Finite element modeling of anterior cruciate ligament reconstructed knee joints : toward clinical use via fast motion implementation and verification against MRI follow-up

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Dissertations in Forestry and Natural Sciences

PAUL OCTAVIAN BOLCOS

Finite Element Modeling of Anterior Cruciate Ligament Reconstructed Knee Joints

Toward Clinical Use via Fast Motion Implementation and Verification Against MRI Follow-up PUBLICATIONS OF

THE UNIVERSITY OF EASTERN FINLAND

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PUBLICATIONS OF THE UNIVERSITY OF EASTERN FINLAND DISSERTATIONS IN FORESTRY AND NATURAL SCIENCES

N:o 417

Paul Octavian Bolcos

FINITE ELEMENT MODELING OF ANTERIOR CRUCIATE LIGAMENT RECONSTRUCTED KNEE JOINTS.

TOWARD CLINICAL USE VIA FAST MOTION IMPLEMENTATION AND VERIFICATION AGAINST MRI

FOLLOW-UP

ACADEMIC DISSERTATION

To be presented by the permission of the Faculty of Science and Forestry for public examination in the Auditorium SN201 in Snellmania Building at the University of Eastern Finland, Kuopio, on February 19, 2021, at 15 o’clock.

University of Eastern Finland Department of Applied Physics

Kuopio 2021

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Grano Oy Jyväskylä, 2021

Editors: Pertti Pasanen, Raine Kortet, Jukka Tuomela, and Matti Terde

Distribution:

University of Eastern Finland Library / Sales of publications julkaisumyynti@uef.fi

http://www.uef.fi/kirjasto

ISBN: 978-952-61-3718-6 (print) ISSNL: 1798-5668

ISSN: 1798-5668 ISBN: 978-952-61-3719-3 (pdf)

ISSNL: 1798-5668 ISSN: 1798-5676

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Author’s address: University of Eastern Finland Department of Applied Physics P.O.Box 1627

70211 Kuopio Finland

email: paul.bolcos@uef.fi

Supervisors: Professor Rami Kristian Korhonen University of Eastern Finland Department of Applied Physics P.O.Box 1627

email: rami.korhonen@uef.fi

Adjunct Professor Mika Einari Mononen University of Eastern Finland

Department of Applied Physics P.O.Box 1627

email: mika.mononen@uef.fi Professor Juha Töyräs University of Eastern Finland Department of Applied Physics P.O.Box 1627

email: juha.toyras@uef.fi

Reviewers: Professor Ruth Wilcox

University of Leeds

Department of Engineering and Physical Sciences Woodhouse Ln, Woodhouse

LS29JT Leeds United Kindom

email: r.k.wilcox@leeds.ac.uk Professor Marcus Pandy University of Melbourne

Department of Mechanical Engineering Biomedical Engineering

The University of Melbourne 3010 Victoria

Australia

email: pandym@unimelb.edu.au Opponent: Associate Professor Amy Lerner

University of Rochester

Department of Biomedical Engineering P.O. Box 270168

14627, Rochester, NY United States of America

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Paul Octavian Bolcos

Finite element modeling of anterior cruciate ligament reconstructed knee joints. Toward clinical use via fast motion implementation and verification against MRI follow-up

Kuopio: University of Eastern Finland, 2021 Publications of the University of Eastern Finland

Dissertations in Forestry and Natural Sciences, 2021, 417

ABSTRACT

Osteoarthritis (OA) is a common musculoskeletal joint disease, in which primarily the articular cartilage and subchondral bone are altered. OA affects more than 250 million people and places a huge financial burden on both the patient and the society. The most common knee joint injury is anterior cruciate ligament (ACL) rupture. This type of injury is typically seen in sport activities, such as football, basketball or downhill skiing. Given the role of the ACL in knee joint stability, it is often reconstructed using tendon grafts. However, ACL reconstruction (ACLR) has not been shown to significantly reduce the risk for the onset and development of OA.

Currently, there is no cure for knee OA and only very limited treatment options exist. The patient’s quality of life is largely maintained through rehabilitation, analgesic medication or lifestyle modifications. At the end stages of OA, the articular cartilage surface is fully degraded, with bone-on-bone contact. This leads to severe pain and joint stiffness. At this point, the only solution is total knee replacement. However, a much more cost-effective option would be disease prevention. Thus, clinical assessment tools for predicting the onset and development knee OA are urgently needed. To achieve this, finite element (FE) models have been used to simulate the risks for knee cartilage degeneration due to excessive tissue stresses and/or strains. Magnetic resonance imaging (MRI) and motion capture can be combined to create subject-specific FE models. However, the most suitable method to implement the joint motion which would combine rapid analysis with a reliable clinical application is still not known.

The greatest limitation in these models is related to the verification of the simulation results. Typically, FE modeling studies verify the results against literature values, which only reveal whether the simulation results are within physiological limits. An alternative is to compare simulation results against follow-up information from radiological scans. Due to better soft tissue contrast in MRI than from radiological scans, improved assessment methods have been developed. Semi-quantitative methods, such as Whole-Organ Magnetic Resonance Imaging Score (WORMS), can be used to evaluate structural alterations in the knee joint tissues by assigning a score based on established criteria. Quantitative methods rely on the quantification of tissue-specific MRI parameters, such asT2or T1ρrelaxation times.

The goals of this thesis were to: (1) develop a method for fast and reliable motion implementation in FE models; (2) develop a methodology to identify patients with ACLR at high-risk of developing knee OA; (3) verify FE model results against local compositional changes, evaluated using changes in T2 or T1ρ relaxation times; (4) develop a mechanobiological model to predict the effects of focal cartilage defects

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in patients with ACLR on the development of OA.

In this thesis, subject-specific FE models were created. The knee joint geometry was obtained by manually segmenting the soft tissues from MRIs. The knee joint motion was obtained from motion capture or in combination with musculoskeletal modeling. First, we created four FE models for the same patient to compare different methodologies of implementing the motion; namely, using kinetics (forces and moments) or kinetic-kinematics (forces and angles). Two models used a more complex geometry, which includes the patello-femoral joint, and two used a simplified geometry. We then identified the method most-suitable for use in a clinical setting. Second, we generated subject-specific FE models for seven patients with ACLR and six controls using the previously developed methodology. Then we investigated two potential mechanisms for OA onset and development: collagen network degeneration due to excessive maximum principal stresses and proteoglycan (PG) loss as a result of excessive absolute maximum shear strains. We verified the FE model predictions against local changes in collagen-sensitive T₂ relaxation times and proteoglycan-sensitive T_1ρ relaxation times. Third, we qualitatively compared the collagen-specific and proteoglycan-specific parameters.

