 123(5), 519-522.
• Li, L. P., Buschmann, M. D., & Shirazi-Adl, A. (2002). The role of fibril reinforcement in
the mechanical behavior of cartilage. Biorheology, 39(1, 2), 89-96.
80
• Li, L. P., Buschmann, M. D., & Shirazi-Adl, A. (2003). Strain-rate dependent stiffness of
articular cartilage in unconfined compression. Journal of biomechanical engineering,
125(2), 161-168.
• Li, L. P., Cheung, J. T. M., & Herzog, W. (2009). Three-dimensional fibril-reinforced finite
element model of articular cartilage. Medical & biological engineering & computing,
47(6), 607.
• Li, L. P., Soulhat, J., Buschmann, M. D., & Shirazi-Adl, A. (1999). Nonlinear analysis of
cartilage in unconfined ramp compression using a fibril reinforced poroelastic model.
Clinical Biomechanics, 14(9), 673-682.
• Loeuille, D., Gonord, P., Guingamp, C., Gillet, P., Blum, A., Sauzade, M., & Netter, P.
(1997). In vitro magnetic resonance microimaging of experimental osteoarthritis in the rat
knee joint. The Journal of rheumatology, 24(1), 133-139.
• Mcleod, W. D., Moschi, A., Andrews, J. R., & Hughston, J. C. (1976). Tibial plateau
topography. The American journal of sports medicine, 5(1), 13-18.
• Meachim, G., Bentley, G., & Baker, R. U. T. H. (1977). Effect of age on thickness of adult
patellar articular cartilage. Annals of the rheumatic diseases, 36(6), 563-568.
• Modest, V. E., Murphy, M. C., & Mann, R. W. (1989). Optical verification of a technique
for in situ ultrasonic measurement of articular cartilage thickness. Journal of biomechanics,
22(2), 171-176.
• Moon, K. L., Genant, H. K., Davis, P. L., Chafetz, N. I., Helms, C. A., Morris, J. M., ... &
Bovill, E. G. (1983). Nuclear magnetic resonance imaging in orthopaedics: principles and
applications. Journal of orthopaedic research, 1(1), 101-114.
81
• Moskowitz, R. W. (Ed.). (2007). Osteoarthritis: diagnosis and medical/surgical
management. Lippincott Williams & Wilkins.
• Mow, V. C. (1989). Biomechanics of articular cartilage. Basic biomechanics of the
musculoskeletal system.
• Mow, V. C. (2005). Structure and function of articular cartilage and meniscus. Basic
orthopaedic biomechanics and mechano-biology, 181-258.
• Mow, V. C., Ateshian, G. A., & Spilker, R. L. (1993). Biomechanics of diarthrodial joints:
a review of twenty years of progress. Journal of biomechanical engineering, 115(4B), 460-
467.
• Mow, V. C., Kuei, S. C., Lai, W. M., & Armstrong, C. G. (1980). Biphasic creep and stress
relaxation of articular cartilage in compression: theory and experiments. J Biomech Eng,
102(1), 73-84.
• Mow, V., & Roth, V. (1981). Effects of nonlinear strain-dependent permeability and rate
of compression on the stress behavior of articular cartilage. Journal of biomechanical
engineering, 103, 61.
• Muehleman, C., Majumdar, S., Issever, A. S., Arfelli, F., Menk, R. H., Rigon, L., ... &
Kuettner, K. E. (2004). X-ray detection of structural orientation in human articular
cartilage. Osteoarthritis and cartilage, 12(2), 97-105.
• Okabe, A. (1992). Spatial tessellations. John Wiley & Sons, Ltd.
• Parsons, F. G. (1899). The joints of mammals compared with those of man: a course of
lectures delivered at the Royal College of Surgeons of England. Journal of anatomy and
physiology, 34(Pt 1), 41.
82
• Parsons, J. R., & Black, J. (1977). The viscoelastic shear behavior of normal rabbit articular
cartilage. Journal of Biomechanics, 10(1), 21-29.
• Piscaer, T. M., Waarsing, J. H., Kops, N., Pavljasevic, P., Verhaar, J. A. N., van Osch, G.
J. V. M., & Weinans, H. (2008). In vivo imaging of cartilage degeneration using μCT-
arthrography. Osteoarthritis and cartilage, 16(9), 1011-1017.
