Mechanical and tribological investigation on natural and artificial human joints

Università di Pisa

Scuola di Dottorato in Ingegneria “Leonardo da Vinci”

Dottorato di Ricerca in Ingegneria Meccanica

Mechanical and tribol

natural and artificial human joints

Tesi svolta per il conseguimento del titolo di Dottore di Ricerca

Settore Scientifico Disciplinare: ING

Allievo: Ing. Lorenza Mattei

Tutori:

Prof. Enrico Ciulli

Prof. Bruno Piccigallo

Ing. Francesca Di Puccio

Università di Pisa

Scuola di Dottorato in Ingegneria “Leonardo da Vinci”

Dottorato di Ricerca in Ingegneria Meccanica

Mechanical and tribological investigation on

natural and artificial human joints

Tesi svolta per il conseguimento del titolo di Dottore di Ricerca

Settore Scientifico Disciplinare: ING-IND/13

Mattei

Prof. Enrico Ciulli

Prof. Bruno Piccigallo

Ing. Francesca Di Puccio

IX Ciclo

Anno 2011

A

BSTRACT

Synovial joints of the human lower limb, e.g. the knee and the hip, are characterized by a wide mobility and can support very high loads, up to several times the body weight. Their functionality is fundamental to guaranteeing the main daily activities, such the walking, and, as a consequence, when affected by pathological conditions, can strongly limit the normal life. Unfortunately, very often these joints can lose their functionality and needs to be replaced by joint implants by means of an orthopaedic surgery called arthroplasty.

The osteoarthritis (OA), which consists in a damage of the articular cartilage (AC), is the most frequent cause of disability affecting the elderly population and is also the primary cause of joint replacements. A type of OA, called secondary OA, has generated much interest in the engineering world since it is thought to be mechanically induced, due to abnormal and excessive stresses and strains occurring in the tissue. During the stage period at the Imperial College London (London, UK), the mechanical behaviour of the AC has been investigated with the final aim of evaluating critical conditions likely causing cartilage damage [1, 2]. Since the knee meniscectomy is recognized as a possible cause of OA, the mechanical response of the AC in a patient specific model of a meniscectomised knee has been studied. Advanced material models of the AC have been implemented in a realistic 3D geometry for the first time in the literature, showing the importance to model the AC as a biphasic tissue in order to accurately predict the AC mechanical response and the onset of AC damages.

As mentioned above, in case of osteoarthritic joints, typically affected by chronic pain and reduced mobility, the arthroplasty is considered necessary. Many researchers have been investigating the optimization of the design of joints implants, in order to guarantee the same biomechanics of the natural joints, and to avoid the wear of the articulating surfaces, which is considered one of the main causes of implant failures. In the last few years there has been an increasing amount of literature on the biotribology of hip implants, whose structure, geometry and kinematics, more simple in respect with the knee replacements, make easier both theoretical and experimental studies. On the basis of an in-depth review of the lubrication models in the literature, the elasto-hydrodynamic lubrication (EHL) model of hard-on-hard hip implants has been developed to predict and compare in-vivo performances of hip implants with different geometry and materials: the results obtained can explain the actual clinical trends and provide a sensitivity analysis of lubrication to some design parameters. Also some preliminary theoretical and experimental studies (in collaboration with the Istituti Ortopedici Rizzoli, Bo, IT) on the wear of hip implants have been carried out. In the future the combination of the wear and the lubrication analyses will be addressed.

C

ONTENTS

PART I: MECHANICAL BEHAVIOUR OF THE ARTICULAR CARTILAGE IN

THE KNEE JOINT

1

CHAPTER 1 State of the art 3

1.1 Introduction 3

1.2 Articular cartilage 3

1.2.1 Role 3

1.2.2 Structure and composition 4

1.3 Mechanical properties of articular cartilage 5

1.3.1 Compressive behaviour 6 1.3.2 Swelling behaviour 6 1.3.3 Tensile behaviour 7 1.3.4 Viscoelasticity behaviour 7 1.3.5 Anisotropy behaviour 7 1.4 Ostheoarthritis 8

1.4.1 Aetiology and epidemiology 8

1.4.2 Cartilage damage caused by OA 9

1.5 Modelling of articular cartilage 10

1.5.1 Monophasic AC model 10

1.5.2 Biphasic linear AC model 11

1.5.3 Biphasic non linear AC model 11

1.5.4 Biphasic poro-viscoelastic linear AC model 12 1.5.5 Biphasic transversely isotropic linear AC model 13

1.5.6 Fibril reinforced AC model 14

1.5.7 Swelling AC model 14

1.6 Secondary osteoarthritis in the knee joint 14

1.6.1 The knee joint 14

1.6.2 FEM models of the meniscectomised knee joint 19

CHAPTER 2 Mechanical behaviour and modelling of the articular cartilage 21

2.1 Introduction 21

2.2 Simplified model of the tibio-femoral joint 21

2.2.1 Geometry 21

2.2.2 Mesh 22

2.2.3 Materials 22

2.2.4 Boundary conditions 25

2.3 Results and discussion 26

2.3.1 Mesh convergence analysis 26

2.3.2 Comparison of AC material models 29

2.4 Conclusion 35

CHAPTER 3 3D FEM model of a patient specific meniscectomised knee joint 37

3.1 Introduction 37

3.2 Patient specific model of a meniscectomised knee joint 37

3.2.1 Geometry 37

3.2.2 Mesh 38

3.2.3 Materials 38

3.2.4 Boundary conditions 40

3.3 Results and discussion 42

3.4 Conclusion and future developments 50

PART II: BIOTRIBOLOGY OF HIP IMPLANTS

53

CHAPTER 4 Lubrication and wear modelling of hip implants: a review 55

4.1 Introduction 55

4.2 Artificial hip joint 57

4.3 Lubrication models: main characteristics 60

4.3.1 Geometry 60

4.3.2 Lubricant rheology 61

4.3.3 Operating conditions 62

4.3.4 Elastic deformation 65

4.3.5 Governing equations and numerical methods 66

4.3.6 Lubrication regimes 66

4.4 Elastohydrodynamic lubrication modelling 68

4.4.1 Validity of simplifying modelling hypotheses 69

4.4.2 Effect of the geometrical parameters 69

4.4.3 Effect of the bearing construction 71

4.4.4 Effect of the operating conditions 71

4.4.5 Prediction of lubrication regimes 72

4.5 Mixed lubrication modelling 73

4.6 Wear models: main characteristics 76

4.6.1 Geometry 76

4.6.2 Operating conditions 77

4.6.3 Governing equations and numerical approach 77

4.6.4 Wear modelling 78

4.7 Conclusions 86

CHAPTER 5 EHL model of hip implants 87

5.1 Introduction 87

5.2 EHL model: main characteristics 88

5.2.1 Geometry and materials 88

5.2.2 Lubricant rheology 89

5.2.3 Operating conditions 89

5.2.4 Elastic deformation 89

5.3 Equations governing the EHL model 89

5.3.1 Equations 89

5.3.2 Dimensionless equations 92

5.3.3 Discrete equations 93

5.4 Numerical methods 95

5.4.1 Introduction to the multilevel techniques 95

5.4.2 Multilevel multigrid method 96

5.4.2.1 Coarse grid correction cycle 96

5.4.2.2 Full Multigrid 99

5.4.2.3 Relaxation 99

5.4.3 Multilevel multi-integration method 103

5.4.4 Errors and convergence checks 105

5.4.5 Numerical implementation of the EHL model 107

5.5 Results and discussion 110

5.5.1 Numerical convergence 110

5.5.2 Validity of simplifying modelling hypotheses 113 5.5.2.1 Model validation for hard-on-hard hip implants 113 5.5.2.2 Model validation for soft-on-hard hip implants 115