We also quantitatively assessed the relationship between the simulated biomechanical and MRI parameters. Fourth, we investigated the effect of cartilage defects on the onset and development OA, as a result of fixed charge density (FCD) content changes around lesions. To achieve this a cartilage degeneration algorithm controlled by either absolute maximum shear strains, deviatoric strains or fluid velocity was used, and was based on a previous study. The predictions of the mechanobiological model were compared against longitudinal changes in T2 and T1ρrelaxation times.

The results of this thesis indicated that all motion implementation approaches produced similar articular cartilage mechanical responses. The total modeling time was almost 10 times shorter in the kinetic-kinematic approach with a simpler geometry than the kinetic approach with the more complex geometry. Therefore, the kinetic-kinematic approach would be more suitable in clinical settings. The identified locations and volumes at risk of collagen network degeneration (element exceeding maximum principal stress threshold) matched locations with large changes in collagen-sensitive T2 relaxation time. The predicted and measured degenerated volumes were significantly correlated (r= 0.711). There was also a strong positive correlation between the maximum principal stresses and the change inT2relaxation time (r=0.649). This suggests that maximum principal stresses, at a compartment level, are indicative of the risk for collagen degeneration. The locations and volumes susceptible to PG loss identified by the FE models and measured from T1ρ information did not match. We did not find any significant correlation between the absolute shear strain and the change inT1ρrelaxation time.

These results indicated that at a compartment level, absolute maximum shear strain does not indicate PG loss or thatT_1ρ does not reflect PGs very well. However, the mechanobiological model was able to predict FCD loss around cartilage defects as a function of time due to either absolute maximum shear strains, deviatoric strains or fluid velocity. Moreover, the size and location of the defect may directly contribute to cartilage degeneration through a decrease in the FCD content around the lesion. Therefore, highly localized strain levels indicate highly localized cell death and PG loss, even though global strain levels do not change.

In conclusion, the methodology presented in this thesis could be used to postoperatively assess patients at risk of biomechanically driven osteoarthritis after

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surgical interventions, such as ACLR. The modeling methodology presented in this thesis could be used to evaluate optimal non-surgical options for avoiding or delaying the onset and development knee OA and could provide a pathway towards clinical application.

Universal Decimal Classification:53.084.85, 535.3, 535.4, 681.7.02 OCIS codes:050.1960, 130.4815

Medical Subject Headings: Anterior Cruciate Ligament Reconstruction, Anterior Cruciate Ligament Rupture, Cartilage, Articular; Collagen; Computer Simulation;

Diagnostic Imaging; Finite Element Analysis; Gait Analysis; Knee Injuries; Knee Joint;

Magnetic Resonance Imaging; Models, Theoretical; Osteoarthritis; Stress, Mechanical;

Verification

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ACKNOWLEDGMENTS

This study was carried out during 2016-2021 in the Department of Applied Physics at the University of Eastern Finland. The study was financially supported by the Doctoral Programme in Science, Technology and Computing (SCITECO), in the Department of Applied Physics, as well as research grants from the Academy of Finland, the European Research Council, the National Institutes of Health. I would like to thank the Sigrid Juselius and Orion Research foundations for financial support. CSC-IT Center for Science Ltd, Finland is acknowledged for it’s excellence in finite-element modeling software.

First and foremost, I would like to express my sincere gratitude to my supervisors Professor Rami Korhonen, Professor Juha Töyäs and Adjunct Professor Mika Mononen for their guidance and support during this project. Especially, I would like to thank Rami for creating a great working environment. It has been an honour to work with him in the last 4 years and I am grateful that he shared his vast expertise in biomechanics with me. I am deeply grateful for Juha, whose enthusiasm for science and research as well as work ethics have been an inspiration. Thank you Rami and Juha for hosting parties that have helped me feel a part of the BBC. A special thanks to Mika for showing me the secrets of knee joint finite element modeling. Without your help, the numerous battles with the models would have been lost.

I wish to thank the official reviewers Professor Ruth Wilcox and Professor Marcus Pandy, for providing constructive criticisms and suggestions on the thesis, that have helped create a more coherent narrative. I’d also like to thank Ewen MacDonald for linguistic review of my thesis.

I would like to express my deepest gratitude to all the co-authors for their valuable work and critical review of the manuscripts. In particular I would like to thank the people at UCSF and Cleveland Clinic for their exceptional professionalism. Special thanks to Mikko Nissi for offering his expertise with MRI.

I warmly thank my roommates Gustavo, Atte, Amir, Elvis and Ari for making our office more entertaining. Many thanks to all current and emeritus members of the Biophysics of Bone and Cartilage group and to all people with whom I have had the opportunity to interact with. You all have created an outstanding work environment and it has been a privilege to work with such a devoted and professional group. A very warm thank you to Marlitt and Lukas for your continual friendship.

I wish to express my most sincere thanks to my parents Adriana and Adrian for your physical, financial and moral support during my long years in education.

Finally, I owe my deepest gratitude to Defne for her love, patience and understanding during this journey. Also, thank you for finding me this opportunity and sorry for dragging you around. It is a privilege to have you by my side.

Kuopio, January 20, 2021 Paul Octavian Bolcos

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LIST OF PUBLICATIONS

This thesis consists of the review of the author’s work in the field of computational modeling of articular cartilage in the knee joint and the following selection of the author’s publications:

I P.O. Bolcos, M.E. Mononen, A. Mohammadi , M. Tanaka, M. Samaan, X. Li, R. Souza, J.S. Jurvelin, J. Töyräs, R.K. Korhonen, "Comparison between kinetic and kinetic-kinematic driven knee joint models,"Sci. Rep.,8, 17351 (2018) II P.O. Bolcos, M.E. Mononen , M. Tanaka, M. Yang, J-S. Suomalainen, M.J.

Nissi, J.Töyräs, X. Li, B. Ma, , R.K. Korhonen, "Identification of locations susceptible to osteoarthritis in patients with anterior cruciate ligament reconstruction: Combining knee joint computational modeling with follow-up T1ρand T2 imaging",Clin. Biomech.,79, 104844 (2020).

III P.O. Bolcos, M.E. Mononen, Koren E. Roach, M. Tanaka, J-S. Suomalainen, S.

Mikkonen, M.J. Nissi, J. Töyräs, T.M. Link, R. Souza, S. Majumdar, B. Ma, X. Li, R.K. Korhonen, "Subject-specific biomechanical analysis to estimate locations susceptible to osteoarthritis - finite element modeling and MRI follow-up of ACL reconstructed patients", submitted.

