Mechanobiological modeling of articular cartilage : predicting post-traumatic knee osteoarthritis

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Dissertations in Forestry and Natural Sciences

GUSTAVO ALEJANDRO OROZCO GRAJALES

MECHANOBIOLOGICAL MODELING OF ARTICULAR CARTILAGE

PUBLICATIONS OF

THE UNIVERSITY OF EASTERN FINLAND

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Gustavo Alejandro Orozco Grajales

MECHANOBIOLOGICAL MODELING OF ARTICULAR CARTILAGE

PREDICTING POST-TRAUMATIC KNEE OSTEOARTHRITIS

ACADEMIC DISSERTATION

To be presented by permission of the Faculty of Science and Forestry for public examination at the University of Eastern Finland, Kuopio,

on May 22^nd, 2020, at 17 o’clock.

University of Eastern Finland Department of Applied Physics

Kuopio 2020

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

No 377

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Grano Oy Jyväskylä, 2020

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

Distribution:

University of Eastern Finland Library / Sales of publications www.uef.fi/kirjasto

ISBN: 978-952-61-3390-4 (print) ISSNL: 1798-5668

ISSN: 1798-5668 ISBN: 978-952-61-3391-1 (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: gustavo.orozco@uef.fi Supervisors: Professor Rami K. Korhonen

University of Eastern Finland Department of Applied Physics Kuopio, Finland

email: rami.korhonen@uef.fi University Researcher Petri Tanska University of Eastern Finland Department of Applied Physics Kuopio, Finland

email: petri.tanska@uef.fi

Academy Research Fellow Mika E. Mononen University of Eastern Finland

Department of Applied Physics Kuopio, Finland

email: mika.mononen@uef.fi Reviewers: Professor Nele Famaey

Katholieke Universiteit Leuven

Department of Mechanical Engineering Leuven, Belgium

email: nele.famaey@kuleuven.be Professor Noailly Jérôme

Universitat Pompeu Fabra

Departament de Tecnologies de la Informació i les Comunicacions - SIMBIOsys

Barcelona, Spain

email: jerome.noailly@upf.edu Opponent: Professor Yasin Y. Dhaher

Northwestern University

Physical Medicine and Rehabilitation Department 345 East Superior Street

IL 60611, Chicago, USA

email: y-dhaher@northwestern.edu

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Gustavo Alejandro Orozco Grajales

Mechanobiological modeling of articular cartilage: predicting post-traumatic knee osteoarthritis

Kuopio: University of Eastern Finland, 2020 Publications of the University of Eastern Finland

Dissertations in Forestry and Natural Sciences, 2020, 377

ABSTRACT

Articular cartilage is a versatile biological material which provides low-friction and exceptional mechanical properties in the diarthrodial joints. However, traumatic knee joint injuries, such as rupture of the anterior cruciate ligament (ACL), may cause cartilage damage, leading later to changes in tissue composition and eventually to post-traumatic osteoarthritis (PTOA). Specifically, osteoarthritis is characterized by the deterioration of articular cartilage and a decline in the properties of subchondral bone, evoking pain, joint stiffness, and disability. Indeed, this musculoskeletal disease is a leading cause of disability and its incidence continues to increase. As well as the reduction of thickness and tensile strength, another sign of PTOA is the loss of fixed charge density (FCD) of the proteoglycans in cartilage. However, the fundamental mechanisms leading to these local degenerative processes remain unclear.

Imaging techniques such as magnetic resonance imaging (MRI) have been used for detecting alterations in cartilage in patients with early knee OA. Specifically, variations in T^1ρ and T² relaxation times following knee injuries have revealed the importance of detecting cartilage degeneration after ACL injury and its subsequent reconstruction. However, the risk factors for cartilage degeneration in PTOA after trauma and the subsequent effects on the response of the knee tissue are poorly understood.

Although abnormal loading in injured joints has been suggested to be partly responsible for the development of PTOA, cartilage lesions might also contribute to the progression of the cartilage breakdown. Since the inherent limitations to obtain in vivo measurements in clinical trials, finite element (FE) models have been developed to elucidate the biomechanical response of the knee joint tissues under normal and altered loading. Similarly, in vitro mechanobiological models have been shown to be able to simulate adaptation processes in tissue as a function of time.

In this thesis, numerical knee joint models were created for evaluating the effect of different ligament geometrical representation on the biomechanical response of the knee joint, especially on the articular cartilage response during the stance phase of gait. A cartilage degeneration algorithm was also developed and validated based on experimental measurements of FCD content changes in injured cartilage subjected to a moderate dynamic loading. Different degenerative mechanisms such as

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excessive deviatoric and maximum shear strain, as well as fluid velocity, were evaluated to simulate the deterioration of cartilage. Finally, this validated degenerative algorithm was implemented into 3D patient-specific FE knee joint models to simulate cartilage adaptation (alterations in the FCD content) after an ACL injury and subsequent reconstruction during the stance phase of the gait.

The findings in this thesis revealed that compression-tension relationship in knee ligaments due to the fibril-reinforcement properties contributes substantially to the knee joint motion and forces as well as to the cartilage mechanical response during motion. In addition, the results from the degeneration algorithm indicated that fluid velocity, as well as maximum shear and deviatoric strain mechanisms, predicted changes in cartilage compositions (e.g. FCD loss and cell death) around the articular cartilage defects similarly with the degradation observed in the experiments.

Furthermore, mechanobiological knee joint models were able to predict subject- specific FCD decrease in similar locations to those revealed by MRI maps after ACL reconstruction.

In conclusion, this thesis offers new insights into the interplay between the cartilage mechanical response and the development of PTOA after joint trauma. This information can provide potential opportunities in which mechanobiological models could be implemented as a clinical tool for diagnosing and improving surgical procedures and treatments of PTOA in the knee. Future numerical models might provide new information in the design of optimal interventions intended to prevent or minimize the progression of the disease.

National Library of Medicine Classification: QT 34.5, WE 300, WE 348, WE 870, WE 872

Medical Subject Headings: Biomechanical Phenomena; Cartilage, Articular; Knee Joint;

Knee Injuries; Ligaments; Osteoarthritis, Knee; Gait; Numerical Analysis, Computer- Assisted; Computer Simula-tion; Algorithms

Yleinen suomalainen ontologia: biomekaniikka; nivelrusto; nivelet; polvet; vammat;

nivelsiteet; nivelrikko; numeerinen analyysi; mallintaminen; simulointi; algoritmit

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ACKNOWLEDGEMENTS

This study was carried out during the years 2016-2020 in the Department of Applied Physics at the University of Eastern Finland.

