Multiscale computational modeling of the knee joint: Development, validation, and application of musculoskeletal finite element modeling methods

(1)

Dissertations in Forestry and Natural Sciences

DISSERTATIONS | AMIR ESRAFILIAN | MULTISCALE COMPUTATIONAL MODELING OF THE KNEE JOINT: ... | No 428

AMIR ESRAFILIAN

Multiscale computational modeling of the knee joint:

Development, validation, and application of musculoskeletal finite element modeling methods PUBLICATIONS OF

THE UNIVERSITY OF EASTERN FINLAND

(2)

(3)

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

No: 428

Amir Esrafilian

MULTISCALE COMPUTATIONAL MODELING OF THE KNEE JOINT:

DEVELOPMENT, VALIDATION, AND APPLICATION OF MUSCULOSKELETAL FINITE ELEMENT MODELING

METHODS

ACADEMIC DISSERTATION

To be presented by the permission of the Faculty of Science and Forestry for public examination in the Auditorium MS300 in Medistudia Building at the University of Eastern Finland, Kuopio, on September 3^rd, 2021, at 12 o’clock.

University of Eastern Finland Department of Applied Physics

Kuopio 2021

(4)

PunaMusta Oy Joensuu, 2021

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

Distribution:

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

http://www.uef.fi/kirjasto

ISBN: 978-952-61-4280-7 (print) ISSNL: 1798-5668

ISSN: 1798-5668 ISBN: 978-952-61-4281-4 (pdf)

ISSNL: 1798-5668 ISSN: 1798-5676

(5)

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

70211 Kuopio, Finland email: amir.esrafilian@uef.fi

esrafilian@gmail.com Supervisors: Professor Rami K. Korhonen

University of Eastern Finland Department of Applied Physics P.O.Box 1627

70211 Kuopio, Finland email: rami.korhonen@uef.fi Senior researcher Lauri Stenroth University of Eastern Finland Department of Applied Physics P.O.Box 1627

70211 Kuopio, Finland email: lauri.stenroth@uef.fi Professor Pasi A. Karjalainen University of Eastern Finland Department of Applied Physics P.O.Box 1627

70211 Kuopio, Finland email: pasi.karjalainen@uef.fi

Reviewers: Assistant professor Corinne R. Henak University of Wisconsin-Madison Department of Mechanical Engineering Madison, USA

email: chenak@wisc.edu Assistant professor Hans Kainz University of Vienna

Department of Biomechanics Vienna, Austria

email: hans.kainz@univie.ac.at Opponent: Professor Elena Gutierrez Farewik

KTH Royal Institute of Technology Department of Engineering Mechanics Stockholm, Sweden

email: lanie@kth.se

(6)

(7)

Amir Esrafilian

Multiscale computational modeling of the knee joint: Development, validation, and application of musculoskeletal finite element modeling methods

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

Dissertations in Forestry and Natural Sciences, 2021, 428

ABSTRACT

The knee joint facilitates locomotion and provides stability, force transmission, and shock absorption for the lower extremity of the human body. Within the knee, articular cartilages and menisci act as load-bearing tissues with nearly frictionless contact surfaces. Nonetheless, the high loading and the intrinsic complexity of the knee joint mechanism make the joint potentially vulnerable to injuries and degen- erative joint diseases, e.g., knee osteoarthritis (KOA). Several studies have shown compelling evidence that altered joint mechanics (such as stress and strain within the knee load-bearing tissue) contribute to the onset and progression of KOA. In addition, the effectiveness of knee rehabilitation protocols and post-surgery joint recovery are often influenced by localized knee joint loading. Hence, a thorough knowledge of the joint kinematics, kinetics, and soft tissue mechanical responses is essential in the assessment of mechanically-induced KOA and restoration of knee joint functionality.

Several experimental studies have been conducted to investigate knee joint mechanics in different activities. In this regard, in vivo and in situ knee joint contact forces, contact areas, and contact pressures have been measured. Furthermore, the interrelationship of mechanical loading, cartilage composition, and tissue-level mechanics of cartilage have been experimentally investigated. Although these experimental studies have provided fundamental knowledge of joint- and tissue-level mechanics of the knee joint, they are either limited to specific subjects (e.g., subjects with instrumented implants) or require highly invasive procedures. Also, experimental approaches typically cannot measure tissue-level mechanics such as stress and strain within the tissue.