IV G.A. Orozco,P.O. Bolcos, A. Mohammadi, M.S. Tanaka, M. Yang, T.M. Link, B.

Ma, X. Li, P. Tanska, R. K. Korhonen, "Prediction of local fixed charge density loss in cartilage following ACL injury and reconstruction: A computational proof-of-concept study with MRI follow-up", J. Orthop. Res., article in press, (2020).

Throughout the overview, these papers will be referred to by Roman numerals.

AUTHOR’S CONTRIBUTION

The publications in this dissertation are original research articles on the development of subject-specific finite element models for rapid assessment of knee joint osteoarthritis susceptibility in patients with anterior crucial ligament reconstruction. The ideas behind studiesI -III originated in discussions between the author and supervisors. The ideas behind study IV developed between the main author and co-authors during weekly meetings. The author has contributed to each method presented in the publications, except for the magnetic resonance imaging and motion capture acquisition. The author carried out all of the simulations, analyses and writing related to studiesI-III. In studyIV, the author collaborated directly with the main author of the study in the simulation and analysis aspects and provided critical review of the submitted manuscript. In all studies the contribution of co-authors has been significant.

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1 Introduction

The knee is the largest synovial joint in the body. It contains two main compartments: the tibiofemoral compartment, connecting the tibia and the femur, and the patellofemoral compartment, connecting the patella and femur. These compartments act as a hinge mechanism, allowing movement of the lower leg relative to the femur. The joint-forming surfaces are covered with a layer of articular cartilage, which provides near frictionless movement between the bones and plays an important role in transferring and distributing loads within the joint [1].

Osteoarthritis (OA) is a common degenerative joint disease [1, 2]. This long-term chronic disease commonly affects the knee and hip joint, but can also be present in hands, feet or spine [1, 3, 4]. OA is mainly characterized by deterioration of articular cartilage. Once OA has developed, the articular cartilage continues to wear away, leading to bone-on-bone contact, which results in joint stiffness, pain and impaired movement [1–4]. These symptoms decrease the patient’s quality of life due to an inability to cope with daily activities, causing social isolation, a decrease in their ability to work and/or depression [2, 4, 5].

The major factors that increase the risk for the onset and development of OA are joint injury, age, gender and obesity [2, 4, 5]. In particular, knee joint injuries, such as anterior cruciate ligament (ACL) rupture, increase the risks of OA onset and development even in young adults [6–12]. Furthermore, studies reported that even knee joint surgeries, such as ACL reconstruction (ACLR), are not effective in preventing OA [8,13,14]. One of the mechanisms leading to OA in both ACL rupture and patients with ACLR may be altered joint biomechanics, leading to excessive stresses and strains experienced by articular cartilage [15–17].

The World Health Organisation (WHO) ranks OA as the leading cause of disability, with over 250 million cases worldwide in 2010 alone [4]. The prevalence of OA is expected to increase due to several factors: population growth, aging as a result of increased life expectancy and the increased prevalence of obesity [2, 4, 18].

OA poses a huge economic burden on societies. Recent estimates place the total costs for hip and knee OA per patient at around $12 thousand per year [18]. In 2013, the total cost for OA were near 1 billion EUR in Finland [19] and more than

$303 billion in US [20]. Despite research efforts, there is no cure for OA and thus by far the best and most cost effective treatment option would be prevention.

Traditional methods for OA diagnosis are based on evaluating patient symptoms and traditional radiography [21]. They rely on evaluating joint space narrowing and the presence of osteophytes. Traditional radiography, however, is unable to offer sufficient information on soft tissue integrity and composition.

Magnetic resonance (MR) images have been used to evaluate structural changes and compositional changes in articular cartilage [22–32] and improve diagnosis.

However, MR imaging cannot assess altered joint biomechanics or excessive joint and tissue loading.

Finite element (FE) modeling is a computational method that can be used to evaluate the effects of altered knee biomechanics and simulate joint and local tissue

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loading. High maximum principal stresses have been linked with collagen matrix degeneration in articular cartilage [33–36]. High tissue deformations have been linked with proteoglycan (PG) loss. This has enabled the use of FE modelings in assessing personalized biomechanical risks for the onset and development of knee OA [37–40]. To generate subject-specific FE models, the knee joint geometry is obtained from medical images, while the subject motion is obtained from motion capture techniques. However, it is unclear whether simpler models in terms of knee geometry and motion implementation can produce the same results as more complex FE models in terms of contact mechanics and mechanical response of articular cartilage. Finally, to increase the trustworthiness of the generated FE models, the simulated results need to be verified against subject follow-up data.

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2 Knee joint

In synovial joints, smooth articular cartilage covers the surface of bones in contact with each other, allowing for increased joint mobility and frictionless motion between articulating bones. The knee joint is one of the largest and most complex synovial joints in the human body. It consists of two articulating joints, namely the tibio-femoral and patello-femoral. The main components of the knee joint are bones (tibia, femur, patella), articular cartilage (femoral -, tibial - and patellar cartilage), menisci (medial and lateral), ligaments, tendons and muscles (see Fig.

2.1). The bones form the body of the knee joint. Articular cartilage distributes and transmits forces across the joint, minimizing stress and strain concentrations within the joint. Together with menisci and synovial fluid, articular cartilage provides almost frictionless contact between joint surfaces. Ligaments attach one articulating bone to the other across the joint. Their role is to restrict and guide the joint movement and ensure joint stability. Tendons are dense fibrous tissues that connect muscles to bones, transmitting the force of its associated muscle to the bone [1, 41, 42]. The main soft tissues used inStudies I-IV are presented in more details below.

2.1 ARTICULAR CARTILAGE

Articular cartilage is a thin, avascular, heterogeneous, fibrous connective tissue covering the articulating surfaces of bones. Typically, the thickness of articular cartilage is∼1.5-5 mm in healthy human joints. The thickness varies depending on location (tibia, femur or patella) and site (anterior, middle or posterior) [43–45].

Cartilage is thickest at the contact regions (i.e. high load areas) [46] and thinnest in regions without contact [45, 47]. The structure and functional properties of articular cartilage are highly depth-dependent. Articular cartilage can be divided in four zones: superficial, middle, deep and calcified cartilage zone [1, 48].