5.5.3 Effect of the stationary conditions 116

5.5.4 Effect of the geometry and the materials: comparison of hip

implant performances 117

5.6 Conclusion and future developments 120

CHAPTER 6 Preliminary studies on the wear of hip implants 123

6.1 Introduction 123

6.2 Analytical wear modelling of hip implants 123

6.2.2 Materials and Methods 124

6.2.2.1 Hip implants 124

6.2.2.2 Basic assumptions 124

6.2.2.3 Wear descriptors 124

6.2.2.4 Contact and pressure distribution 126

6.2.2.5 Sliding distance 126

6.2.2.6 Boundary conditions 127

6.2.3 Results and discussion 127

6.2.4 Conclusion 129

6.3 Experimental Investigations on the wear of hip implants 130

6.3.1 Introduction 130

6.3.2 Materials and methods 130

6.3.3 Results and discussion 131

6.3.4 Conclusion 133

P

REFACE

The present thesis deals with the bioengineering of the human articular joints and is focused on two different aspects and consequently organized in two main parts: t: the bioengineering of the human articular joints: the first part (Part I) focuses on natural synovial joints, i.e. the knee, whilst the second one (Part II) on artificial synovial joints, i.e. the hip implants.

Even if these subjects can appear isolated, they have the same medical background and very similar medical motivations. Part I is dedicated to the investigations of the mechanical behaviour of the articular cartilage (AC), aimed to a deeper understanding of the secondary osthearthritis initiation. This pathology, mechanically induced, is one of the main causes of the arthroplasty, both of knee and hip. As a consequence the first part of the thesis naturally introduces to the second one which concerns biotribological problems related to joint implants. Evidently the analyses carried out in the two parts involved mechanical problems of very different nature and needed to exploit different tools/software. In Part I a 3D patient specific FEM of a meniscetomised knee has been developed with particular attention to the implementation of advanced biphasic material models for the AC. Consequently this part has involved aspects of computational structural and continuum mechanics, faced by means of the FE code ABAQUS. In Part II an elasto-hydrodynamic model of hard-on-hard hip implants has been developed. Therefore, this work deals with theoretical and numerical aspects of fluid dynamics and tribology, which has required user define programming to solve systems of non linear equations ( in C++ and Mathcad).

The choice of the two research themes is partially due to the collaborations established during the PhD. Part I was indeed carried out with the Tribology group of the Imperial College London (ICL) (London, UK), during the last year of the Ph.D. The theoretical investigations on AC described in the thesis has provided supplement to the experimental studies led by the Tribology laboratory of ICL, and to the medical research conducted by the Orthopaedic Surgery section of the Charing Cross Hospital (London, UK). Part II was performed at the University of Pisa, under the supervision of the Tribology Group. Some experimental activity on the assessment of wear in hip prostheses, was carried out in collaboration with the medical technology laboratory of the Istituti Ortopedici Rizzoli (Bo, IT).

The studies presented in Part I have been making part of an ongoing research on the AC and could be focused on a specific objective right from the beginning. On the other side there was not ongoing research on the specific matter of Part II and thus an extensively review of the state of the art was necessary before developing a good research project. As a consequence each part of the thesis has been given a structure which reflects the given context and topic. To note that the order used to present the two parts does not correspond to the chronological one.

Part I is subdivided in three chapters, strictly related to each other and of which a consecutively reading is suggested. Chapter 1 is dedicated to a general introduction to the topic, providing an overview on the main medical and bioengineering issues related to the AC and its modelling. In the Chapter 2 investigations on the AC mechanical behaviour by means FEMs are described. Several biphasic advanced AC material models have been implemented in a simplified tibio-femoral joint model and compared. This chapter represents the basis for the development of the 3D patient specific knee FEM, described in Chapter 3. The model, adopting biphasic AC models firstly in the literature, allows to explore the transient response of the soft tissue during daily activities, such as walking.

Part II is composed of three chapters as well. Each chapter can be read separately because dealing with distinct topics and complete in themselves. Chapter 4 is dedicated to a literature review of lubrication and wear modelling of hip implants. This study was very important since allowed to well focus the research objectives. In particular the development of a steady state EHL model of hard on hard implants is described in Chapter 6. The combination of the simplified ball-on-plane geometry, the steady state analysis and the stable and efficient multi level methods has allowed to develop a low computational cost model which provides reliable information on hip implant in-vivo behaviour, very useful for the optimization of prosthesis design. Also some preliminary experimental and theoretical studies on the wear of hip implants have been carried out. The main findings are detailed in Chapter 7. The final goal would be to combine the lubrication and the wear analyses in order obtain more reliable information on the interaction of the articulating surfaces.

PART

I

MECHANICAL

BEHAVIOUR

OF

THE

ARTICULAR

CARTILAGE

IN

CHAPTER 1

S

TATE OF THE ART

1.1

I

NTRODUCTION

In the last decades the study of the mechanical behavior of the articular cartilage (AC) in the knee joint has been very attractive and motivating, as a significant amount of literature in this field can confirm [3]. Both theoretical [3-15] and experimental studies have been addressed to investigate the mechanical behavior of this complex soft tissue and to characterize its mechanical properties [5-7, 13, 16-18]. This research has a strong medical motivation. Indeed the AC damage, such as osteoarthritis (OA), is nowadays considered as the main cause of disability affecting the elderly population and is also the primary cause of joint replacements. A particular type of OA, called secondary OA, has generated much interest in the engineering world since it is thought to be mechanically induced due to abnormal and excessive stresses and strains occurring in the AC. As a consequence the mechanical characterization of the soft tissue could help in the understanding the onset of the secondary OA and eventually in finding out clinical solutions to this pathology.

This chapter provides a general introduction to the modeling of AC in both simplified and patient specific knee models, which are described in the following chapters. An overview of the AC biological and mechanical properties, the AC FEM models reported in literature and some aspects of the biomechanics of the knee is also presented.

1.2

A

RTICULAR CARTILAGE

1.2.1

R

The AC is a soft tissue found in many parts of the human skeletal system such as larynx, ears, thorax and joints. There are four main types of cartilage in the human body: hyaline cartilage, fibrocartilage, yellow elastic fibrocartilage and cellular

cartilage. This study focuses on hyaline cartilage also known as articular cartilage which covers the articulating surfaces of synovial joints such as hip and knee. The AC is an essential constituent of joint biomechanics. Its purpose is to support a significant part of the body weight whilst providing a smooth surface to facilitate movement and avoid wear between joints. Thus the AC has two main roles. The first role is to allow joint movement whilst at the same time reducing friction to a minimum by providing excellent and incomparable lubrication properties. It thus protects the joint articulation by avoiding abrasion and wear of the bone surfaces. The second role of articular cartilage is to absorb and attenuate stresses due to body movements, by means of its deformation.