Alternatively, numerical simulations such as musculoskeletal finite element (MSFE) models have been developed to estimate detailed joint loading and tissue- level mechanical responses. There are, however, limitations associated with the current approaches. Sequentially linked MSFE models benefit from the highly-detailed representation of the joint; however, the musculoskeletal (MS) model’s estimations are independent of the FE model calculations. Moreover, the sequentially linked MSFE models are subjected to poor couplings at the interface between the MS and the FE models. On the other hand, concurrent (embedded) MSFE models provide a nested coupling between the MS and FE models’ loading and boundary conditions. Nonetheless, embedded MSFE models are limited to simple soft tissue material models and also cannot estimate tissue-level mechanical responses (e.g., stress, strain, and fluid flow) due to the enormous computational costs of concurrent simulations.

The purpose of this thesis was to develop, evaluate, and validate different MSFE modeling workflows of the knee joint, firstly to improve the coupling of the sequential MSFE models and secondly to introduce a multiscale knee joint modeling

(8)

pipeline potentially feasible for both research purposes and clinical assessments. To this end, a total of three state-of-the-art sequential and concurrent MSFE models of the knee joint were developed, with each of the models incorporating different levels of subject-specificity and complexity of the knee joint mechanism. The MS models provided the FE models with loading and boundary conditions and consisted of a static-optimization based model, an electromyography (EMG) assisted model (both with a 1 degree of freedom knee joint), and a concurrent 12 degrees of freedom (DoFs) knee MS model with elastic cartilages. The muscle-force driven FE models were identical between the three approaches and incorporated subject-specific joint geometries, a 12 DoFs knee joint, and a fibril-reinforced poro(visco)elastic material model for knee cartilages and menisci. The three developed MSFE models were first used to analyze joint mechanics during walking of a healthy individual, and then the results were compared and validated against the experimental data. Finally, a semi-automated atlas-based EMG-assisted MSFE modeling pipeline of the knee joint was devised and utilized to investigate knee joint mechanics of individuals with KOA in different functional activities, and also to demonstrate subject-specific design of rehabilitation protocols.

The results of this thesis indicate that a static-optimization based MS model with a 1 DoF knee joint may be too simple to be used as part of the MSFE modeling pipeline to estimate reliably knee joint mechanics consistent with experiments.

Moreover, the results suggested that assisting the MSFE analysis with the subjects’

muscle activation patterns rather than using a complex multi-DoFs knee MS model but with no information on the subject’s muscle activation patterns might effectually improve the accuracy of the estimated joint mechanics. The sequential EMG-assisted MSFE model with the muscle-force-driven approach outperformed the MSFE model with the concurrent 12 DoFs knee, expanding the capability to use cutting-edge MS and FE models with minimal computational cost when compared to concurrent MSFE models. Moreover, the knee joint mechanics in different functional activities estimated by the developed atlas-based MSFE analysis pipeline provided insights which could be adopted to design more efficient and subject-specific rehabilitation protocols as well as improved daily activity routines to optimally load specific knee joint regions.

In conclusion, the developed muscle-force driven approach could provide a nested coupling at the interface between the sequentially-linked MSFE models.

More importantly, the developed pipeline, i.e., the atlas-based MSFE modeling pipeline, showcased the potential to become the tool of choice for research purposes with large cohorts as well as supporting clinical decision-making with subject- specific design of rehabilitation protocols, considering tissue remodeling and degeneration response. In the future, this novel devised pipeline may be integrated into tissue adaptation algorithms involving subject-specific mechanically-induced knee joint degeneration and adaptation.

Universal Decimal Classification:53.072; 538.951; 539.319; 616.728.3

Library of Congress Subject Headings: Knee, Osteoarthritis, Electromyography, Model- ing, Simulation , Finite element method, Musculoskeletal system – Analysis, Motion – Anal- ysis, Tissues – Mechanical properties, Cartilage, Collagen, Ligaments, Meniscus (Anatomy), Kinematics, Kinetics, Dynamics, Elasticity, Strains and stresses.

(9)

ACKNOWLEDGEMENTS

This study was carried out during 2017-2021 in the Department of Applied Physics at the University of Eastern Finland. The study was financially supported by the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant No 713645, Academy of Finland (grants no. 286526, 324529, 324994, 328920), Sigrid Juselius Foundation and Business Finland (grant no. 3455/31/2019), the Finnish Cultural Foundation and the Päivikki and Sakari Sohlberg Foundation, the European Regional Developments Fund and the Uni- versity of Eastern Finland under the project: Human measurement and analysis - research and innovation laboratories (HUMEA, project identifiers: A73200 and A73241). This study was also partly funded by a grant from the National Health and Medical Research Council of Australia (NHMRC GNT2001734). CSC-IT Center for Science Ltd, Finland is acknowledged for providing software and computing resources.

First and foremost, I would like to express my sincere gratitude to my supervisors, professor Rami Korhonen, senior researcher Lauri Stenroth, and professor Pasi Karjalainen for their guidance and support during this project. Especially, I would like to thank Rami for creating a great working environment, for all his kind guides and supports, and for transmitting not only his knowledge but also the critical thinking ability through many thoughtful discussions. Special thanks go to Lauri for all the patience, help, and the great critical discussions during these years.