2.1.1 Structure and composition

Articular cartilage consists of cells called chondrocytes and a multicomponent extracellular matrix (ECM), composed of a fluid and a solid matrix [1]. The fluid matrix (i.e. interstitial fluid) is the most abundant component of cartilage accounting for ∼80% of its wet weight. The fluid content linearly decreases from the superficial zone to the deep zone [49–52]. Most of the fluid and ions can flow freely in and out of the tissues; this process is important in transporting and distributing nutrients to the cells [53]. The solid matrix consists of collagen and proteoglycans.Collagenis a rod-shaped protein contributing from 15 to 22% of the wet weight of cartilage [1, 41]. Articular cartilage contains mainly type II collagen (90-95%) as well as other types (e.g. types VI, IX, XI, XII or XIV) [54]. Type II collagen forms an organized network, helping to create and stabilize the network.

The collagen fibrils have an arcade-like orientation throughout cartilage depth. In the superficial zone, the collagen fibrils are densely packed and run in parallel to

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Figure 2.1: Schematic illustrations of the main components of the knee joint from:

a) anterior view in the coronal plane; b) cross-sectional view in the sagittal plane;

and c) axial view of the menisci and tibia.

the joint surface. In the middle zone, the fibrils start to turn towards the bone, while in the deep zone and calcified cartilage zone they are perpendicular to the bone [55]. Proteoglycans (PGs) consist of a core protein covalently bonded to glycosaminoglycan (GAG) side chains, contributing to∼5-10% of the wet weight of cartilage [1, 41]. The PG content increases from the superficial zone to the deep zone [56, 57]. The most prominent PG in cartilage is negatively charged aggrecan.

Aggrecan is embedded in the collagen mesh causing fixed charge density (FCD) in cartilage. This induces fluid flow to the tissues through increased osmotic pressure.

The increased fluid flow results in tissue swelling, which is resisted by the collagen network [1, 58].The cells(i.e. chondrocytes) develop, maintain and repair the ECM by producing collagen and PG [1]. The shape, function and metabolic activity of chondrocytes depend on the location in cartilage. In the superficial zone the cells are flat and parallel to the surface, while in the deep zone they are spherical and oriented in columns perpendicular to the surface. Since the chondrocyte content is

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∼1% of the total volume in humans, cartilage regeneration and repair is slow [59–61].

2.1.2 Biomechanical properties

Articular cartilage has to withstand various loading conditions during normal daily activities. The ability to withstand these conditions is attributable to the poroelastic properties of cartilage [1, 62, 63]. The biomechanical properties of cartilage are determined by its major constituents: interstitial fluid, collagen and PGs; each having its own properties, functions and characteristics [58, 64].

Additionally, the mechanical response of articular cartilage is highly depth-dependent as a result of the depth-dependent variations in articular cartilage structure and composition [57, 65, 66].

The permeability of cartilage, and thus the fluid flow, is determined by the fluid content, PGs and collagen network [48–50, 67–69]. Fluid flow has a significant role in the impact- and dynamic-loading properties of cartilage [70, 71]. During impact loading, the fluid flow out of the tissue is minimal, due to very low permeability and incompressibility of cartilage . In static (creep) load, fluid flows out of the tissue, resulting in higher tissue deformations. When unloaded, the fluid will flow back into the tissue. Thus, the interstitial fluid governs the articular cartilage viscoelastic stress-relaxation and creep behavior [1, 70, 72, 73].

The tensile, dynamic and shear properties of cartilage are primarily determined by the collagen network [65, 74, 75]. The collagen fibrils resist tissue deformations in the direction of the fibrils [76]. The tensile stiffness is highest at the superficial zone, since the fibrils are oriented parallel to the surface [65]. The orientation of collagen fibrils in the middle zone is optimal for resisting shear forces, resulting in a shear modulus distinctly higher than in the superficial zone [65]. In the deep zone, the vertical orientation of the fibrils help the tissue resist cartilage swelling and protect it from high tensile and shear strains, especially at the cartilage-bone interface [74].

The PGs primarily determine the compressive stiffness of cartilage tissue in a mechanical equilibrium [63, 74]. When cartilage is compressed, PGs become more densely packed, resulting in higher FCD, which then leads to increased osmotic pressure and swelling. The swelling pressure from the FCD allows the collagen network to withstand tensile loads by pre-stressing the collagen fibril network.

2.2 MENISCUS

Menisci are two crescent-shaped fibrocartilage tissues located between the femoral and tibial articular cartilage. Their wedge-like shape is adjusted to match the femoral and tibial surfaces. Both meniscal horns insert into the tibial bone in the intercondylar area, via insertion ligaments. The lateral meniscus is almost circular and covers a larger area of the lateral tibial plateau, while the medial meniscus is semicircular and covers ∼60% of the medial tibial plateau. The main role of menisci is to distribute the knee joint load, thus decreasing articular cartilage stresses. Further, they contribute to joint stability [1, 77–80].

In terms of composition, menisci contain ECM and cells, similar to articular cartilage [1, 77]. The ECM is composed of water (∼60-70% wet weight), collagen (∼10-15% wet weight) and PGs (∼1-2% wet weight) [1, 78]. Menisci contains mainly collagen type I, as well as other types (e.g. types II, III, IV, VI or XVIII). The collagen fibres are oriented mainly in the circumferential direction, resulting in a high tensile

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strength in that direction. Further vertical compressive loads are transformed into circumferential "hoop" stresses [79, 81]. Most of the PGs are located in the inner two-thirds of the menisci, and play a role in increasing the compressive strength of the menisci [82]. Despite being fully vascularized at birth, only∼10-30% of the peripheral menisci have blood vessels and nerves. Thus, only this peripheral area of the menisci has healing capabilities [1, 83, 84].

2.3 LIGAMENTS

Ligaments are connective tissues mainly consisting of collagen fibres [1, 77]. They are bone-to-bone connections that contribute to load transfer, guiding joint motion and provide joint stability [1, 77]. The primary ligaments of the knee joint are the cruciate and collateral ligaments. The anterior cruciate ligament (ACL) connects the inner surface of the lateral femoral condyle to the medial part of the anterior intercondylar eminence of tibia. The posterior cruciate ligament (PCL) connects the posterior part of the intercondylar notch of the medial femur to the posterior intercondylar eminence of the tibia. The medial collateral ligament (MCL) connects the medial femoral epicondyle to the medial tibia∼5 cm below the joint line. The lateral collateral ligament (LCL) connect the lateral femoral epicondyle to the head of the fibula.

The ACL primarily restricts anterior translations and internal rotations of the tibia with respect to the femur, while the PCL restricts posterior translations and external rotations [1, 85]. The dynamic stability of the knee joint is provided by the interaction between ACL and PCL. MCL and LCL primarily restrict valgus and varus rotations respectively [42, 51, 86].

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3 Knee osteoarthritis

Knee OA is characterized by a progressive loss of articular cartilage from joint surfaces, subchondral bone cysts and osteophyte formation. OA affects all synovial joint tissues: cartilage, menisci, ligaments, synovium, muscles, bones and causes joint pain during motion. Typically, OA is seen in the foot, knee, hip, spine or hand joints, but can affect all synovial joints.