1.2.2

S

The AC can be described as a biphasic material as it consists of two distinct phases [9]: a solid matrix and an interstitial fluid Fig. 1.1 shows a diagram of the cross section of AC showing the main constituents and structure of the tissue.

The solid matrix, often referred to as the Extra Cellular Matrix (ECM), contains the following constituents.

• Proteoglycans. Around 20% - 35% of the weight of the solid phase of AC can be accounted by the PG’s [19]. These are macromolecules with a protein backbone (Hyaluranic acid) with negatively charged GAG’s repeating units. The negative charge forms the so called Fixed Charge Density (FCD), which accounts for the swelling behaviour of the tissue (i.e. the Donnan effect) [20].

• Collagen. The main type of collagen present in AC is Type II which accounts for 50% - 75% of the tissue’s dry weight [21, 22]. These fibrils are a key determinant in the mechanical properties of AC in that they can only resist tension but not compression and, moreover, they have strain dependent stiffness [21].

• Chondrocytes. These are mature cartilage cells which form the ECM of the tissue. As shown from Fig. 2, in the superficial zone of the tissue they have an ovoid shape and are oriented parallel to the surface. In the deeper zones of the tissue, the cells are more spherical in shape and are present in groups of 4 to 8 cells called isogenous groups.

The interstitial fluid phase contains approximately 75% saturated water containing mobile ions [23].

The structural arrangement of the various constituents of the tissue is highly anisotropic and inhomogeneous. Furthermore, there is a depth dependent variation in the constituents and hence depth dependent material properties in the tissue [20]. If a cross section is cut through the tissue four distinct zones can be identified: the superficial zone, the middle zone, the deep zone and the calcified zone. The superficial zone is the thinnest of the four [19]. Here there is a high collagen fibril content, and unlike in the other 3 zones, the fibrils are actually aligned parallel to the surface. This

is because a better resistance to shear stresses arising from knee motion can be achieved. In this zone, the PG content is lowest whilst the water content is highest (85% by mass [21]). The PG content increases with depth, whilst the water content decreases with depth reaching a value of 70% in the subchondral bone. In the middle zone the collagen fibrils are randomly arranged, whilst in the deep zone the fibrils are perpendicular to the calcified cartilage surface.

A layer termed surface amorphous layer (SAL) also exists on the surface of AC. This is a superficial layer extending from the superficial tangential zone. It contains no fibrils or chondrocytes [24] and it has been suggested to contain proteins and glycoproteins, PG’s, chondroitin/keratin sulphates, (phospho)lipids and/or hyaluronic acid-protein complexes [25].

It is worth noting to that, unlike most biological tissues, AC is aneural, avascular and alymphatic [20]. Due to lack of blood supply, the nutrients needed by the tissue are transported by the synovial fluid. The combination of this together with the fact that chondrocytes have a very low mitogenic potency means that AC has indeed very low regenerative capabilities [20]. This is why damaged tissue can hardly be repaired to restore the correct functioning of the joint and why artificial replacements are sometimes the only viable option.

Fig. 1.1. Schematic illustration of the architecture and the components of articular cartilage.

1.3

M

ECHANICAL PROPERTIES OF ARTICULAR CARTILAGE

The mechanical properties of the tissue are influenced by three main structural components: the biphasic nature of the tissue, the collagen content and the PG content. These three factors each contribute in a very specific way to the mechanical behaviour of the tissue and it is the combined effect of these that makes AC such a complex material. Subchondral Bone Chondrocyte Surface Amorphous Layer Collagen Proteoglycan Calcified Bone

1.3.1

C

Cartilage is a very unique material due to its time and stress dependent mechanical properties when subject to compression [26]. This is because the solid matrix found within cartilage is a deformable, permeable and porous material where the fluid trapped within it can exude out and back into the tissue thereby varying its mechanical properties [27]. The water which enters and leaves the tissue comes from the surrounding synovial fluid however the permeability of cartilage is such that it is permeable to water and impermeable to all the other constituents of the synovial fluid.

Due to the biphasic nature of the tissue, it behaves in two very distinct mechanical ways depending on the stress at a given time. In the first stages of the application of the load is supported by the interstitial fluid pressurisation hence initially preserving the solid constituent. Gradually, as interstitial fluid exudes out of the contact area, the hydrostatic pressure drops and a load shift to the solid constituent occurs. This fluid exudation hence causes an additional secondary deformation phase. The permeability and hydrophilicity of the ECM is responsible for the control of the flow exudation from the tissue. This second phase of deformation is stiffer than the original phase as this phase now contains a higher proportion of ECM to water as some of it has been exuded out. Hence, as the load on the cartilage is increased, an increase in stiffness occurs, as water is forced out of the tissue. The quantity of water present at a given time is in fact one of the major determinants of the mechanical properties of cartilage, together with the quantity of collagen fibrils and PGs. The quantity of water within the tissue thus determines the stiffness of the material.

The permeability and therefore the amount of water present in the cartilage is related to the swelling pressure and the fluid motion under compression. This swelling pressure and fluid motion is dictated by the FCD and thus by the PG content. Thus, the compressive behaviour of the tissue is governed by the PGs and it is not surprising that the compressive modulus of the tissue increases with depth, due to the PG content increasing [28].

1.3.2

S

PGs determine the swelling characteristics of the tissue through two distinct phenomena, namely chemical expansion and the osmotic swelling (Donnan effect).

As previously mentioned, an FCD arises due to the negative charge of the GAG repeating units in the PGs. To meet the electroneutrality, a large number of of counter ions (e.g. Na+_{) must be present within the interstitial matrix. There seems to be a}

higher concentration of ions within the ECM than outside of the tissue in the synovial fluid. This different ionic concentration causes a Donnan osmotic pressure to arise, and hence causing fluid to flow into the tissue and hence swelling to occur. The swelling of the tissue comes to a halt when an ionic balance between the ions within the solid matrix and the ions in the synovial fluid surrounding the tissue is created.

Furthermore, because the PGs are so tightly packed in a matrix arrangement, a repulsion force between them arises causing the material to swell and draw in fluid as a pressure difference is developed; this is the chemical expansion phenomenon.

1.3.3

T

The collagen fibrils are responsible for the tensile behaviour of the material. The amount of collagen fibrils present, as well as the fibril orientation and collagen cross linking, is a major determinant in the tensile properties of the tissue [19]. However, a fundamental characteristic of the tensile properties of the collagen fibrils is the strain dependent stiffness where the stiffness of the fibrils increases with tensile strain. When cartilage is tested in tension, the collagen fibrils and entangled PG molecules are aligned and stretched along the axis of loading. For small deformations, when the tensile stress in the specimen is relatively small, a nonlinear toe-region is seen in the stress–strain curve, due to realignment of the collagen fibrils, rather than their stretching. For larger deformations, and after realignment, the collagen fibrils are stretched and therefore generate a larger tensile stress due to the intrinsic stiffness of the collagen fibrils themselves. Due to this phenomenon the tensile stiffness of AC is highly strain-dependent.