First, I would like to express my gratitude to my supervisors Professor Rami Korhonen, University Researcher Petri Tanska and Academy Research Fellow Mika Mononen for their support and guidance during this thesis. I would especially like to thank Rami for permitting me to work in such a great research group and for transmitting his knowledge through many fruitful discussions. I would like to thank Pete for all the patience and help during these years. I would also like to express my gratitude to Mika for his help and technical guidance, adding always the magic seasoning of modeling.

Second, I would like to thank the official reviewers, Professor Nele Famaey and Professor Noally Jérôme for their exceptional suggestions and constructive recommendations. Furthermore, I thank Dr. Ewen MacDonald for the linguistic review.

I want to thank all my co-authors for their significant contributions to the investigations., It has certainly been a privilege to work with all of you. Special thanks to Professor Alan Grodzinsky for his brilliant suggestions and unquenchable enthusiastic personality.

A big thanks to my office mates Elvis Danso, Atte Eskelinen, Ari Ronkainen, Amir Esrafilian, and Paul Bolcos for all those marvelous conversations and debates. I want to extend my acknowledgments to all the current and emeritus members of the Biophysics of Bone and Cartilage (BBC) research group; It has been an honor to work with such an extraordinary professional team. Special gratitude goes to Abhisek Bhattarai, Atte Eskelinen, Elvis Danso, and Ari Ronkainen. This voyage would have been tortuous without your wonderful friendship. A huge thank you goes to Professor Juha Töyräs; your ski and skate lessons helped me enormously during those cold and dark winters. I really appreciated it.

My warmest gratitude is expressed to my parents Nazly and José, my lovely sister Juliana, and my aunt Magda. Mamá, thanks for your support during all these years.

Finally, my deepest gratitude to my beloved wife, Alejandra for her love, immeasurable support and patience during this adventure called Finland. It would not have been possible without your divine existence by my side.

The study was financially supported by the Doctoral Programme in Science, Technology, and Computing (SCITECO), the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant No 713645, Academy of Finland (286526, 324529), ERC Grant No 281180, Sigrid Juselius foundation, Päivikki ja Sakari Sohlberg foundation and NIH/NIAMS No P50 AR065645.

Gustavo A. Orozco G.

Kuopio, 2020

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To the memory of Edilma Quintero

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LIST OF PUBLICATIONS

This thesis is based on the present review of the author´s work in the field of computational biomechanics of articular cartilage and the knee joint, and the following author´s publications:

I Orozco GA, Tanska P, Mononen ME, Halonen KS, Korhonen RK. “The effect of constitutive representations and structural constituents of ligaments on knee joint mechanics.” Sci Rep. 8, 2323 (2018).

II Orozco GA, Tanska P, Florea C, Grodzinsky AJ, Korhonen RK. “A novel mechanobiological model can predict how physiologically relevant dynamic loading causes proteoglycan loss in mechanically injured articular cartilage.”

Sci Rep. 8, 15599 (2018).

III Orozco GA, Bolcos P, Mohammadi A, Tanaka MS, Yang M, Link TM, Ma BC, Li X, Tanska P, Korhonen RK. “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. (resubmitted after major revision, 07/04/2020)

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

AUTHOR'S CONTRIBUTION

The publications in this dissertation are original research papers on the development of finite element models for investigating the mechanical response of articular cartilage and the knee joint, predicting post-traumatic osteoarthritis. The author was the main contributor to all studies. The author was involved in the planning, and design of each study, and was the main writer of each publication. In all papers, the co-operation with the co-authors has been significant.

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ABBREVIATIONS

2-D Two-dimensional

3-D Three-dimensional

ACL Anterior cruciate ligament ANOVA Analysis of variance

CPE4P First-order, 4-node continuum quadrilateral pore pressure element C3D4P First-order, 4-node continuum tetrahedral pore pressure element C3D8P First-order, 8-node continuum hexahedral pore pressure element CMC Computed muscle control

CUBE 3D intermediate-weighted, fat-saturated fast-spin-echo DD Digital densitometry

DWI Diffusion-weighted imaging ECM Extracellular matrix

EMG Electromyography FSE Fast-spin echo FCD Fixed charge density

FE Finite element

FOV Field of view

FRPES Fibril-reinforced poroelastic swelling FRPHE Fibril-reinforced porohyperelastic FRPVE Fibril-reinforced poroviscoelastic

FRPVES Fibril-reinforced poroviscoelastic swelling GAG Glycosaminoglycan

GRE Gradient-recalled echo GRF Ground reaction force IK Inverse kinematics JSW Joint space width

LCL Lateral collateral ligament MCL Medial collateral ligament MRI Magnetic resonance imaging MSM Musculoskeletal model OA Osteoarthritis

PTOA Post-traumatic osteoarthritis PCL Posterior cruciate ligament

PG Proteoglycan

PT Patellar tendon

QT Quadriceps tendon

ROM Range of motion

RRA Residual reduction algorithm

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SE Spin-echo

T1ρ Longitudinal relaxation time in the rotating frame T2 Transverse or spin-spin relaxation time

TE Echo time

TR Repetition time

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SYMBOLS AND NOTATIONS

𝑎𝑖 Muscle activation parameter

𝑎̇ Time rate of change of muscle activation 𝑎₀ Material constant

𝑎_𝑖

⃑⃑⃑ The acceleration of the center of mass of the ith body segment

𝑏 Material constant

𝑐⁻ Mobile anion concentration 𝐶₁ Neo-Hookean material constant

𝐶 Density ratio of primary to secondary fibrils 𝑐ext External salt concentration

𝑐FCD Fixed charge density content at equilibrium 𝑐_FCD0 Initial fixed charge density

𝐂 Elastic stiffness matrix

𝐷𝑟 Non-fibrillar matrix degeneration rate factor

E Young's modulus

𝐸₀ Initial fibril network modulus

𝐸_ε Strain-dependent fibril network modulus 𝐸_f Elastic fibril network modulus

𝐸m Nonfibrillar matrix modulus

𝐸_nf Young’s modulus of the non-fibrillar matrix

𝑒 Void ratio

𝑒0 Initial void ratio 𝑒

⃗

f,i Fibril orientation vector 𝑒

⃗

_f,0 Initial fibril orientation vector 𝑒⃗_f,i^p Primary fibril orientation vector 𝑒⃗_f,i^s Secondary fibril orientation vector 𝛆 Infinitesimal strain tensor

𝜀_f Fibril strain

𝜀_ḟ Time derivative of fibril strain 𝜀eq,f Equivalent fibril strain 𝜀j Principal strains 𝜀dev Deviatoric strain 𝜀_shr Maximum shear strain 𝜀_max Maximum principal strain 𝜀min Minimum principal strain 𝜀𝛽 Strain variable

𝜀_𝛽,thres Strain threshold at which the non-fibrillar matrix damage initiates 𝜀_𝛽,failure Strain value for failure of the ground substance