Some of the first signs of OA are collagen network fibrillation, elevated water content, and a reduction of PG content starting initially from the superficial zone.

These compositional and structural changes lead to increased permeability and decreased cartilage stiffness. As the disease progresses, the cartilage surface starts to degrade, which leads to the presence of cartilage fragments in the joint space. At the end stage of OA, the articular cartilage surface has become fully degraded, causing bone-on-bone contact. This leads to severe pain and joint stiffness.

3.1 RISK FACTORS FOR OA

There are several risk factors that increase the likelihood of knee OA. A brief overview of some of the risk factors and their relationship to knee OA is presented below.

Aging. There is increased incidence of OA in subjects over 50 years. With age, there are morphological and biomechanical changes in articular cartilage, such as decreased collagen concentration, shorter protein cores and increased water and PG content [1, 77, 87]. In addition to alterations in articular cartilage, with age there are changes in muscle (sarcopenia), bone (osteopenia, osteoporosis), fat (increased fat deposits) and the nervous system (altered proprioception). These all play a role in the development of knee joint OA [88].

Gender. There are studies reporting markedly higher incidence of OA in female subjects, especially when associated with age [89, 90], weight [91] and knee injuries [12, 92]. Moreover, there are gender-based gait differences in osteoarthritic knee joints [93, 94].

Knee alignment. In theory, a shift from a neutral alignment of the knee affects the load distribution. Some studies identified knee malalignment as a potential predictor of OA [95–99]. However it is still unknown whether malalignment precedes the development of OA or it is a direct result of cartilage loss [100, 101].

To achieve a neutral alignment in patients with knee malalignment, tibial osteotomy is typically performed [102, 103]. In tibial osteotomy, a wedge of tibial bone is surgically removed. However, several factors such as gender, surgeon experience, pain medication intake and postoperative complications greatly affect the individual’s susceptibility to OA onset and development [104–106].

Disuse. As a result of chronic bed rest or sedentary life, there is a drastic change in articular cartilage, bones and muscles. In articular cartilage, there is a reduction in the amount, size and synthesis of PGs, as well as an increase in cartilage surface fibrillation [77]. In weight-bearing bones, there are substantial changes in bone geometry and composition [77]. As a result of disuse, the muscles

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undergo a rapid and significant atrophy with no change in muscle fibre numbers [77]. When combined together, these changes can lead to abnormal joint biomechanics and stresses and strains experienced by articular cartilage [15–17, 107].

Weight. A high body mass index (BMI) and obesity significantly increase the risk of developing OA [108–110]. Excessive impact and cumulative loads, decreased muscle strength and altered biomechanics may promote structural joint damage [111]. Moreover, obesity can trigger metabolic responses that inhibit matrix synthesis and induce cartilage degeneration [112]. The primary and most effective intervention for obese OA patients is weight loss. It was shown that a reduction of 2 units of BMI could decrease the probability of developing knee OA by 50%.

Weight loss has been shown to reduce both the extent of inflammation and the magnitude of knee joint loads [113].

3.1.1 Traumatic joint injuries and surgery

Traumatic knee joint injuries include meniscal tears and cruciate and collateral ligament ruptures. These injuries are typically seen in sports activities that involve jumping, pivoting or cutting, such as football, basketball or downhill skiing.

Treatment. Since the ACL plays and important role in knee joint stability, ACL injuries are commonly treated conservatively or surgically. In the conservative treatment, several weeks of rehabilitative therapy (exercises) are performed under the supervision of a physical therapist. The goal of rehabilitative therapy is to reduce pain, swelling, strengthen muscles and restore the knee joint’s full range of motion. In the surgical treatment, the damaged ACL is replaced with a tendon graft, which can be harvested from another part of the patient knee (autograft) or from a deceased donor (allograft). Typical tendon grafts used in ACLR are from the patellar, quadriceps or Achilles tendons. To accurately position the tendon graft, tunnels are drilled into the femur and tibia. The graft is then secured to the bone using fixation devices, such as screws. ACLR surgery is usually followed by progressive physical therapy to strengthen the muscles and improve flexibility [12, 114–116]. The aim of surgical treatment is to restore the knee joint stability and full range of motion, as the ACL resists anterior translations and internal rotations of the tibia.

There are two main treatment options for meniscal tears, depending on their severity, size and location: conservative and surgical. Conservative treatment is prescribed when the damaged tissue is small and is located on the outside edge of the meniscus. Conservative treatment techniques often include resting, ice (cold packs several times a day), using compression bandages and keeping the leg elevated to reduce additional swelling. The surgical treatment of meniscal injuries is typically done using arthroscopic surgery. Miniature surgical instruments and a miniature camera are inserted through small incisions in the knee. Then the damaged tissues are either trimmed away (partial meniscectomy) or repaired using sutures (meniscus repair).

Relationship with OA. There is an increased risk for the onset and development of knee OA as a result of meniscal tears [117–121] or ACL ruptures [122–124]. Furthermore, ACL injuries are often accompanied by meniscal tears, subchondral bone or cartilage damage, which further increase the susceptibility of OA [6, 7, 9, 114, 125–128]. It is unclear if conservative treatment of ACL ruptures is more suitable than surgical treatment, with several studies

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showing little difference in OA susceptibility between the two [12, 115, 116].

Postoperatively, the incidence of knee OA was notably higher in patients with ACLR in both long-term [9, 11, 128–130] and short-term follow-ups [8, 14, 126] than in healthy controls. For meniscal tears, both conservative and surgical treatment options lead to increased OA susceptibility [119]. Postoperatively, the incidence of knee OA was higher in partial meniscectomy patients both in short and long-term follow-ups than in healthy controls [80, 131–136].

The increased susceptibility for OA development in patients with traumatic joint injuries may be due to altered biomechanics. The surgical interventions may not instantaneously stabilize the joint back to normal and may also cause short-term inflammation. For instance several studies report significantly different gait characteristics [92, 94, 137–139] in patients with ACLR than in healthy controls.

For ACLR patients, there are significant differences in both knee joint contact forces [140, 141] and muscle forces [16, 94, 142, 143] even at 1-year post ACLR surgery. This altered joint motion may lead to abnormal stresses and strains experienced by articular cartilage [15–17], which may increase the risk for the onset and development of knee OA.

3.2 RADIOLOGICAL EVALUATION METHODS OF OSTEOARTHRITIS There are several methods for evaluating the severity of knee joint OA (ex.Fig. 3.1).