1.3.4

V

AC has highly viscoelastic behaviour. There are two distinct mechanisms responsible for this behavior:

1. flow-dependent viscoelasticity: Mow et al. demonstrated that the main mechanism driving creep and stress relaxation in the tissue is the diffusional drag caused by the flow of the interstitial fluid relative to the solid matrix. Fluid movement in loaded cartilage is governed by permeability of the solid matrix which varies both with the pore size and the FCD of PGs which limits the fluid flow. Since when the tissue is deformed both the FCD and the void ratio change, the permeability of AC is strain dependent.

2. flow-independent viscoelasticity: it is due to the time-dependent deformability of the solid matrix accounted by both the viscoelasticity of the PGs and the collagen fibrils.

1.3.5

A

AC results highly inhomogeneous and anisotropic material because of its structural and compositional variation through the depth of the tissue.

The through depth variation of the PG content causes a through depth variation in the FCD and hence in the osmotic swelling pressure, whilst the through depth variation and change in orientation of the collagen fibrils causes a variation in the material stiffness [19, 29]. The collagen fibrils further on contribute to the anisotropy of the AC by their compressive and tensile non linearity.

1.4

O

STHEOARTHRITIS

1.4.1

A

Arthritis (from Greek arthro-, joint + -itis, inflammation) is a group of conditions involving damage to the joints of the body and causing a constant and localized joint pain. There are over 100 different forms of arthritis such as osteoarthritis (OA), rheumatoid arthritis, psoriatic arthritis, septic arthritis. Among all arthritis types, OA is certainly the most frequent.

Osteoarthritis (OA) is a degenerative joint condition which causes an irreversible damage to the integrity of AC and leads to pain and disability. It is the most common joint disorder in the world and thus it results having a significant impact sociologically, economically and on the well being. In western populations it is one of the most frequent causes of pain, loss of function and disability in adults. Radiographic evidence of OA occurs in the majority of people by 65 years of age and in about 80% of those aged over 75 years [30].

OA commonly affects the hands, feet, spine, and mainly the large weight bearing joints, such as the hips and knees, although in theory, any joint in the body can be affected.

The main symptom is the pain, causing loss of ability and often stiffness. Pain is generally described as a sharp ache, or a burning sensation in the associate muscles and tendons. OA can cause a crackling noise (called crepitus) when the affected joint is moved or touched, and patients may experience muscle spasm and contractions in the tendons. Occasionally, the joints may also be filled with fluid.

Aetiology of OA has caused a lot of debate in the recent years [31]: many answers regarding the nature and progression of the pathology are still unknown [19]. However, the OA classification in primary and secondary OA [32] is largely well accepted. Primary OA occurs in apparently intact joints (e.g. healthy ligaments and meniscii) and in absence of predisposing factors [32]. It is related (but no caused) to the aging process and typically occurs in older individuals. Secondary OA develops as a consequence of injury, obesity, genetics or inflammatory factors and can strike at an earlier age. Secondary OA is commonly considered mechanically induced and associated to abnormal or excessive stresses in the AC.

Either for primary and secondary OA, the diagnosis is made with reasonable certainty based on history and clinical examination. X-rays may confirm the diagnosis. The typical changes seen on X-ray include: joint space narrowing, subchondral sclerosis (increased bony formation around the joint), subchondral cyst formation, and osteophytes.

As a treatment lifestyle modification is suggested right from the beginning. This include both daily exercise and weight loss, for overweight people. In addition analgesics are the mainstay of treatment. Paracetamol is used first line and NSAIDS

are only recommended as add on therapy if pain relief is not sufficient. Physical therapy has also been shown to significantly improve function, decrease pain, and delay need for surgical intervention in advanced cases [33]. Exercise prescribed by a physical therapist has been shown to be more effective than medications in treating osteoarthritis of the knee. Functional, gait, and balance training has been recommended to address impairments of proprioception, balance, and strength in individuals with lower extremity arthritis as these can contribute to higher falls in older individuals [34].

However if the above management is ineffective, joint replacement surgery or resurfacing may be required in advanced cases. Nowadays OA is considered the primary cause of joint replacements. By considering population aged from 60 up to 90 years, the prevalence of hip and knee OA is 7.4% (8% in women, 6.7% in men) and 12.2% (14.9% in women, 8.7% in men), respectively. For this group of patient affected by OA, the rate for hip replacement is 37.7% in men and 52.7% in women, while for knee replacement is 11.8% in men and 17.9% in women [35].

1.4.2

C

OA

OA is well known to cause irreversible structural damage to the AC: changes in AC composition induce changes in the mechanical properties of the tissue leading to a loss of its functionality. Softening, fibrillation and ulceration of the AC are observed in damaged OA, finally leading to cartilage loss. An example of damaged AC is shown in Fig. 1.2.

Fig. 1.2. Macroscopic AC damage caused by OA: Cartilage abrasion on the medial humeral condyle (a); cartilage ulceration on the medial femoral condyle (b); cartilage repair of the

medial femoral condyle (c); marginal osteophytes on processus anconeus of ulna (d) (Permission of reproduction by Creative Commons Attribution 2.0).

Early cartilage degeneration is primarily related to PGs depletion and to alteration of the collagen fibrils network [36-38]. The most supported hypothesis [37] states that the degeneration of the cartilage initiates with the disintegration of collagenous fibril network in the transitional zone, caused by the loosing of the cross-links between the large collagen fibrils. As a consequence PG retention decreases and causes a reduction of PG concentration [36]. These structural changes lead to cartilage swelling and increased water content, increasing tissue permeability and allowing free water and other molecules to flow in and out of the tissue. This process also cause a decrease in the mechanical stiffening of the tissue.

In severely damaged AC, cracks initiate at the articular surface, and extend downwards at approximately 45° into the superficial, middle and eventually deep zone [39]. Cracks along the tidemark can be observed as well.

In addition also the chondrocytes contribute to the AC degeneration, during all OA stages. Healthy AC is characterized by a delicate balance between synthesis and degradation of chondrocytes. The settle of OA disturbs this balance, with both degradation and synthesis usually enhanced [40]. Obviously dead cells can no longer maintain or repair the tissue. Moreover, because dead cells are not removed effectively from the cartilage, their products may contribute to pathologic cartilage degradation and inflammation.

1.5

M

ODELLING OF ARTICULAR CARTILAGE

AC is a very complex material to model mainly because of its biphasic nature. Hence when modelling the tissue one must take into account both the solid matrix and the fluid component and all the very unique mechanical properties previously described. Over the past years, different attempts towards the modelling of cartilage behaviour have been made. Some are more realistic than others, whilst some are very simple, as the monophasic one.

A brief review of the theory of the AC numerical models used to investigate the mechanical behaviour of the tissue is given in this section.

1.5.1

M

AC

The monophasic (M) model in the simplest model, typically adopted for 3D complex geometries, such as for modelling the AC in 3D knee FEMs [41-49]. Basically the AC is modelled as a made of only the solid phase, considered as linear elastic, homogenous, isotropic and incompressible. The model is so characterized by only two mechanical parameters: Young’s modulus and Poisson ratio. More exactly the Young’s modulus used in this model is referred to dynamic Young’s modulus (or instantaneous Young’s modulus), which is extrapolated from the indentation curves data obtained instantaneously after the load application on the AC. The instantaneous Young’s Modulus takes into account both the response of the solid and the fluid constituent. At high strain rates the tissue appears to be much stiffer than at low strain rates. The

reason for this being that at high strain rates most of the load is supported by the hydrostatic pressure of the interstitial fluid which develops due to the low permeability of the tissue. At lower strain rates, as more time is given for the interstitial to exude out of the contact zone, load shift from the fluid to the solid constituent occurs and the tissue appears to be less stiff due to the low stiffness of the solid constituent. Hence dynamic Young’s modulus is lower at low strain rates. Once all the interstitial fluid has exuded out of the contact zone, the instantaneous compressive modulus is equal to the compressive modulus at equilibrium, i.e. the Young’s modulus of the solid matrix.