𝑓 Force-length-velocity surface

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𝐅 Deformation tensor

𝐹⃗_i⁰ Maximum isometric muscle force

𝐺nf Shear modulus of the non-fibrillar matrix h Normalized tissue depth

𝜂 Viscoelastic damping coefficient

𝐈 Identity tensor

𝐽 Jacobian determinant of the deformation gradient 𝐾nf Bulk modulus of the non-fibrillar matrix

k Spring constant

𝑘 Hydraulic permeability 𝑘0 Initial hydraulic permeability 𝜅𝑧 Tissue damage parameter 𝑚_𝑖 Mass of the ith body segment

𝑀 Permeability strain-dependency coefficient or molar mass

𝑙_i Muscle length

𝑛fl Fluid volume fraction and depth-wise fluid fraction distribution 𝑛_fl0 Initial fluid volume fraction

𝑛_s Solid volume fraction

𝜈 Poisson’s ratio

𝜈m Poisson’s ratio of the non-fibrillar matrix 𝜈nf Poisson’s ratio of the non-fibrillar matrix 𝑥𝑖

⃑⃑⃑ ^subject Position of the i^th marker of the subject 𝑥_𝑖

⃑⃑⃑ ^model Position of the i^th marker of the model 𝜃𝑗subject

Joint angle of the j^th for the subject 𝜃𝑗model

Joint angle of the j^th for the model

𝑤_𝑖 Weight function for markers in the model 𝑤𝑗 Weight function for joint angles in the model

𝑊 Objective function

𝜋 Osmotic swelling pressure

𝑝 Fluid pressure

𝑝 Formulation constant

𝜌𝑧 Relative collagen density

𝑞 Rate of fluid flow or electric charge 𝜙_int Internal osmotic coefficient

𝜙_ext Internal osmotic coefficient 𝛾_int^± Internal activity coefficient 𝛾_ext^± External activity coefficient 𝑟i,j Moment arm

𝑅 Molar gas constant

𝝈fl Fluid stress tensor

𝝈eff Effective solid stress tensor 𝝈_nf Non-fibrillar matrix stress tensor

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𝝈tot Total stress tensor 𝝈_s Solid matrix stress tensor 𝜎_f Fibril stress

𝜎ḟ Time derivative of fibril stress

𝜎_f,i^p Fibril stresses for primary collagen fibrils 𝜎_f,i^s Fibril stresses for secondary collagen fibrils 𝜎f,i Fibril stress in the i:th fibril

𝜏 j Generalized force in the jth joint axis 𝑣i Shortening velocity of the muscle 𝑣_fl Fluid flow velocity

𝑣fl,thres Fluid velocity value at which the non-fibrillar matrix damage initiates 𝑣_fl,failure Fluid velocity value for failure of the ground substance

∇ Gradient

∆ Change of any changeable quantity

⊗ Outer or tensor product 𝑇c Chemical expansion stress

𝑇 Absolute temperature

𝑇 Transpose

totf Total number of fibrils α Level of statistical significance 𝑢 Muscle excitation parameter 𝜇^f Chemical potential of water

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1 Introduction

The knee joint is essential if an individual is to perform daily activities. This synovial joint connects the hip and ankle joints and allows motion of the lower extremity relative to the thigh. The knee joint has four articulating surfaces between the femur and the tibia. The ends of the bones are covered by a smooth layer of articular cartilage. The smooth articular surfaces, coupled with the lubricating characteristics of the synovial fluid, ensure low-friction movement in the joint. Knee joint stability is provided by different biological structures such as ligaments, menisci, and muscles which show multiplex biomechanical response and influence the articular cartilage behavior under different loading conditions [1,2]. In particular, ligaments and tendons provide firmness in more than one degree of freedom as well as restraining joint motion [3–5]. Research on knee ligament interactions and their effect on the cartilage mechanical response contributes to elucidating musculoskeletal disorders, injury mechanisms as well as improving rehabilitation protocols. Thus, during severe knee joint injuries, such as anterior cruciate ligament (ACL) rupture, other tissues in the joint such as the meniscus, cartilage, and other ligaments are also damage. ACL injury is a condition that generally affects the young and healthy population; ACL rupture leads to pain and instability and can predispose the subject to cartilage degeneration and post-traumatic osteoarthritis (PTOA) [6,7]. Unfortunately, there is a lack of convincing evidence that ACL reconstruction can prevent PTOA progression in the joint [8,9]. Alterations in the knee joint moments and contact forces during walking early after an ACL injury and reconstruction e.g. within 5 years of ACL injury, have demonstrated a linkage with the development of knee osteoarthritis (OA) [10,11].

PTOA is responsible for approximately $3 billion of health care costs in the United States annually [7,12]. Since it would be advantageous to reduce the socioeconomic impact and major health issues associated with PTOA after ACL injury, it would be highly desirable to develop better rehabilitation programs to minimize the risk of OA. However, effective strategies would require the identification of patients with an ACL injury who are most likely to develop PTOA and who would benefit from these kinds of interventions [13]. Currently, there are no clinical predictive tools which can identify patients early after ACL injury who are at a higher risk of developing post-traumatic OA [14]. The signs of PTOA include a PG loss from cartilage and surface disruption with lesions penetrating into the tissue [15].

Consequently, the fixed charge density (FCD) content and cartilage swelling decrease around the lesion, reducing the ability of the collagen network to support the tensile forces. However, the local adaptive and degenerative processes occurring in articular cartilage associated with these physiological changes remain unclear and are difficult to predict [13,16]. It has been suggested that local cartilage lesions might contribute to the development of PTOA following ACL injury and reconstruction [17].

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Finite element (FE) models of the knee joint make it possible to investigate biomechanical responses of articular cartilage and other tissues under both normal and altered loading situations [18–21]. Similarly, in vitro mechanobiological models have been shown to be able to simulate tissue degeneration around cartilage lesions as a function of time [22–24]. The mechanisms leading to PTOA have been related to higher localized tissue stresses, strains (shear, deviatoric) or fluid flow around the lesions, especially to increase cell death and the consequential FCD loss [24,25].

Complementary, experimental studies have indicated that during early PTOA, the collagen content remains unaffected, but there may be other structural and compositional variations [26–28]. Moreover, other investigators have suggested that the FCD decrease appears before collagen damage with a short follow-up time [29,30]

and disorganization of the collagen network is almost negligible around cartilage defects [25]. However, the mechanisms leading to these structural changes remain unclear.

Magnetic resonance imaging (MRI) techniques are widely used in the analysis of early OA in the knee joint. Recent improvements in MRI techniques have made it possible to detect even changes in the biochemical composition in articular cartilage using T^1ρ and T² mapping sequences following an ACL injury and reconstruction [31–

34]. For example, recent investigations have revealed that T1ρ/T2 relaxation times increase after the ACL reconstruction surgery [35,36]. However, there are no studies that have compared MRI follow-up information of ACL reconstructed patients with the mechanobiological predictions of the progression of PTOA utilizing patient- specific knee joint models (including cartilage defects) and local adaptive algorithms.