In this section, an overview of the main methods is briefly presented.

Figure 3.1: Illustrative example of the correspondence between X-Ray radiography [144], semi-quantitative MRI and quantitative MRI methods for healthy and advanced OA patients. Note: Articular cartilage constituents are not in realistic scale, but are used to showcase the difference between healthy and degenerated cartilage.

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3.2.1 Radiographic methods

The level of OA severity can be assessed using conventional radiography (X-Ray) typically in the anterior-posterior direction, which are then classified based on visual inspection by radiologists (Fig. 3.1). The most commonly used classification methods are Kellgren-Lawrence (KL), International Knee Documentation Committee (IKDC), Fairbank and Ahlbäck. These methods focus on joint space narrowing (JSN) and the presence of osteophytes in the entire knee joint. An overview of the KL, IKDC and Fairbank grading systems and classification is provided in Table 3.1. Despite their wide-spread use, the X-Ray based methods have several limitations. These methods are susceptible to both inter- and intraobserver variability [144–148]. Further, they are unable to directly measure cartilage degeneration, since the absorption coefficient of articular cartilage is similar to that of synovial fluid, which renders cartilage invisible in the x-ray images [149]. Changes in articular cartilage composition and integrity precede joint space narrowing or osteophyte formation [35, 64, 150, 151].

Table 3.1: Overview of radiographic grading systems. Note: JSN - Joint Space Narrowing.

System Grade and radiographic interpretation

KL 0: No JSN 1:

Doubtful JSN and possible osteophytes

2: Definite JSN and definite osteophytes

3: Definite JSN, moderate multiple osteophytes, possible bone-end deformity

4: marked JSN, large osteophytes, severe sclerosis, definite bone-end deformity IKDC A: No JSN B: > 4-mm

joint space;

small osteophytes, slight sclerosis, or femoral condyle flattening

C: 2- to 4-mm joint space

D: < 2-mm joint space

-

Fairbank 0: Normal 1:

Squaring of tibial margin

2:

Flattening of femoral condyle, squaring and sclerosis of tibial margin

3: JSN, hypertrophic changes, or both

4: > 75%

JSN with secondary feature

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3.2.2 Magnetic resonance imaging

MRI is based on nuclear magnetic resonance. In the presence of a homogeneous magnetic field (B0), the nuclei of mainly ¹H or ²³Na will begin to precess about the direction ofB0 at Larmor frequency (ω0), and form the net magnetization. If a radio-frequency (RF) pulse is applied, the magnetization will tip away from the direction of the main magnetic field by a specific angle, called flip angle. Tipping the net magnetization also creates longitudinal and transverse vector components of the magnetization (Mzand Mxy, respectively).

After the application of the excitation pulse, the magnetization recovers (i.e relaxation), during which the nuclei exchange energy with each other (T2-, transverse- or spin-spin relaxation) or with their molecular surroundings (T1-, longitudinal- or spin-lattice relaxation) and emit echos. As the relaxation process depends on the chemical and physical environment of nuclei, the relaxation time is specific to individual tissues and fluids in the human body. MRI provides excellent soft tissue contrast. For this reason, various semi-quantitative and quantitative methods have been developed to classify knee joint OA based on MR images.

Semi-quantitative methods

There are three common semi-quantitative scoring methods: Whole-Organ Magnetic Resonance Imaging Score (WORMS) [22], Boston-Leeds Osteoarthritis Knee Score (BLOKS) [25] and MRI OsteoArthritis Knee Score (MOAKS) [152]. In each method, regional abnormalities in various soft tissues, such as cartilage, menisci and bones are assessed. Even though there are some differences in the definition and scoring of regions, all methods evaluate features present in OA: depth and extent of cartilage loss, cartilage defects, meniscal defects and subchondral bone marrow lesions.

Figure 3.2: Regional subdivisions of articular cartilage surfaces according to WORMS [22] system (left) and MOAKS [25] system (right) for the lateral joint compartment.

In the WORMS scoring system (seeFig. 3.1-left), the articular cartilage surfaces are divided as follows: femoral, tibial and patellar cartilage. For femoral and tibial cartilage, a further subdivision into anterior, central and posterior regions is done (Fig. 3.2-left). The patellar cartilage is divided into medial and lateral regions.

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Then each region is scored using an eight-point scoring system. A similar regional subdivision is done in MOAKS (Fig. 3.2-right); however, a four-point scoring system is used. An illustrative example of different categories of WORMS, MOAKS and BLOKS scoring systems is shown inFig. 3.3.

Figure 3.3: Schematic diagram of the correspondence between categories from WORMS, MOAKS and BLOKS for describing articular cartilage loss [152]

In cohort studies with follow-up information, these methods may reveal longitudinal structural changes in the knee joint. Nonetheless, they are still susceptible to misclassification and inter- and intra-observer variability [29, 153].

and still offer insufficient information on the composition of soft-tissues.

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Quantitative methods

T2 mapping. In the presence of an external magnetic field along the z-direction, a net magnetization forms in the z-direction, which is known as the longitudinal magnetization. When a RF-pulse is applied, this magnetization is tipped to the xy- plane or transverse plane. TheT2 relaxation time (spin-spin relaxation ortransverse relaxation) is the process by which these transverse components of the magnetization decay. The process occurs at an exponential decay rate.

By acquiring several MR images at different echo times (TE) and then fitting an exponential signal decay function (S(TE)) to the signal intensity of each pixel, the T2relaxation time can be quantified:

S(TE)∝e^−TE/T² (3.1) The signal intensity ofT2-weighted MR images varies with cartilage tissue depth [152, 154–159]. Although this behavior has been associated with fluid content or PG content [160], it is mainly indicative of the orientation of collagen fibrils in articular cartilage [154,161,162]. In healthy articular cartilage, typical values forT2relaxation times are ∼20-40ms in the superficial zone, ∼50ms in the middle zone and 10- 20ms in the deep zone [163–167]. Therefore it is possible to evaluate the collagen architecture through depth-wise T2 relaxation time profiles throughout the tissue depth [162,166]. Importantly, local increases inT2relaxation times (seeFig. 3.1) have been associated with degeneration of the cartilage matrix, due to collagen fibrillation and increased fluid content [14, 24, 27, 156, 157, 168, 169]. Further,T₂relaxation times have also been associated with the mechanical properties of cartilage [167].