1.5.2

B

AC

The biphasic linear elastic (BL1) model is the most basic biphasic model, developed by Mow et al. in 1980 [9]. The tissue is assumed to be constituted of a linear elastic, homogenous, isotropic and incompressible solid phase and an incompressible fluid phase; the permeability is assumed to be constant and so is the void ratio, defined as the void ratio, defined as the ratio between the fluid fraction and solid fraction. The equilibrium (stationary) mechanical response after full relaxation of the tissue is described by assigning a single Young’s modulus and Poisson ratio to the tissue.

BL model is satisfactory if a very crude and general approximation of the tissue’s mechanical behaviour is needed as it shows the main mechanical behaviour of AC due to its biphasic nature. Moreover, it is fairly simple to model using FEM and not too computationally expensive.

Nevertheless the model can account for the long term creep and stress relaxation during confined compression, significant deviations are seen for the short term response of the tissue. The validity of the model also decreases in case of unconfined compression simulations mainly because the anisotropy of the tissue is not considered. In particular, it is worth noting that the BL model severely underestimates the stresses in the tissue which is a severe drawback when investigating damage.

This model is the “building block” of all the more advanced models which include a more detailed material characterisation, as described in the following.

1.5.3

B

AC

The biphasic non linear model (BNL1_{) combined the biphasic linear elastic (BL)}

model with a non linear permeability, thus reproducing the flow dependent viscoelasticity in a more realistic way. The permeability is in reality strain dependent, thus causing nonlinearity due to flow effects. This is because when the tissue

1

The letter “L”/ “NL” which compare in the acronym of AC material models is referred to

undergoes deformation, the porous elastic matrix deforms, thus changing the pores’ conformation and size and reducing the porous matrix void ratio. Therefore, permeability decreases with increasing strain.

Lai et al., 1981 [50] firstly proposed the non linear law which defines the permeability as a function of the void ratio. This law was revisited by Van der Voet in 1997 [51] assuming the following form:

M e e k k _      + + = 0 0 1 1 (1.1) where k0 and M are material constants, and e and eo the initial and the current void

ratio, respectively. The eq. 1.1 is thus implemented in the BNL model.

1.5.4

B

-

AC

The biphasic poro-viscoelastic (BPVL1_{) linear elastic model takes in account both}

the flow dependent and the flow independent mechanism due to the viscoelastic nature of the collagen fibrils and the proteoglycan matrix. It was first analytically formulated by Mak in 1986 [52] and implemented using two different algorithms: the continuous spectrum and the discrete spectrum algorithm. The continuous spectrum, proposed by Fung [53] uses an integral model to represent viscoelasticity and it provides a more descriptive differential representation of the linear viscoelasticity law than the discrete spectrum does. However, Suh and co- workers [12] discovered that the discrete spectrum algorithm, which is derived from a combination of Kelvin’s viscoelastic discrete models, saves significant CPU time and memory compared to the continuous spectrum algorithm. A slight divergence in predicting creep and stress relaxation of AC between the two algorithms exists, however, it is very small and the overall tissue behaviour is still well described. Hence, the discrete spectrum approximation is more widely used viscoelastic approximation. It has been adopted also in this study §2.2.3.

The discrete spectrum model describes viscoelastic properties using the reduced relaxation function G(t) written as a series of combinations of discrete relaxations function Gi [12], such as:

∑

where τi is the discrete relaxation time constant. To implement of eq. 1.2 it can be

assumed G∞=1 and Gi=Ğ, with Ğ called discrete spectrum magnitude and equal to: 1 1 0 + − = d N G G( (1.2)

where G0= G(0). Furthermore τi can be considered equally distributed with a decadic

interval between the short term and long term relaxation time constants, i.e. τS and τL ,

      = S L d N τ τ log (1.2)

Further details on the implementation of the discrete algorithm can be found in [12]. It is important to note that 3 additional parameters are necessary to characterize a biphasic poro-viscoelastic material: Ğ, τS and τL.

BPVL model is able to simultaneously account for experimental data obtained from confined and unconfined compression, and indentation. In particular it can predict simultaneously lateral displacements and reaction forces during unconfined compression tests [7], also for different strain rates [6]. Although the model includes the fundamental characteristic of the flow-independent viscoelasticity, it lacks the anisotropy and compression-tension non linearity of the tissue.

1.5.5

B

AC

As previously discussed, AC is highly anisotropic. Because complete anisotropy can be very difficult to implement numerically, transverse isotropy can be an acceptable assumption in modelling the AC. The first biphasic transversely isotropic linear elastic (BTIL1_{) model was formulated by Donzelli et al. in 1999 [8].}

Transverse isotropy material can be seen as an orthotropic material with one plane of isotropy [19]. In the AC the plane of isotropy is thought to be oriented with the superficial collagen fibrils, which run parallel in the same direction. Assuming the fibrils running in the 3rd _{direction, the stresses in the tissue are given by the following}

matrix:                                         − − − − − − =                     − 23 13 12 33 22 11 1 23 13 12 3 2 23 1 13 3 32 2 1 12 3 31 2 21 1 23 113 12 33 22 11 / 1 0 0 0 0 0 0 / 1 0 0 0 0 0 0 / 1 0 0 0 0 0 0 / 1 / / 0 0 0 / / 1 / 0 0 0 / / / 1 γ γ γ ε ε ε ν ν ν ν ν ν σ σ σ σ σ σ G G G E E E E E E E E E (1.3)

where Gij are the shear moduli. In reality to characterize the model only 5 independent

material parameters are needed because of the following relation: E1=E2, G13=G23,

G12=E1/(2(1+ν12)), ν13/E1= ν31/E3, )), ν13/E1= ν31/E3, ν23/E2= ν32/E3. To note that E3 is

much greater than E1,2 because of the assumption that all the collagen fibrils are

aligned in the 3rd _direction.

This model, as shown experimentally in [6], is able to account very well for either lateral displacement and reaction force during unconfined tests, but it cannot for both variables simultaneously. This was probably caused by the absence of the viscoelasticity of the solid matrix.

1.5.6

F

AC

The fibril reinforced model was first suggested by Soulhat et al. [54] and it was successfully implemented by Li et al. [21]. It can be considered an advancement of the BTIL model. While in the BTIL model the fibrils are simply modelled by specifying a higher Young’s modulus in the direction where they are assumed to be aligned, in the FR model the fibrils are modelled individually. Thus in the fibril reinforced model the solid phase is represented by two individual constituents with individual mechanical properties: the PG matrix and the collagen fibrils. The PG matrix with the interstitial fluid is modelled exactly as in the biphasic formulation. A linear or a non linear formulation can be applied to the fibrils. In the first case the fibrils are considered as linear elastic springs [21] and the model is simply referred as fibril reinforced (FR). In the second case the strain dependent stiffness of the collagen fibrils is modelled [15]; this more sophisticated model is well known as fibril reinforced poro-viscoelastic model (FRPV). Both in the FR and the FRPV models the compression-tension non linearity of the fibrils is taken into account, assuming zero stiffness in compression. In addition, in the FRPV model, also the compression-tension non linear viscoelasticity of the fibrils can be modelled. This particular detail represents a significant advancement in respect with the BPVL.