• Proctor, C. S., Schmidt, M. B., Whipple, R. R., Kelly, M. A., & Mow, V. C. (1989).
Material properties of the normal medial bovine meniscus. Journal of Orthopaedic
Research, 7(6), 771-782.
• Proffen, B. L., McElfresh, M., Fleming, B. C., & Murray, M. M. (2012). A comparative
anatomical study of the human knee and six animal species. The Knee, 19(4), 493-499.
• Ramaniraka, N. A., Saunier, P., Siegrist, O., & Pioletti, D. P. (2007). Biomechanical
evaluation of intra-articular and extra-articular procedures in anterior cruciate ligament
reconstruction: a finite element analysis. Clinical Biomechanics, 22(3), 336-343.
• Renders, G. A., Mulder, L., Lin, A. S., Langenbach, G. E., Koolstra, J. H., Guldberg, R.
E., & Everts, V. (2014). Contrast-enhanced microCT (EPIC-µCT) ex vivo applied to the
mouse and human jaw joint. Dentomaxillofacial Radiology, 43(2), 20130098.
• Renka, R. J. (1984). Algorithm 624: Triangulation and interpolation at arbitrarily
distributed points in the plane. ACM Transactions on Mathematical Software (TOMS),
10(4), 440-442.
• Rieppo, J., Hyttinen, M. M., Halmesmaki, E., Ruotsalainen, H., Vasara, A., Kiviranta, I.,
... & Helminen, H. J. (2009). Changes in spatial collagen content and collagen network
83
architecture in porcine articular cartilage during growth and maturation. Osteoarthritis and
Cartilage, 17(4), 448-455.
• Roemer, F. W., Mohr, A., Lynch, J. A., Meta, M. D., Guermazi, A., & Genant, H. K.
(2005). Micro-CT arthrography: a pilot study for the ex vivo visualization of the rat knee
joint. American Journal of Roentgenology, 184(4), 1215-1219..
• Rushfeldt, P. D., Mann, R. W., & Harris, W. H. (1981). Improved techniques for measuring
in vitro the geometry and pressure distribution in the human acetabulum—I. Ultrasonic
measurement of acetabular surfaces, sphericity and cartilage thickness. Journal of
biomechanics, 14(4), 253-260.
• Schenk, R., Eggli, P., Fleisch, H., & Rosini, S. (1986). Quantitative morphometric
evaluation of the inhibitory activity of new aminobisphosphonates on bone resorption in
the rat. Calcified Tissue International, 38(6), 342-349.
• Scherrer, P. K. (1977). Determining the distance between the three-dimensional bone
architecture in vitro by computed surfaces of two rigid bodies using parametric surface
patches and an instrumented spatial linkage, Purdue University.
• Scherrer, P. K., & Hillberry, B. M. (1979). Piecewise mathematical representation of
articular surfaces. Journal of biomechanics, 12(4), 301-311.
• Seedhom, B. B., Longton, E. B., Wright, V., & Dowson, D. (1972). Dimensions of the
knee. Radiographic and autopsy study of sizes required by a knee prosthesis. Annals of the
rheumatic diseases, 31(1), 54.
• Semptikovski, S. C. (2013). Geometrical Reconstruction of the Knee Joint.
84
• Shiba, R., Sorbie, C., Siu, D. W., Bryant, J. T., Cooke, T. D. V., & Wevers, H. W. (1988).
Geometry of the humeroulnar joint. Journal of orthopaedic research, 6(6), 897-906.
• Shirazi, R., Shirazi-Adl, A., & Hurtig, M. (2008). Role of cartilage collagen fibrils
networks in knee joint biomechanics under compression. Journal of biomechanics, 41(16),
3340-3348.
• Sim, S., Lavoie, J. F., Moreau, A., Garon, M., Quenneville, E., & Buschmann, M. (2014).
Novel technique to map the biomechanical properties of entire mice articular surfaces using
indentation. Osteoarthritis and Cartilage, 22, S310-S311.
• Soslowsky, L. J., Ateshian, G. A., Pollock, R. G., & Mow, V. C. (1989). An in situ method
to determine diarthrodial joint contact areas using stereophotogrammetry. ASME Advances
in Bioengineering, 15, 129-130.
• Soulhat, J., Buschmann, M. D., & Shirazi-Adl, A. (1999). A fibril-network-reinforced
biphasic model of cartilage in unconfined compression. Journal of biomechanical
engineering, 121(3), 340-347.