T1ρmapping. Relaxation process in a rotating frame of reference can be created by applying a continuous RF pulse or spin-lock pulse. This pulse locks the net magnetization in the transverse plane [31, 155, 157, 163, 170]. T_1ρ describes the longitudinal relaxation process while the spin-lock pulse is kepton. Similar to the T₂ relaxation time, the T_1ρ is directly proportional to the final image signal intensity. By applying several RF pulses at different spin-lock times (TSL) and then fitting an exponential signal decay function (S(TSL)) to the signal intensity of each pixel, theT_1ρrelaxation time can be quantified:

S(TSL)_∝e^−TSL/T^1ρ (3.2) Even though the T_1ρ relaxation time carries some sensitivity to collagen [162, 171, 172], it is mainly indicative of PG content [27, 31, 156, 170, 173–176]. It has also been postulated thatT_1ρmay be indicative on general cartilage state, rather than any particular constituent [177]. In addition, theT_1ρ relaxation time is reportedly more sensitive to cartilage degeneration thanT2[27,178]. Moreover, theT2relaxation time is strongly dependent on the orientation of articular cartilage with respect to the main magnetic field [162].

Combined T₂ andT_1ρ mapping. Recently, an MRI sequence for simultaneous acquisition ofT₂andT_1ρrelaxation times was introduced [163]. Using this sequence, differences in articular cartilage [179], bone [180] and meniscus [26] between ACLR patients and healthy controls could be quantified. BothT2and T_1ρrelaxation times were significantly correlated with patient reported outcomes following ACLR [181]

and also matched structural changes quantified using WORMS grading [26, 179–

181].

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3.2.3 X-ray computed tomography

X-ray computed tomography (CT) is an imaging method where a series of X-rays are acquired at different angles. Using image reconstruction algorithms, cross-sectional views of the scanned object can be obtained. In contrast to conventional 2D X-ray imaging, this method eliminates the superposition of distinct anatomical structures and yields a much higher contrast compared to conventional radiography. Moreover, the scanned object can be visualized in all cross-sectional planes. In contrast to MRI, the signal values depend on the characteristic attenuation coefficient of the material [149, 182] and is expressed as:

I(x) =I0e^−µx (3.3)

where: I0 - initial radiation intensity; x - distance traveled in the medium; µ- the linear attenuation coefficient. The image voxels represent the average attenuation of the tissue at that volume element. Conventionally they are reported in Hounsfield Units (HU). The relationship between the attenuation coefficient and the corresponding HU value is given by:

HU=₁₀₀₀× ^µ−µwater

µ_water−µ_air (3.4)

where: µwater and µ_air - attenuation coefficients of water and air, respectively. By definition, water has an HU value of 0, and air has an HU value of -1000 [149]. Due to their composition, the attenuation values of most soft tissues are within a narrow range (30-100 HU). Because of this, contrast agents have been used to improve the differentiation between soft tissues and the adjacent physiological fluids. Recently, contrast enhanced CT methods have been used to detect early articular cartilage degeneration as a result of PG loss and increased water content. These methods rely on simultaneously using two contrast agents to enhance CT contrast and achieve a quantification of PG loss: 1) a non-ionic gadolinium-based contrast agent and 2) a iodine based cationic agent. Imaging is done at two time points (e.g. immediately after contrast agent injection and 45 minutes later). The non-ionic agent diffuses freely into articular cartilage and is distributed into cartilage in relation to the water content of the tissue, and is unaffected by the negatively charged PGs. The cationic contrast agent has a high affinity for the negatively charged GAG chains, thus it is distributed in proportion to the PG content [183–186].

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4 Finite element modeling of knee joint soft tissues

Finite element (FE) modeling is a computational method, which can simulate the knee joint mechanics using complex materials and can directly estimate stresses and strains within the knee joint. The typical workflow for creating FE models can be divided into six parts: 1) geometry creation, 2) mesh creation, 3) defining material properties,4)defining boundary and loading conditions and5)simulating the model and6)verifying the model results.

4.1 GEOMETRY AND MESHING

Nowadays, the 3D knee joint geometry is obtained from clinical MRI or CT images.

Traditionally this is done by manually segmenting the soft tissues of interest using image processing software. After segmentation, the resulting soft tissue geometries are post-processed. This step is necessary in order to remove sharp edges or inaccuracies. The geometries are smoothed to improve the contact behavior in the FE model between the tissues and the mesh density is reduced. Then the segmented geometries are meshed using third party software or built-in functions in the image processing software. In 3D FE models, tetrahedral or hexahedral elements can be used. Although tetrahedral meshing is more straightforward, hexahedral elements allow for feasible contact mechanics using lesser elements compared to tetrahedral elements and are typically preferred in contact modeling (Fig. 4.1).

Figure 4.1: Example of tetrahedral and hexahedral options for meshing the medial tibial cartilage. The same element size of 0.5 mm is used in both cases.

4.2 MATERIAL PROPERTIES

In FE modeling, several material formulations have been used to characterize soft and hard tissues (Table 4.1). In this section, the theory behind the material

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formulations used in the current thesis is briefly presented.

Table 4.1: Examples of material formulations used in literature to model soft and hard tissues

Tissue Formulations

Articular cartilage Isotropic Poroelastic (IPE) [40, 45, 187–190], Transversely Isotropic Poroelastic (TIPE) [1, 67, 190, 191], fibril-reinforced poroviscoelastic (FRPVE) [39, 67, 190, 192–197], FRPVE with swelling [64, 198–203]

Menisci Isotropic Elastic [45,187–189], Transversely Isotropic Elastic (TIE) [39, 40, 190, 195], Fibril-reinforced Poroelastic (FRPE) [150, 193, 194, 203]

Ligaments Springs [204–211], Hyperelastic [40], Fibril-reinforced [188, 204, 211]

Bones Rigid [39, 64, 187, 190–193, 195–199, 202, 204–208, 212, 213], Homogeneous Isotropic Elastic [40, 214, 215], Inhomogeneous Isotropic Elastic [215, 216]

4.2.1 Biphasic theory

Biphasic and poroelastic theories separate the solid and fluid phases [72, 217].