1.5.7

S

AC

Swelling model is the most advanced model for the AC which replicates the triphasic behaviour of the AC, hence its biphasic nature as well as its swelling behaviour whose contributors are the osmotic swelling and chemical expansion. Firstly in the literature, Wu et al. in 2002 [55] proposed to implement the triphasic theory taking advantage from the mathematical identity between thermal and mass diffusion processes. Thus the triphasic model can be transferred to a convective thermal diffusion phenomenon and can be implemented into commercial FE software. To examine in detail this model is beyond the aim of this study; further details can be found in [3, 56].

1.6

S

ECONDARY OSTEOARTHRITIS IN THE KNEE JOINT

1.6.1

T

The knee joint is one of the largest and more complex articulation of the human body. As shown in Fig. 1.3 the knee consists of three bones: the femur (thigh bone), the tibia (shin bone) and the patella (kneecap). These bones made up two articulations:

1. femoral-patellar joint: consists of the patella and the patellar groove, which is located on the front of the femur, between the two femoral condyles.

2. femoral-tibia joint: links the femur with the tibia; each femoral condyle (medial and lateral) articulates with the correspondent tibial condyle.

Fig. 1.3. Anatomy of the knee: bones and synovial capsule.

The great functionality of these articulations is guaranteed by the joint lubrication, the smooth surface of the articular cartilage, and the bearing support of the meniscii, examined in the following.

The joint is bathed in synovial fluid, a biological lubricant which is contained inside the synovial membrane called the articular or synovial capsule.

The cartilage which, in the knee, can be hayaline (articular cartilage) or fibrous. The articular cartilage covers the surface along which the joints move and makes certain that the joint surfaces can slide easily over each other (see ). The fibrous cartilage, which has a tensile strength and can resist pressure, can be found in the meniscii.

The meniscii are two articular disks, i.e. the lateral and the medial (Fig. 1.4). Each disk consist of connective tissue with extensive collagen fibers containing cartilage-like cells. Strong fibers run along the menisci from one attachment to the other, while weaker radial fibers are interlaced with the former. The menisci are flattened at the center of the knee joint, fused with the synovial membrane laterally, and can move over the tibial surface. The lateral meniscus has a circular shape and is smaller and more mobile of the medial one. The medial meniscus ha a “C” shape and posteriorly it is wider than the lateral one. Their role is fundamental in the biomechanics of the knee: they protect the ends of the bones from rubbing on each other and to effectively deepen the tibial sockets into which the femur attaches (i.e. increase of the contact area ). They also play an important role in shock absorption, and may be cracked, or torn, when the knee is forcefully rotated and/or bent.

Femur Tibia Fibula Synovial Capsule Patella Tibial lateral condyle Tibial medial condyle Patellar groove Femoral medial condyle Femoral lateral condyle

Fig. 1.4. Anatomy of the knee: meniscii and main ligaments.

The articulation is stabilized by several ligaments (Fig. 1.4). The most important are listed below:

• The cruciate ligaments are a pair of ligaments in the center of the knee joint (i.e. intracapsular) that form a cross, and this is where the name “cruciate”comes from. More exactly they are:

– anterior cruciate ligament (ACL): stretches from the lateral condyle of femur to the anterior intercondylar area. The ACL is critically important because it prevents the tibia from being pushed too far anterior relative to the femur. It is often torn during twisting or bending of the knee.

– posterior cruciate ligament (PCL): stretches from medial condyle of femur to the posterior intercondylar area. Injury to this ligament is uncommon but can occur as a direct result of forced trauma to the ligament. This ligament prevents posterior displacement of the tibia relative to the femur. • The collateral ligaments are a pair of ligaments positioned on the two opposite

sides of the knee (extra-capsular); more exactly they are:

– medial collateral ligament (MCL ): stretches from the medial epicondyle of the femur to the medial tibial condyle. It is composed of three groups of fibers, one stretching between the two bones, and two fused with the medial meniscus. It protects the medial side of the knee from being bent open by a stress applied to the lateral side of the knee (i.e. a valgus force). – lateral collateral ligament (LCL): stretches from the lateral epicondyle of

the femur to the head of fibula. It protects the lateral side from an inside bending force (i.e. a varus force).

The kinematics of the knee is quite complex. By simplifying, the knee can be considered a mobile trocho-ginglymus (i.e. a pivotal hinge joint), which permits flexion and extension as well as internal-external rotation. For a more detailed

Lateral meniscus ACL MCL PCL Medial meniscus LCL Patellar tendon Patella Femoral Condyles Quadriceps tendon Lateral meniscus ACL LATERAL MEDIAL POSTERIOR PCL ANTERIOR Medial meniscus

description of the tibio-femoral joint kinematic motions in the sagittal, frontal and transversal planes are described:

• sagittal plane:

– flexion-extension rotation; considering 0° for the full extension, the range of motion is 0°-135°;

– anterior-posterior translation: due to the gliding which take place during the reaching of the full extension (explained below);

• transversal plane:

– internal-external rotation: this motion is allowed only for certain flexion ranges. If the knee is in full extension the internal-external rotation is highly restricted by interlocking of condyes with tibia. Rotation increases as the knee is flexed and maximum for 90° of flexion: in this case the internal and the external rotation can be at maximum of 45° and 30°, respectively. beyond 90° of flexion, the internal external rotation decreases, due to soft tissue restriction.

• frontal plane:

– abduction-adduction rotation is affected by the amount of knee flexion. In particular full extension precludes motion; increased passive abduction and adduction occurs with knee flexion less than 30°.

Particular attention has to be made to the tibio-femoral kinematic during the final phase of the extension (c.a. last 20°), which is depicted in Fig. 1.5. The convex surface of a femoral condyle is wider of the concave surface of the correspondent tibial condyle. Consequently, the pure rolling of the articulating surfaces would yield to the contact loss. To avoid that, the rolling is combined with a posterior gliding of the knee. In addition a small medial rotation of the femur occurs to distribute the load on the whole surface of the medial condyle, being the medial condyle wider than the lateral one, posteriorly.

(a) (b)

Fig. 1.5. Tibio-femoral kinematics in the final phase of the extension (a): combination of rolling, gliding and internal rotation (b).

Internal rotation Rolling

Extension

To complete the overview on the knee biomechanics, a look to the loading conditions of the knee the has to be given. In Fig. 1.6 is illustrated a scheme of the load and torque components which can act in the knee during the motion. As an example the knee vertical load during walking, in according to [57], is reported in Fig. 1.7.

Fig. 1.6. Scheme of the forces and torques which can act on the knee.