• Spandonis, Y., Heese, F. P., & Hall, L. D. (2004). High resolution MRI relaxation
measurements of water in the articular cartilage of the meniscectomized rat knee at 4.7 T.
Magnetic resonance imaging, 22(7), 943-951.
• Steines, D., Cheng, C., Wong, J., Tsai, S., Napel, P., & Lang, P. (2000, April).
Segmentation of osteoarthritic femoral cartilage using live wire. In Proc Int Soc Magn
Reson Med (Vol. 8, p. 220).
85
• Suh, J. K., & Bai, S. (1998). Finite element formulation of biphasic poroviscoelastic model
for articular cartilage. Transactions-American Society of Mechanical Engineers Journal of
biomechanical Engineering, 120, 195-201.
• Suh, J. K., & Mow, V. C. (1990). Effects of friction on the unconfined compressive
response of articular cartilage: a finite element analysis.
• Suh, J. K., Spilker, R. L., & Mow, V. C. (1990, November). Finite element analysis of the
indentation problem for articular cartilage using a finite deformation biphasic model. In
Winter Annual Meeting of the American Society of Mechanical Engineers (pp. 215-218).
• Taylor, Z. A., & Miller, K. (2006). Constitutive modeling of cartilaginous tissues: a review.
Journal of applied biomechanics, 22(3), 212-229.
• Teeple, E., Fleming, B. C., Mechrefe, A. P., Crisco, J. J., Brady, M. F., & Jay, G. D. (2007).
Frictional properties of Hartley guinea pig knees with and without proteolytic disruption
of the articular surfaces. Osteoarthritis and cartilage, 15(3), 309-315.
• Tessier, J. J., Bowyer, J., Brownrigg, N. J., Peers, I. S., Westwood, F. R., Waterton, J. C.,
& Maciewicz, R. A. (2003). Characterisation of the guinea pig model of osteoarthritis by
in vivo three-dimensional magnetic resonance imaging. Osteoarthritis and cartilage,
11(12), 845-853.
• Viceconti, M., Zannoni, C., Testi, D., & Cappello, A. (1999). CT data sets surface
extraction for biomechanical modeling of long bones. Computer methods and programs in
biomedicine, 59(3), 159-166.
86
• Wachsmuth, L., Keiffer, R., Juretschke, H. P., Raiss, R. X., Kimmig, N., & Lindhorst, E.
(2003). In vivo contrast-enhanced micro MR-imaging of experimental osteoarthritis in the
rabbit knee joint at 7.1 T. Osteoarthritis and cartilage, 11(12), 891-902.
• Wang, Z. L., Teo, J. C. M., Chui, C. K., Ong, S. H., Yan, C. H., Wang, S. C., ... & Teoh,
S. H. (2005). Computational biomechanical modelling of the lumbar spine using marching-
cubes surface smoothened finite element voxel meshing. Computer methods and programs
in biomedicine, 80(1), 25-35.
• Weiss, J. A., & Puso, M. A. (1998). Finite Element Implementation of Anisotropic Quasi-
linear Viscoelasticty using a Discrete Spectrum Approximation. Journal of Biomech.
Engng, 120, 62-70.
• Wismans, J. A. C., Veldpaus, F., Janssen, J., Huson, A., & Struben, P. (1980). A three-
dimensional mathematical model of the knee-joint. Journal of Biomechanics, 13(8), 677-
685.
• Woo, S. Y., Akeson, W. H., & Jemmott, G. F. (1976). Measurements of nonhomogeneous,
directional mechanical properties of articular cartilage in tension. Journal of biomechanics,
9(12), 785-791.
• Wu, F. T., Ng-Thow-Hing, V., Singh, K., Agur, A. M., & McKee, N. H. (2007).
Computational representation of the aponeuroses as NURBS surfaces in 3D
musculoskeletal models. computer methods and programs in biomedicine, 88(2), 112-122.
• Xerogeanes, J. W., Fox, R. J., Takeda, Y., Kim, H. S., Ishibashi, Y., Carlin, G. J., & Woo,
S. L. (1998). A functional comparison of animal anterior cruciate ligament models to the
human anterior cruciate ligament. Annals of biomedical engineering, 26(3), 345-352.
87
• Zheng, H. (2011). 3D Knee Joint Model Construction.