Therefore, in this thesis, numerical knee joint models were developed for evaluating the effect of different ligament geometrical representations on the biomechanical response of the knee joint, especially on response of articular cartilage during the stance phase of gait (study I). Another aim was to develop and validate a cartilage degeneration algorithm based on experimental FCD measurements conducted in injured cartilage followed by a physiologically relevant dynamic compression. The FCD content at different follow-up time points was assessed using digital densitometry measurements. This degenerative algorithm was developed by implementing excessive deviatoric and maximum shear strain [24], as well as incorporating fluid velocity controlled mechanisms to simulate the FCD loss as a function of time (study II). Finally, the validated degenerative algorithm was implemented into three-dimensional (3D) patient-specific fibril-reinforced poroviscoelastic knee joint models to simulate alterations in the FCD content around cartilage lesions after an ACL injury and its subsequent reconstruction during the stance phase of the gait (study III).

The findings presented in this thesis contribute to understanding the development of OA after an injury to articular cartilage and the benefits of applying patient-specific mechanobiological models on the prediction of cartilage degradation. The developed methodologies can be utilized for simulating the effect

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of surgical procedures and rehabilitation protocols on the early progression of PTOA.

These simulations can contribute to identifying optimal intervention protocols in order to slow down the progression of the disease and ultimately improving the quality of life of those people who suffer a knee injury.

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2 Knee joint

The knee joint is considered as the largest joint in the human body; it is composed of two distinct articulations: the patello-femoral and tibio-femoral joints. The patellofemoral joint is the articulation between the posterior patella and the femur.

The tibiofemoral joint is the articulation between the distal femur and the proximal tibia. The knee consists of bones which are covered at their ends with articular cartilage, (femoral, tibial and patellar), meniscus, ligaments, and tendons. The knee joint works in association with the hip joint and ankle to support the body’s weight when the individual adopts a static erect posture. During motion, the knee complex is responsible for supporting the body not only during routine motion but also when performing difficult activities. In fact, the knee joint is one of the joints most often injured in the human body. Articular cartilage, relative to other soft tissues, has to support mechanical loads of extreme magnitude, as well as reducing stress concentrations in the articulating surfaces [37,38]. The knee joint stability is provided by different structures such as the menisci, ligaments, and muscles which all participate in the complex mechanical response under different loading conditions (Figure 2.1).

Figure 2.1: Frontal (left) and axial (right) view of the knee joint.

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2.1 Articular cartilage

Articular cartilage is a poroelastic connective tissue composed of chondrocytes and a specialized extracellular matrix (ECM). This matrix is composed of a solid phase (collagen fibrils and proteoglycan (PG)) and a fluid phase (interstitial fluid). One characteristic of cartilage is that it contains mainly type II collagen and a large amount of aggregating PGs. The tissue can be considered as a “prestressed biocomposite” material. The grade of prestressing can be adjusted by varying the aggrecan concentration within the collagen fibril network [39]. This characteristic has similarities with structural engineering applications, i.e. it minimizes deflections and increases shear strength to support the mechanical environment during joint motion.

Additionally, the cartilage structure is highly depth-dependent, being commonly classified into four different zones between the articular surface and the subchondral bone: the superficial, middle, deep, and calcified zone. In human cartilage, the proportion of each zone relative to the total thickness of healthy cartilage in femur is reported to be 7 ± 1% for the superficial zone, 19 ± 7% for the transitional (middle) zone, and 75 ± 9% for the deep (radial) zone [39–41] (Figure 2.2).

2.1.1 Structure and composition

If one considers whole cartilage, then 70 to 85% of its weight is water. The remainder of the cartilage is composed mainly of PGs, collagen and a small number of lipids.

The porous medium of the solid phase permits fluid to become entrapped in the tissue, and to be exchangeable. The fluid content in the tissue is controlled by the concentration of PGs, by the swelling pressure, and by the interplay of the collagen fibrillar network preventing the swelling of the tissue. Nearly 30% of the dry weight of cartilage is composed of PGs. The water content and the PG concentration vary throughout the depth of the tissue. Near the surface, the concentration of PGs is relatively low, and the water content is at its highest (Figure 2.2). In contrast, in deeper regions of the tissue, the PG concentration is highest, and the water content is lowest [37,38,42]. PGs in articular cartilage consist of a core protein with covalently attached glycosaminoglycan (GAG) chains. The negatively charged GAG side chains of PGs result in a depth-wise distribution of FCD in articular cartilage [43–45].

Collagen protein makes up 60 to 70% of the dry weight of cartilage. Collagen type II is predominant in the tissue, although other collagen types are also present in small proportions. In the superficial zone, the collagen content decreases towards the middle zone and increases again towards the deep zone [37,42,43]. Furthermore, the collagen architecture varies throughout the depth of the cartilage (Figure 2.2). In the superficial zone, a densely packed collagen fibril network is oriented in parallel to the articular surface. In the middle zone, the collagen fibrils are more randomly oriented. In the deep zone, fibrils are arranged perpendicular to the subchondral bone [37,46,47].

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Chondrocytes are the only cell type found in normal cartilage. Their main role is to develop, maintain and repair the ECM. In cartilage, these cells produce collagen, PG, and other components to preserve the tissue. Chondrocytes differ in size, shape, and action in the different cartilage zones. In the superficial zone, the cells are flat and lie in parallel to the articular surface. In the middle zone, chondrocytes have a spherical shape and they are assembled in groups. Finally, in the deep zone, chondrocytes are spheroidal and they are located in columns which run perpendicular to the tissue surface [43,48–50]. Importantly, chondrocytes are considered as the primary functional, structural, and metabolic unit in cartilage.

Alterations in their functions are responsible for causing disturbances in the ECM, leading to cartilage degeneration which impairs the normal function of the joint [51–

56].

Figure 2.2: Depth-wise structure and composition of the cartilage of the knee joint.

2.1.2 Biomechanical properties

Articular cartilage is a versatile material since it has to support both static and dynamic mechanical loads. The interstitial fluid, collagen fibrils network and PG influence the biomechanical properties of cartilage. The biomechanical response under loading states is primarily related to the depth-dependent composition and structure of the components. PGs are mainly responsible for the compressive mechanical properties of cartilage [2,39,42]. The interstitial fluid and its interactions with the ECM components confer the ability to resist fast-rate compression and return the tissue to its normal shape after a strain. On the other hand, the collagen

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network primarily alters the dynamic (instantaneous) compressive, shear and tensile properties of articular cartilage and also defines the nonlinear behavior of fibril reinforced properties in tension [26,37,43]. Cartilage has a high tensile strength in the superficial zone since collagen fibrils are oriented in parallel to the surface. The random orientation of the collagen fibrils in the middle zone provides excellent strength to withstand shear loads. In the deep zone, collagen fibrils oriented perpendicular to the surface and which extend across the cartilage-bone interface have an enhanced resistance to the swelling of the tissue and provide protection under conditions of high shear deformations [42,57,58].