Abaqus (the FE simulation software used in the current thesis) utilizes a poroelastic theory, with the distinction from the biphasic theory that the solid phase contains a continuous distribution of pores, but the results produced by these two theories are the same. The total stress (σt) is expressed as the sum of the solid matrix (σs) and the fluid matrix (σ_{f l}) [72, 218, 219]. The constitutive equation, i.e. stress-strain relation, for the entire tissue is given by:

σt=σs+σ_{f l}=−φspI+σ_{e f f}+ (−φ_{f l}pI)

=−(φs+φ_{f l})pI+σ_{e f f}

(4.1) where: φs - solid volume fraction;φ_{f l} - fluid volume fraction; p - pore pressure;I- unit tensor;σ_{e f f} - effective solid stress. Asφ_{f l}+φs=_1,_{Eq. 4.1}reduces to:

σt=σ_{e f f}−pI (4.2)

For linear elastic materials, the effective solid stress tensor (σe f f) can be written as:

σ_{e f f} =C·ε (4.3)

where: C- stiffness matrix;ε- total elastic strain tensor. By reorderingEq. 4.3, the elasticity tensor (D) can be found:

D·σ_{e f f} =C⁻¹σ_{e f f} =ε (4.4)

The fluid flow inside the cartilage matrix is modelled with Darcy’s law [219], in which the fluid velocity is described by a constant permeability (k):

q=−k∇p (4.5)

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where: q - rate of fluid flow; ∇p - hydraulic pressure gradient. The void ratio (e) of the material is the fluid (φ_{f l}) to solid (φs) ratio and can be expressed in terms of porosity (n):

e= ⁿ

1−n (4.6)

FromEqs. 4.5and 4.6, the fluid velocity (ν_{f l}) through the porous medium can be expressed as:

ν_{f l}=q e+1

e

=−k∇p e+1

e

(4.7) As both fluid and solid phases are incompressible and homogeneous, the continuity equation (balance of mass) is given by:

∇ ·(φsνs+φ_{f l}ν_{f l}) =₀ _(4.8) whereν_s - velocity vector for solid phase; ν_{f l} - velocity vector for fluid phase. The momentum equations for solid and fluid phases, neglecting inertia effects (acceleration = 0), are:

φs∇p+∇σ_{e f f}+K(ν_{f l}−νs) =0 φs∇p−K(ν_{f l}−νs) =0

∇ ·σ_t=0

(4.9)

with the diffusive drag coefficient (K) being related to the permeability (k) by:

k= (φ_{f l})²/K (4.10)

4.2.2 Isotropic material behavior

The simplest form of linear elasticity is the isotropic case. The elastic properties are defined by the Young’s modulus (E) and the Poisson’s ratio (ν). The shear modulus (G) can be expressed asG=E/2(1+ν). In this caseEq. 4.4can be written as:









 ε₁₁ ε₂₂ ε₃₃ γ₁₂ γ₁₃ γ23











=







1/E −ν/E −ν/E 0 0 0

−ν/E 1/E −ν/E 0 0 0

−ν/E −ν/E 1/E 0 0 0

0 0 0 1/G 0 0

0 0 0 0 1/G 0

0 0 0 0 0 1/G















 σ₁₁ σ₂₂ σ₃₃ σ₁₂ σ₁₃ σ23











(4.11)

4.2.3 Orthotropic material behavior

In orthotropic materials, each principal direction (1, 2 and 3) has a corresponding elastic modulus (E1,E2andE3), a Poisson’s ratio (ν1,ν2andν3) and a shear modulus (G1,G2andG3). The stress-strain relationship fromEq. 4.4reduces to:

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







 ε11

ε22

ε₃₃ γ₁₂ γ₁₃ γ₂₃











=







1/E1 −ν21/E2 −ν31/E3 0 0 0

−ν12/E1 1/E2 −ν32/E3 0 0 0

−ν₁₃/E1 −ν₂₃/E2 1/E3 0 0 0

0 0 0 1/G12 0 0

0 0 0 0 1/G13 0

0 0 0 0 0 1/G23















 σ11

σ22

σ₃₃ σ₁₂ σ₁₃ σ₂₃









 (4.12) where ν_ij - is the Poisson’s ratio that characterizes the transverse strain in the j- direction, when the material is stresses in thei-direction andν_ij/E_i =ν_ji/E_j. 4.2.4 Transversely isotropic material behavior

Transverse isotropy is a special subclass of orthotropy characterized by a plane of isotropy at every point in the material. Let the 1-2 plane be the plane of isotropy at every point, E₁ = E₂ = Ep,ν₃₁ = ν₃₂ = νtp,ν₁₃ = ν₂₃ = νpt and G₁₃ = G₂₃ = Gt. ThenEq. 4.4reduces to:









 ε₁₁ ε₂₂ ε₃₃ γ₁₂ γ₁₃ γ23











=







1/Ep −νp/Ep −νtp/Et 0 0 0

−νp/Ep 1/Ep −νtp/Et 0 0 0

−νpt/Ep −νpt/Ep 1/Et 0 0 0

0 0 0 1/Gp 0 0

0 0 0 0 1/Gt 0

0 0 0 0 0 1/Gt















 σ₁₁ σ₂₂ σ₃₃ σ₁₂ σ₁₃ σ23











(4.13)

withνpt/Ep = νtp/Et and Gp = Ep/2(₁+νp). For material stability the following must hold: (1) Ep,Et,Gp,Gt > 0; (2) |νp| < 1; (3) |νpt| < (Ep/Et)^1/2; (4) |νtp| <

(Et/Ep)^1/2; and (5) 1−ν²_p−2νtpνpt−2νpνtpνpt > 0. For instantaneous loading, the transversely isotropic poroelastic material behaves like an incompressible elastic material [220–222], similarly to the isotropic model.

4.2.5 Fibril-reinforced material behavior

The biphasic fibril-reinforced poroelastic (FRPE) material model is currently one of the most advanced material models and takes into account poroelasticity, anisotropy and non-linearity. In this material formulation, the tissue is divided into a fibrillar part, representing the collagen fibrils, and non-fibrillar part, mirroring the porous material (PGs and fluid) [62–64, 67, 221, 221, 222]. The total stress (σt) in this material model can be expressed as:

σt=σ_{n f} +σ_f+pI (4.14)

where: σ_{n f} - non-fibrillar matrix stress;σ_f - fibril network stress; p- fluid pressure;

I- unit tensor.

Non-fibrillar matrix

The non-fibrillar part is typically modeled with the Neo-Hookean hyperelastic model, due to highly non-linear tissue behavior during large

Finite element modeling of anterior cruciate ligament reconstructed knee joints : toward clinical use via fast motion implementation and verification against MRI follow-up

Dissertations in Forestry and Natural Sciences

PAUL OCTAVIAN BOLCOS

Finite Element Modeling of Anterior Cruciate Ligament Reconstructed Knee Joints

Paul Octavian Bolcos

FINITE ELEMENT MODELING OF ANTERIOR CRUCIATE LIGAMENT RECONSTRUCTED KNEE JOINTS.

TOWARD CLINICAL USE VIA FAST MOTION IMPLEMENTATION AND VERIFICATION AGAINST MRI

FOLLOW-UP

TABLE OF CONTENTS

1 Introduction

2 Knee joint

3 Knee osteoarthritis

4 Finite element modeling of knee joint soft tissues