Fig. 1.7. Physiological vertical load acting on the knee joint during the walking [57]. Anterior-posterior Translating Force Distracting Interpenetration Force Medial-Lateral Subluxing Force Internal-external Rotating Torque Abducting-Adducting Torque Flexing-Extending Torque

Time (s)

V

e

rt

ic

a

l l

o

a

d

(%

B

W

)

1.6.2

FEM

In the last decades secondary OA has generated much interest in the engineering world since it is thought to be mechanically induced due to abnormal and excessive stresses and strains occurring in the AC. As a consequence, in order to investigate the initiation of secondary OA, finite element models of damaged joint could be very useful. They would allow to determine loading and kinematic conditions which cause abnormal stresses/strains in the tissue and hence their location, considered likely OA onset. Obviously it would be necessary a very complex joint FEM which theoretically should account for the realistic material behaviour (AC and bone), realistic 3D geometry of AC and bone, and realistic boundary conditions (lubrication, load and motion).

With this regard some papers have addressed the question to what kind of stresses and strains are to be expected in knee prone to OA. As reference case, meniscectomised knee joint has been mostly taken in consideration. Indeed it has been well accepted that meniscectomy cause a severe instability of the knee joint resulting in initiation of OA [58]. Both partial and total menisectomies cause a dramatic increase in articular surface contact stress because of the decrease of the contact area due to a direct contact femur-tibia. This results in tissue damage after only a few years [59, 60].

FEMs of meniscectomised knee joint which can be found in the literature present some drawbacks. Some of them consider the biphasic nature of the AC, adopting BL, BNL or BTIL models, but, as drawback, simulates simplified 2D/axysymmetric geometries of only one knee compartment (i.e. medial or lateral) [61, 62](Fig. 1.8a). In the opposite, other studies propose 3D patient specific FEMs, with accurate geometries reconstructed by medical images, but model the AC as monophasic, homogeneous, isotropic, linear elastic [48, 49] (Fig. 1.8b). This is probably due to the difficulty of implementing advanced AC models in more complex geometries.

In conclusion AC has never been modelled accurately in a 3D tibiofemoral joint contact. However, the M model is a long way away from an accurate representation of AC. It fails to account for the time/strain dependent nature of the tissue and the stress and strain distribution would indeed be very different from the reality [3, 63].

Hence, for more realistic studies in AC mechanics within a tibiofemoral contact and for the investigation of the development and the origin of damage in the joint, more realistic AC models would be necessarily required [19].

(a) (b)

Fig. 1.8. An example of knee FEMs used for investigations on OA: (a) simplified axy-symmetric geometry combined with biphasic models of the AC [61]; (b) realistic knee geometry combined

with monophasic modeling of the AC [48].

1.7

R

ESEARCH GOAL

The basic research goal of Part I of the thesis consists in the development of a patient specific finite element model of the tibio-femoral joint following bicompartmental total meniscectomy, in which advanced AC models are implemented. Although bicompartmental total meniscectomy is clinically rare, it does stimulate particular interest from a biomechanical perspective since it causes severe changes in the stress and strain distribution and results a favorable condition for the OA initiation.

The final objective of this research is firstly to demonstrate the importance and the necessity of implementing advanced AC models if an accurate representation of the tibiofemoral joint contact problem has to be made. Secondly, to investigate the mechanical response of the AC during daily activities, such as the walking, and finally to evaluate AC contact areas subjected to the highest stress, where AC damage and OA might take origin.

The content of Part I of the thesis has been described in a paper actually submitted to a peer reviewed journal [2].

CHAPTER 2

M

ECHANICAL BEHAVIOUR AND MODELLING OF

THE ARTICULAR CARTILAGE

2.1

I

NTRODUCTION

The first step towards the implementation of an advanced AC material model in realistic 3D knee geometries consists in the development of a 3D simplified model of the tibio-femoral joint. The present chapter reports the development of a FEM model reproducing the realistic local anatomy of the tibio-femoral joint. To the aim of investigating the mechanical behaviour of the AC, several biphasic constitutive models were used and their mechanical responses compared.

The most proper model for the AC, out of the examined ones, was identified and subsequently used in development of a patient specific FE model of the knee described in the next Chapter. In addition the simplified knee model has been used for mesh convergence analysis aimed at evaluating the mesh properties which guarantee accuracy even in a more complex geometry.

2.2

S

IMPLIFIED MODEL OF THE TIBIO

-

FEMORAL JOINT

2.2.1

G

The simplified model of the tibio-femoral joint (Fig. 2.1) consisted in two interacting spherical layers of AC which covered two correspondent bone structures. The dimensions of the model were chosen coherently with the realistic tibio-femoral contact geometry, assuming the thickness of the AC layers of 2.5 mm and curvature radius of 200 mm (measured in a tibio-femoral contact region). Only a quarter of the model was simulated since kinematic symmetry boundary conditions (BCs) were exploited (see §2.1.4).

Fig. 2.1. Simplified model of the tibio-femoral joint: geometry and two examples of mesh.

2.2.2

M

Both for the AC and for the bone meshing hexahedral elements were employed. Indeed hexahedral elements are requested by ABAQUS for soils consolidation analysis with biphasic materials. In Fig. 2.1 two examples of mesh for the AC are showed. They correspond to the mesh properties used for the study of the AC mechanical behaviour described in the present chapter, and for the implementation of AC advanced models in a realistic geometry (Chapter 5), respectively.

These meshes, the former more accurate and with a higher computational cost than the second one, were identified by means of a mesh convergence study described in §2.2.1.

2.2.3

M

The anatomical structures considered in the model consist of the cortical bone, the trabecular bone and the AC.

The femur and the tibia extremities are assumed to be composed of only ephiphyseal surface bone, which has mechanical properties in the range between the cortical and the trabecular bone’s properties. The femur and the tibia were hence modelled as homogeneous, isotropic and linear elastic material, with the Young’s modulus of 5 GPa and the Poisson’s ratio of 0.3[64].

It is worth noting that the focus of this investigation was however on the AC material model rather than on the bone, since the latter is much stiffer than the AC (2-3 orders of magnitude); modelling the bone with accurate mechanical properties will not play a dominant role in the analysis. In this study biphasic AC models were used which on one hand are more advanced than the elastic monophasic one, and on the

3D Mesh Femoral Bone Tibial Bone AC Rigid surface AC AC Mesh • AC elements: 2x2x0.625 mm3 • Element layers: 8 • AC elements: 2x2x0.3125 mm3 • Element layers: 4 AC Mesh 3D Geometry

other side they are relatively simple to implement in respect with triphasic models. Models such as the fibril reinforced model [21] would have involved many difficulties mainly in a complex 3D mesh/geometries and however it would have gone beyond the interest of the present study. The material models implemented for the AC are listed below (see §1.5.1), whilst the correspondent mechanical properties are summarized in Table 2.1-3:

1) M: monophasic model whose parameters, i.e. the Young’s modulus and Poisson ratio, were obtained by averaging the range reported in the literature of 4-20 MPa and 0.45-0.48, respectively [41-49];

2) BL: biphasic linear model [7]; 3) BNL: biphasic non linear model[7];

4) BPVNL: biphasic poroviscoelastic non linear model. This model combines the BPVL with the non linear relation between the permeability and the void ratio. Three different BPVNL models were implemented in order to investigate the role of the poroviscoelasticity in the mechanical response of the AC:

a. BPVNLa [7]; b. BPVNLb [65]; c. BPVNLc [12];

The relaxation functions of the three models, all extrapolated experimentally, are compared in Fig. 2.2. As it can be observed there is a significant dispersion of data between Di Silvestro et al. [7, 65] and Suh et al. [12] which will reflect in the mechanical response of the correspondent materials (see §2.2.1). 5) BTINL: biphasic transversely isotropic non linear model [7]. This model

integrates the non linear relation between the permeability and the void ratio in the BTIL model.