Cartilage displays a remarkable design combination of form and function. The aggregating PGs attract water, generating an increment of the osmotic swelling pressure related to an increase in the FCD. As the nonfibrillar matrix expands, pre- tension is created in the collagen network. Equilibrium is attained between the swelling pressure and the load on the knee joint, and no further strain takes place.

During compressive loading in cartilage, fluid flows through pores in the solid medium [59–63]. The fluid flows back into the tissue due to osmotic pressure after the load is removed.

2.2 Meniscus

Knee menisci are smooth and lubricated crescent-shaped wedges of fibrocartilage located on the lateral and medial sides of the joint between femoral condyles and tibial plateaus (Figure 2.1). Similarly to articular cartilage, the meniscus is a dense ECM composed mainly of water (60-70% of the wet weight), PGs (1-2% of the wet weight) and collagen (15-22% of wet weight), interposed with cells [64,65].

The collagen network is primarily responsible for the tensile strength of the menisci; they contribute up to 65-75% of the dry weight of the ECM. Collagen type I is the most abundant component of the ECM (90% dry weight) with variable amounts of types II, III, V, and VI. The predominance of collagen type I distinguishes the fibrocartilage of menisci from articular cartilage. The collagens are heavily cross- linked by hydroxy-pyridinium aldehydes [66]. The collagen fibrils are oriented circumferentially in the deeper layers of the meniscus, conferring high tensile mechanical properties in the circumferential direction and transferring vertical compressive loads. In the superficial region of the menisci, the collagen fibers are oriented in a more radial direction, providing better structural integrity [67,68].

Moreover, PGs in the ECM are responsible for hydration and provide the tissue with a high capacity to resist compressive loads [67,69].

The medial meniscus has a semicircular shape; it is approximately 35 mm in diameter covering 60% of the medial tibial cartilage surface, while the lateral meniscus is virtually circular, with a uniform width from anterior to posterior,

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occupying 80% of the lateral tibial articular surface. The horns of the meniscus are attached to the tibial bone through ligaments [65,68,70].

The biomechanical role of the meniscus is distributing weight-bearing loads, reducing the friction between femur and tibia, providing impact absorption, stability, joint lubrication and reducing the contact stresses, as well as increasing the contact area in the tibiofemoral knee joint [64,71].

2.3 Ligaments and tendons

Ligaments and tendons are dense connective tissues which provide support to the knee joint. The fibrillar bundles in tendons and ligaments are densely packed and oriented in parallel to one another in the direction in which the tensile forces are frequently applied. The fibrous component of ligaments and tendons is primarily collagen whereas the ground substance is composed of structural glycoproteins, PGs, and water. Ligaments link bones to bones and tendons connect bone to muscles in the joint. The main ligaments in the knee are anterior cruciate ligament (ACL), posterior cruciate ligament (PCL), medial collateral ligament (MCL) and lateral collateral ligament (LCL). The ACL is attached to the posterior part of the inner surface of the lateral femoral condyle and to the medial part of the anterior intercondylar eminence of the tibia (Figure 2.1). The ACL functions as the primary restraint to anterior translation of the tibia on the femur [72,73]. The PCL is attached distally to the posterior tibial spine to the condyle of the femur. The PCL is a shorter and less oblique than the ACL. The PCL is the main restraint to posterior displacement, or posterior shear, of the tibia below the femur as well as inhibiting lateral rotations [74,75]. The MCL arises proximally from the medial femoral epicondyle and from where it is distally inserted into the medial of the proximal tibia.

The MCL acts to restrain excessive abduction (valgus) and the lateral rotation at the knee [76,77]. Furthermore, the LCL is located on the lateral side of the tibiofemoral joint, starting proximally from the lateral femoral condyle from where it travels distally to the fibular head. The LCL is mainly responsible for limiting varus motion in the knee [78–81].

The quadriceps tendon (QT) connects the quadriceps muscles to the top of the patella bone. The patella is attached to the tibia by the patellar tendon (PT). The primary function of QT and PT in combination with the patella bone is to act together as an extensor mechanism of the lower leg. The patella functions as an anatomic pulley, deflecting the action line of the quadriceps femoris muscle away from the joint center, increasing the angle pull and elevating the capacity of the muscle to produce an extension torque [82–84].

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3 Knee joint injury and its assessment

3.1 Joint disorders and cartilage degradation

The knee is prone to develop defects, injuries and disease processes; these are not unexpected when one considers the properties of the joint. It is essential that the knee should dissipate the huge loads to which it is subjected. Since the joint is one of two of the longest levers in the human body, these forces contribute significantly to various injuries and degenerative damage. Severe knee injuries might involve menisci, ligaments, bones, or articular cartilage. For instance, meniscal injuries are common and can be due to an excessive rotation of the femur on the fixed tibia during the knee flexion. In addition, ligamentous trauma due to an ACL rupture may occur as a result of an applied load that forces the joint to exceed its normal range of motion (ROM), usually the translational ROM. While larger forces may cause ligamentous tears, lower-level forces can similarly cause disruption in ligaments weakened by aging, immobilization, or vascular deficiencies [33,85].

Articular cartilage is vulnerable to experience injuries either by the application of a higher direct load or by abnormal muscular and ligamentous forces. OA is the most common musculoskeletal disorder which causes a degeneration of articular cartilage.

Severe traumatic joint injury may produce deep defects in articular cartilage and initiate erosion under cycling loading. These effects may contribute to the development of OA in the joint. Moreover, obesity, malalignment, muscular weakness or instability in the joint have been considered as potential contributors to the development of degenerative conditions in articular cartilage [86,87].

3.1.1 ACL rupture and reconstruction

An ACL rupture is a common and severe knee injury that occurs during certain physical activities such as running and jumping. This traumatic injury is often accompanied by meniscal tears, other ligament injuries and chondral lesions, increasing the risk of knee OA. A knee with an ACL rupture becomes less stable with increased translation in the anterior direction and rotation in the interior direction [88,89]. The incidence of ACL has been estimated to be between 30 and 78 per 100,000 people every year. Sports activities are the primary source of ACL injuries requiring surgery (65%) [90–92]. For this reason, ACL reconstruction is one of the most common orthopedic procedures, with an increasing incidence across the world, with over 43.5 per 100,000 people in the US being operated every year [91,93,94]. In this surgical technique, a graft tissue is harvested from the patient (autograft) or a donor (allograft) and this is utilized to replace the ruptured ligament [95,96]. Typically, the patellar, achilles and tibialis tendons are chosen as an allograft for ACL

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reconstruction. In most cases, allografts offer some advantages such as shorter surgical time and a better guarantee of adequate graft/tissue for replacement.

However, a risk of disease transmission has been associated with allografts and there may also be immune response complications. In addition, increased costs have also been reported by some health care systems [95]. Moreover, the correct positioning of the ACL graft after rupture has been reported to be extremely important for replacement longevity. Previous studies have reported malposition of the ACL graft as a significant factor that leads to graft failure [95,97–99].

Despite the continuing improvements in ACL reconstruction procedures, one of the main questions is whether ACL reconstruction is able to reduce the risk of OA progression. Some researchers have detected a higher incidence of OA development in patients after ACL reconstruction (44%) than in those who remained ACL deficient (37%) [13,100]. Moreover, there are numerous reports of OA prevalence after ACL injury and reconstruction but nonetheless the effects of ACL replacement remain unclear in terms of preventing OA [7,101,102]. In fact, when the incidence of OA after ACL injury and reconstruction is compared between the healthy and the reconstructed knee, most reports have shown more cases of OA in the reconstructed knees [103–105].

3.1.2 Articular cartilage injuries

Articular cartilage is prone to suffer severe injuries after constant and repetitive impact and torsional joint loading during the performance of sporting activities. In some cases, cartilage lesions can appear after some soft tissue injuries of the knee joint including ligament rupture, meniscus tear and severe patellar dislocation [106–

109]. These situations often cause damage to the articular surface, causing incapacity, chronic pain, and finally, tissue degeneration. Depending on the intensity and the type of loading, the lesions can be either chondral or osteochondral fissures; these injuries can be classified based on the damage generated by the trauma: i) cartilage matrix and cell injuries (damage to the joint surface that does not cause any evident disruption of the articular cartilage response); ii) chondral defects (visible mechanical disruption of articular cartilage); and iii) osteochondral injuries (shown mechanical disruption of articular cartilage and bone in the joint) [110,111]. It should be noted that damage to chondral matrix causes no symptoms such as pain or inflammation.

Hence, this type of lesion is not readily detectable by current clinical imaging methods. In contrast, chondral fractures often cause mechanical symptoms such as synovitis, pain and joint effusions. Nevertheless, severe cartilage and bone ruptures such as osteochondral injuries may cause hemorrhage, inflammation of the joint, and hematomas soon after injury. Depending on the location of the lesion and size of the defect and the structural stability of the joint, the lesion may progress and potentially degenerate the articular cartilage [112,113].

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In most cases, damage in the joint can be repaired if both the integrity of the articular surface of the cartilage is intact and further conditions protect against additional trauma scenarios. However, when the mechanical integrity of the cartilage is affected, chondrocytes are stimulated to repair chondral and bony lesions, but it infrequently restores the tissue. Since both the mechanisms behind cartilage disruption and the ways in which the tissue defects can propagate after an injury remain unclear, current efforts have been focused on developing strategies to terminate or decelerate cartilage degenerative conditions. Some of these approaches have involved predictive numerical models. On the other hand, recently a novel approach has been introduced of grafting tissue lesions with regenerated cartilage.

Applications of tissue engineering such as biomaterial scaffolds and artificial cartilage grafts have been tested in animal models and in preclinical procedures, but the overall findings suggest that more improvements in those procedures should be done before embarking on extensive clinical trials [114–116].

3.1.3 Post-traumatic osteoarthritis

Millions of people around the world suffer from OA; it is the leading cause of mobility-related disability in the United States and Europe. Currently, there are no medications or treatments that can reverse the progression of OA. Due to the aging of the population and the increasing obesity rate, the prevalence of OA is expected to double in the coming years [13]. On the other hand, another cause of the increasing OA rate is joint trauma. Millions of healthy young to middle-aged individuals develop PTOA as a result of a traumatic joint lesion, such as a tear of the anterior cruciate ligament or meniscus; these injuries occur particularly in the young population during recreational activities [117–119]. Unfortunately, joint injury can cause permanent structural changes in cartilage which might lead to cell death, PG depletion and bruising of the subchondral bone, contributing further to PTOA progression [120] (Figure 3.1). Moreover, the injury can induce an increase in the levels of pro-inflammatory cytokines within the joint such as TNF-α, IL-6, and IL-1β.

These cytokines diffuse into cartilage and trigger an upregulation of protease activity, leading to the degradation of the cartilage ECM [121–123]. When the mechanical trauma to cartilage caused by the initial injury is accompanied by cytokine penetration, degradation of cartilage and subchondral bone over weeks and months often progresses to full-blown, painful PTOA within 10-15 years.

The mechanisms which cause PTOA after an injury remain largely unresolved.

Many experimental [124–132] and numerical [22,133–138] analyses have examined cartilage changes in knee PTOA, such as the decrease in the PG content, the increases in permeability and water content, and the disruption of the collagen network. After the initial cartilage injury, the FCD (associated with GAG chains of PGs) and swelling properties may decrease, reducing the stiffness of the tissue and the ability of the

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organized fibril collagen network to support tensile forces [15]. In addition, previous investigations indicate that FCD decreases earlier than collagen matrix damage over a short period of follow-up time [16,29,30] and that the main alterations in the organization of the collagen network are prone to occur at the cartilage-bone interface [25]. However, more investigations are needed to clarify the initial alterations in the tissue.

Alterations in the knee biomechanics after surgical procedures might contribute to the progression of PTOA. For instance, changes in ACL-deficient and - reconstructed individuals may modify the contact area in the tibiofemoral joint, thereby loading cartilage regions that were previously unloaded and reducing loads on areas of cartilage normally experiencing larger loads during locomotion [12,118,139,140]. Moreover, connections have been reported between abnormalities in knee kinematics after surgery and early degenerative alterations in the biochemical composition of cartilage [33,141,142].

Figure 3.1: Progression of osteoarthritis with structural changes in articular cartilage [120].

3.2 Knee joint imaging

Early detection of OA is essential if one wishes to improve medical procedures and increase our understanding of degenerative processes and treatment options.

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Currently, techniques such as magnetic resonance imaging (MRI), ultrasound, and radiography have improved OA diagnosis and clinical decision-making.

Radiography is the most accessible technique to evaluate the knee utilizing osteophyte formation and joint space width (JSW); grading schemes such as the Kellgren-Lawrence score and the Osteoarthritis Research Society International classification score provide guidelines for the diagnosis of OA [143,144]. In addition.

conventional radiography techniques offer an indirect measure of cartilage through an assessment of JSW but they are not able to detect early chondral damage in the joint. Thus, arthrography plus either x-ray or computed tomography can assess cartilage surface boundaries, however, limitations mean that they cannot provide specific tissue information [145,146].