The parameters of the biphasic models BL, BNL, BPVNLa, BTINL, were taken according to Di Silvestro et al. [7] who extrapolated them by unconfined compression tests on AC. This allowed us to use data relative to the same AC sample for different AC models, and consequently to compare the considered models in a more rigorous way.

Table 2.1. Mechanical parameters for the M [41-49], the BL and the BNL model of AC [7].

M Dynamic Young’s modulus, Edyn (MPa) 5

Poisson ratio νdyn 0.46

BL

BNL

Equilibrium Young’s modulus Eeq (MPa) 0.421

Equilibrium Poisson ratio νeq 0.28

Density ρ (g/cm3_{) 1}

Initial Permeability k0 (m4/N s) 6.9 10-15

Initial void ratio e0 4

Table 2.2. Mechanical parameters for the BPVNL models of AC: BPVNLa [7]; BPVNLb [65]; BPVNLb [12].

BPVNLa BPVNLb BPVNLc

BPVNL

Equilibrium Young’s modulus Eeq (MPa) 0.55 0.63 0.7

Equilibrium Poisson ratio νeq 0.24 0.1 0

Density ρ (g/cm3_{) 1}

Initial Permeability k0 (m4/N s) 0.9 10-15 1.72 10-15 2 10-15

Initial void ratio e0 4

Non linear permeability parameter M 5

Viscoelasticity parameters

Ğ 15.180 17.692 10.21

τS (s) 0.45 0.62 0.001

τL (s) 95.9 85.1 10

Table 2.3. Mechanical parameters for the BTINL model of AC [7].

BTIE

Equilibrium Young’s modulus Eeq (MPa)

E1=E2 1.7

E3 0.421

Equilibrium Poisson ratio νeq (MPa)

ν12=ν21 0.09

ν13= ν23 0.01

ν31=ν32 0.002

Equilibrium Shear modulus Geq (GPa)

G12=G21 0.78

G13=G23 0.84

G31=G32 0.21

Density ρ (g/cm3_{) 1}

Initial Permeability k0 (m4/N s) 0.5 10-15

Initial void ratio e0 4

Non linear permeability parameter M 5

It is worth noting that in reality, both Di Silvestro et al. [7, 65] and Suh et al. [12], considered the permeability constant with the void ratio in their speculations. Consequently M was estimated from other literature works [62, 66, 67], as well as the initial void ratio, not reported in [6, 12, 65].

0 100 200 300 400 500 600 700 800 900 1000 0 5 10 15 Time (s) G (t ) BPVNLa BPVNLb BPVNLc

Fig. 2.2. Viscoelastic relaxation function plotted versus time for three different BPV model [6, 12, 65] .

2.2.4

B

The boundary conditions required by the model can be grouped as it follows. • Surface interactions. Two different surface interactions occur in the

tibio-femoral joint: the bone-AC and the AC-AC interactions Fig. 2.3. The bone-AC interaction was modelled as a tie constraint. This was thought to be a good assumption since the internal AC surface is tied to the correspondent bone surface and no relative motion between the two is possible. The AC-AC interaction was modelled as a frictionless (considering only normal stress at the contact surface) and a hard contact (the contact is “total” when it is detected), taking into account the possibility of a finite sliding between the surfaces.

• BCs for the pore pressure of the biphasic AC. When the AC is modelled as a biphasic material, a pore pressure BC is necessary at each surface of the AC layers. As indicated in Fig. 2.3, sealed condition (i.e. no fluid flow) at the bone-AC interfaces, and free draining condition (i.e. zero pore pressure) on the external AC surfaces were adopted.

• Kinematic and loading conditions. As depicted in Fig. 2.3 the lower transversal section of the tibial bone was built in (i.e. encastre) while the femur kinematics was driven by the kinematics of a reference point (RP) of the rigid surface (Fig. 2.3). The vertical translation of the RP and was free since this DoF is indirectly controlled by the vertical load; also the rotation around y axis was considered free whilst all the other DoFs were fixed. The loading cycle consisted in a vertical load of 10 N applied to the rigid surface by a ramp of 15 s and kept on for 450 s, as shown in Fig. 2.4.

Fig. 2.3. Scheme of the boundary conditions applied to simplified knee model. The global coordinate system is shown on the left-bottom.

0 50 100 150 200 250 300 350 400 450 0 2 4 6 8 10 L o a d ( N ) Time(s)

Fig. 2.4. Vertical load applied to the rigid surface of the femur part (see Fig. 2.1) and used for the exploration of the mechanical response of AC models.

2.3

R

ESULTS AND DISCUSSION

2.3.1

M

The mesh convergence analysis represented a remarkable part of this study. Several simulations were carried out for different AC meshes. Two mesh properties were considered separately: the number of element layers in the AC thickness and the aspect ratio of AC elements. In particular the element aspect ratio was defined as the ratio between the element base and height dimensions, always considering elements with square bases. The convergence of the mesh was evaluated considering the

Boundary conditions RP- reference point Encastre Kinematic simmetry BCs Tied bone/AC surfaces Free draining Sealed Frinctionless contact

Pore Pressure BCs for AC

following mechanical local and global variables: the maximum values, in the AC part, of the Von Mises stress and of the contact pressure, the contact area, the reaction force. The main results are reported in Fig. 2.5-6. They were obtained by means of displacement controlled simulations: a physiological compression (0.5 mm) was applied in 1 s to the rigid surface, in absence of any external loads. Moreover these simulations assumed the BPVNLb model of the AC, which was subsequently used for the development of the patient specific knee model. Fig. 2.5 illustrates the mesh convergence in function of the number of element layers in the AC thickness; the curves relative to two different dimensions of the element (2.5 and 1.6 mm) are reported. The mesh convergence was clearly higher when more element layers were used, whilst it appeared to be independent on the element base dimension given that the curves for the elements with bases of 2.5 and 1.6mm were nearly overlapped.

Fig. 2.5. Results of the mesh convergence analysis: investigations on the number of element layers in the AC thickness, adopting two different dimensions for the element base (2.5 and 1.6

mm).

The role played by the aspect ratio on the mesh convergence was demonstrated to be less significant by the results of Fig. 2.6. Varying the aspect ratio between 0.5 and 4,

0 2 4 6 8 10 0 2 4 6 8 10 V o n M is e s (M P a ) Number of layers base 2.5 mm base 1.6 mm 0 1 2 3 4 5 6 7 0 2 4 6 8 10 C o n ta ct p re ss u re ( M P a ) Number of layers base 2.5 mm base 1.6 mm 0 20 40 60 80 100 120 0 2 4 6 8 10 C o n ta ct A re a ( m m 2) Number of layers base 2.5 mm base 1.6 mm 0 100 200 300 400 500 0 2 4 6 8 10 R e a ct io n fo rc e ( N ) Number of layers base 2.5 mm base 1.6 mm