In the evaluation of cartilage defects, MRI techniques offer a high standard contrast to detect both morphological and physiological details directly. The most common methods in imaging of cartilage are spin-echo (SE) and gradient-recalled echo (GRE) sequences, fast SE, and 3D SE and GRE. In addition, physiological imaging alternatives such as T2 mapping, delayed gadolinium-enhanced MRI (dGEMRIC), T^1ρ mapping, sodium MRI, and diffusion-weighted imaging (DWI) might offer detailed information about the biochemical composition of cartilage. In particular, SE and Fast-spin echo (FSE) are helpful for assessing tissue defects [147].

Moreover, some semiquantitative MRI techniques, such as whole-organ assessment, provide certain reliability and sensitivity for identifying tissue lesions and its progression after injury [145,148,149].

In particular, T^1ρ mapping has shown a correlation with FCD and PG content in articular cartilage and therefore it can be very effective in imaging early changes in OA [150,151]. When a PG decrease occurs in the earliest phases of OA, the biochemical interactions are disrupted and T^1ρ allows measurement of the interaction between water molecules and assessments of their extracellular interactions. Various studies have reported elevated T1ρ relaxation times in knee OA cartilage contrasted with normal articular cartilage [145,152,153]. Additionally, variations in the T2

relaxation times are related to the quantity of water and the integrity of the PG- collagen matrix. Collagen disruption in articular cartilage has been significantly correlated with alterations in the spatial distribution of T2 relaxation times [154–156].

In recent biomechanical applications, MRI has been used for developing patient- specific knee joint models [20,21,137] and for predicting cartilage degeneration in numerical models [157–159].

3.3 Gait cycle analysis

Analysis of the gait cycle is a well-known method for the quantitative evaluation of locomotion disorders providing a functional diagnosis, assessment for treatment planning, and monitoring the progress of the disease. Gait cycle analysis is relevant since gait disorders affect a high number of people around the world, including

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individuals suffering neurodegenerative diseases such as sclerosis, Parkinson's disease, spinal deformation, brain tumors, neuromuscular ailments or musculoskeletal disorders, and physiological aging [160,161].

The gait cycle describes two successive events in the same limb, frequently starting with the initial contact of the lower extremity with the ground or heel strike.

During the cycle, each extremity describes two major phases: a stance phase (which makes up about 60% of the gait) and the swing phase (the last 40% of the gait cycle).

There exist two periods of double support occurring between the time one limb makes initial contact and the other one leaves the floor at toe-off. The body is supported by only one limb for around 80% of the cycle. Eight particular events can be described from the gait cycle: 1) initial contact (0%), 2) loading response (0-10%), 3) midstance (10-25%), 4) terminal stance (20-48%), 5) preswing (48-60%), 6) initial swing (60-73%), 7) midswing (73-87%), and 8) terminal swing (87-100%) (Figure 3.2) [162].

Figure 3.2: The gait cycle progression (stance and swing phases) from the initial contact of right extremity and the following contact of the same extremity [162].

The aim of gait analysis techniques is to capture and measure data with respect to different gait parameters based on the use of various devices. These methodologies can be classified into three groups: based on image processing, sensors located on the body, if they are carried by the users (wearable sensors) or utilizing floor sensors.

First, a common image processing system is composed of several digital or analog cameras with the lens to obtain the gait-related information. Some methods such as threshold filtering convert images into black and white images, then calculate the number of light or dark pixels, segmenting parts of the body and removing the background of the image [160,163]. In addition, infrared thermography methods

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have been utilized to recognize human gait patterns based on the thermal intensity of the skin [164]. Second, wearable sensors typically placed on pre-selected parts of the patient's body, such as the feet, knees or hips to measure characteristics of the gait. They consist of skin markers, accelerometers, gyroscopes, extensometers, goniometers, electromyography, etc. In fact, the most common technique to measure knee joint kinematics is based on skin markers, which represent the motion on bones in the body. The main limitations associated with wearable sensors are marker misplacements on the skin, variations during the marker tracking, and tissue artifacts [165,166]. In order to avoid these errors, fluoroscopic techniques have been developed. For instance, single or multiple fluoroscopic methods are employed for checking the bone motions in the joints [167–169]. Finally, systems based on floor sensors, also called “force platforms”, analyze the gait by pressure or force transducers when the subjects walk on these devices. These systems make it possible to examine external forces (ground reaction forces (GRFs)) which can be used to diagnose gait problems of the patient. Furthermore, inertial sensors are used to measure the patient body´s velocity, acceleration, orientation, and gravitational forces, using a combination of accelerometers and gyroscopes.

The knee joint moments can be calculated based on kinematic data obtained from wearable sensors combined with GRFs using inverse dynamics. Complementary, electromyography (EMG) is often used for measuring the electrical activity of the muscles during the gait cycle and includes this information during the optimization process of the inverse dynamics analysis [170,171]. Moreover, various musculoskeletal models have been developed to determine subject-specific locomotion including anatomical geometries, relevant bio-signals (EMG data) and human body rigid structures driven by muscle-activation. However, general uniform scaling methodologies [172,173] are not accurate enough to capture the anatomically realistic geometric effect on the joint contact forces in the knee [174,175].

Gait cycle particularities have been related to medial tibial cartilage knee PTOA.

The external varus and flexion moment in the knee, suggested that alterations for mediolateral load distribution and total muscle contribution, respectively, have been connected to structural variations in knee OA [176,177]. Furthermore, there are recent studies reporting differences in the frontal plane of the knee during walking between patients with post-traumatic versus nontraumatic medial compartment knee OA, showing relatively decreased adduction moments in PTOA knees [10,14,178].

Knee joint contact forces and moments may be estimated utilizing patient-specific gait analysis to evaluate alterations after knee injuries such as ACL rupture or meniscectomy and how orthopedic interventions mitigate these abnormalities after joint trauma. Similarly, the information on the patient´s gait when combined with anatomical geometries of the joint is relevant when developing biomechanical models for evaluating the mechanical response of the articular cartilage as well as predicting potential locations for the development of PTOA.

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4 Computational modeling of the knee

4.1 Musculoskeletal modeling

The musculoskeletal system of the human body is mechanically elaborate since all of it elements are involved during its locomotion. Typically, musculoskeletal models (MSM) are intended to simulate the human skeleton as a rigid system allowing multibody dynamic methods for solving the differential equations which describe the motion of each segment. However, sophisticated representations are needed to simulate the mechanical contribution of soft tissues such as muscles, tendons, and ligaments. In addition, the control mechanisms mediated by the central nervous system on muscles are not fully understood for specific modeling problems.

Therefore, the implementation of these mechanisms is based on assumptions; these are determined mainly by different optimization formulations [179,180].

MSMs are classified into two groups: forward and inverse dynamic models. In the first group, forward dynamic models are intended to simulate the motion of the human body based on a computed muscular activation. These models are computationally demanding and require higher resources for the optimization process to predict the motion of the physical system [181,182]. In the second group, inverse dynamic models describe muscle activation in the system based on the predefined body motion. These models are considered to be more computationally efficient and capable of simulating the effect of the muscles of the joints during locomotion [183,184].

In general terms, MSMs designate the elements of the musculoskeletal system such as muscle construction, geometries, and body segments into differential equations. These formulations can represent the time-dependent behavior of the body in response to muscular excitation. These excitations are obtained after computing a loop optimization strategy in which the aim is to track the experimental motion of the body. Then, calculations obtained from MSMs are contrasted with the acquired locomotion, ground reaction forces, and EMG physical data.

Many MSMs have been developed to simulate the motion of the human body, in fact, many researchers have created their own codes, but the limited sharing of this kind of in-house software has been limited and this makes it difficult for others to compare them with their own results. In contrast, one of the most widely utilized pieces of software for biomechanical analysis is OpenSim (SimTK, Stanford, CA, USA) which is defined as an open-source platform for modeling, simulating, and analyzing the musculoskeletal problems based on quasi-static inverse dynamics optimization. This software utilizes four general steps when performing a musculoskeletal simulation (Figure 4.1). In this example, a general OpenSim workflow is presented to highlight the main parts during the generation of an MSM

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simulation. For more information, please consult the material available in the software documentation [185,186].

Figure 4.1: General workflow for generating an MSM simulation. The inputs used by MSM are experimental motion and experimental ground reaction forces. In Step 1, the experimental motion is used to scale the MSM to adjust the dimensions of the subject. In Step 2, inverse kinematics (IK) is applied to find the model joint angles that reproduce the measured locomotion. In Step 3, inverse dynamics are utilized to calculate the forces and moments in the joints. In Step 4, a static optimization algorithm is utilized to solve the equations of motion for unknown generalized forces related to each muscle activation conditions.

The four steps are described as follows:

In Step 1, a scaling procedure is executed to match the anthropometric information of the subject in the model. The body segments are scaled based on relative distances between pairs of markers acquired from the motion-capture system and the corresponding marker locations in the model. If the joint angles are included in the experimental motion, e.g. if they have been obtained by motion analysis, these can be added into the calculation process (Equation 4.1). Similarly, the mass properties are scaled to correspond to the subject’s information. Then, in Step 2, joint angles and translations are determined based on an inverse kinematics (IK) solution which reproduces the best replication of the measured marker data. In addition, a least-squares function is utilized to minimize the differences between the subject´s marker locations and the corresponding markers of the MSM. Thus, for each frame in the experimental motion, an inverse kinematics approach is applied in order to minimize the weighted squared error; this is defined by

Squared error = ∑ 𝑤_𝑖 (𝑥⃑⃑⃑ _𝑖^subject− 𝑥⃑⃑⃑ _𝑖^model)²+

markers

𝑖=1

∑ 𝑤_𝑗 (𝜃_𝑗^subject− 𝜃_𝑗^model)²

joint angles

𝑗=1

, (4.1)

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where 𝑥⃑⃑⃑ _𝑖^subjectand 𝑥⃑⃑⃑ _𝑖^modelare the positions of the i^th marker of the subject and model, 𝜃_𝑗^subjectand 𝜃_𝑗^modelare the values of the j^th joint angle for the subject and model, and 𝑤𝑖 and 𝑤𝑗are factors that allow markers and joint angles to be weighted differently.

In Step 3, the inverse dynamics algorithm calculates the forces at each joint. Based on the motion and the ground reaction forces, the MSM computes an inverse dynamic analysis utilizing Newton’s second law which mathematically expresses the mass-dependent connection between force and acceleration of the body segments.

Finally, in Step 4, since the motion of the system is defined by the positions, the velocities and the accelerations, the MSM utilizes a static optimization tool which solves the equations of motion for unknown generalized forces related with each muscle activation-to-force conditions. The ideal force generators are defined as

τ⃑ _j= ∑(𝑎_𝑖𝐹⃗_i⁰) ∙

n

𝑖=1

𝑟_i,j ,

(4.2) or constrained by force-length-velocity properties:

τ⃑ j= ∑[𝑎𝑖 𝑓(𝐹⃗_i⁰, 𝑙i, 𝑣i)] ∙

n

𝑖=1

𝑟i,j ,

(4.3) while minimizing the objective function 𝑊

W = ∑ 𝑎_𝑖^p

𝑛

𝑖=1

, (4.4)

where 𝑛 is the number of muscles, 𝑎𝑖 is the muscle activation parameter at a discrete time step, 𝐹⃗_i⁰ is the maximum isometric force, 𝑙_i is the muscle length, 𝑣_i is the shortening velocity, 𝑓(𝐹⃗_i⁰, 𝑙_i, 𝑣_i) is the force-length-velocity surface, 𝑟_i,j is the moment arm, 𝜏 _j is the generalized force in the j^thjoint axis and p is a user-defined constant.

After these four steps, an additional post-processing step can be performed to obtain relevant information such as joint contact forces, tendon lengths, and muscle forces during the motion.

4.2 Finite element models of the knee joint

Computational models help to overcome inherent limitations during experimental and clinical studies such as expensive experimental set-ups, complications in obtaining accurate measures in vivo but they can also be used to reproduce degenerative situations in the knee. For instance, MSMs have been utilized to estimate the muscle and joint contact forces during locomotion. However, these

Mechanobiological modeling of articular cartilage : predicting post-traumatic knee osteoarthritis

Dissertations in Forestry and Natural Sciences

GUSTAVO ALEJANDRO OROZCO GRAJALES

MECHANOBIOLOGICAL MODELING OF ARTICULAR CARTILAGE

Gustavo Alejandro Orozco Grajales

MECHANOBIOLOGICAL MODELING OF ARTICULAR CARTILAGE

ABSTRACT

ACKNOWLEDGEMENTS

To the memory of Edilma Quintero

LIST OF PUBLICATIONS

AUTHOR'S CONTRIBUTION

TABLE OF CONTENTS

ABBREVIATIONS

SYMBOLS AND NOTATIONS

⃗

⃗

1 Introduction

2 Knee joint

2.1 Articular cartilage

2.2 Meniscus

2.3 Ligaments and tendons

3 Knee joint injury and its assessment

3.1 Joint disorders and cartilage degradation

3.2 Knee joint imaging

3.3 Gait cycle analysis

4 Computational modeling of the knee

4.1 Musculoskeletal modeling

4.2 Finite element models of the knee joint