I wish to thank the official reviewers, assistant professor Corinne R. Henak and assistant professor Hans Kainz for giving me professional and constructive feedback on my thesis. I would also like to thank Ewen MacDonald for linguistic review of my thesis.

I would like to express my gratitude to all the co-authors for their invaluable work and critical review of the manuscripts. In particular, I would like to thank professor David G. Lloyd and professor Ilse Jonkers for their brilliant suggestions and exceptional professionalism. I would like to also express my gratitude to Mika and Petri for all their help and technical guidance during these years. I also warmly thank my roommates Gustavo, Ari, Paul, and Atte for all those marvelous conversa- tions and debates. Warm thanks go to Ali and Mohammad for the intimate friend- ship. Also, 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. You all have created an outstanding work environment and it has been a privilege to work with such a devoted and professional group.

My warmest gratitude is expressed to my parents, my sister, and my brother for their support during all these years. Last but not least, I want to express my deepest gratitude to my beloved wife Rezvan for her love, immeasurable support and patience during this journey. Finally, our little daughter Helena, even after the roughest times, you never fail to cheer me up and make me smile. I love you both.

Kuopio, August 4, 2021 Amir Esrafilian

(10)

(11)

LIST OF PUBLICATIONS

This thesis consists of the present review of the author’s work in the field of biomechanical modeling of the human knee joint and the following selection of the author’s publications:

I A. Esrafilian, L. Stenroth, M. E. Mononen, P. Tanska, J. Avela, and R. K. Korho- nen, "EMG-Assisted Muscle Force Driven Finite Element Model of the Knee Joint with Fibril-Reinforced Poroelastic Cartilages and Menisci,"Sci. Rep., 2020, 7(1):3026, doi:10.1038/s41598-020-59602-2.

II A. Esrafilian, L. Stenroth, M. E. Mononen, P. Tanska, S. V. Rossom, D. G.

Lloyd, I. Jonkers, and R. K. Korhonen, "12 Degrees of Freedom Muscle Force Driven Fibril-Reinforced Poroviscoelastic Finite Element Model of the Knee Joint,"IEEE Trans. on Neural Sys. and Rehab. Eng., 2021,29:123, doi:10.1109/

TNSRE.2020. 3037411.

III A. Esrafilian, L. Stenroth, M. E. Mononen, P. Vartiainen, P. Tanska, P. Kar- jalainen, J. S. Suomalainen, J. Arokoski, D. S. Saxby, D. G. Lloyd, and R. K. Ko- rhonen, "An EMG-assisted muscle-force driven finite element analysis pipeline to investigate joint- and tissue-level mechanical responses in functional activities: towards a rapid assessment toolbox", submitted.

IV A. Esrafilian, L. Stenroth, M. E. Mononen, P. Vartiainen, P. Tanska, P. Kar- jalainen, J. S. Suomalainen, J. Arokoski, D. S. Saxby, D. G. Lloyd, and R. K.

Korhonen, "Towards Tailored Rehabilitation by Implementation of a Novel Musculoskeletal Finite Element Analysis Toolbox", submitted.

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

AUTHOR’S CONTRIBUTION

The publications selected in this dissertation are original research papers on the development, validation, and application of the musculoskeletal linked with finite element models of the knee joint. The author was the main contributor to all studies. The author participated in study planning, performed the development of the models and the corresponding musculoskeletal and finite element analyses within studies I, II, III, and IV. The author did not collect magnetic resonance imaging data. The author participated in motion data collection for studiesIandII. In stud- iesIIIandIV, the motion data were collected by the co-authors. The author was the main writer in all studies.

(12)

(13)

θ˙ Estimated joints’ angular velocity θ˙exp Estimated joints’ angular velocity θ¨ Estimated joints’ angular acceleration θ¨_des Desired joints’ angular acceleration θ¨exp Experimental joints’ angular acceleration ν_{n f} Poisson’s ratio of the non-fibrillar matrix ρ_z depth-dependent collagen fibrils density

τi External moment applied to thei^thdegree of freedom

τ(a,u) Time constant according to the activation or deactivation status of the muscle

σ_f_,p Tensile stress within and alongside the primary collagen fibrils σ_f_,s Tensile stress within and alongside the secondary collagen fibrils σ_f Fibrillar matrix stress

σ_i^f Tensile stress within and alongside thei^thcollagen fibril

σ_i^f^,p Tensile stress within and alongside thei^thprimary collagen fibril σ_i^f^,s Tensile stress within and alongside the i^th secondary collagen

fibril

σ_i,global^f Stress within the i^th fibril of the network represented in the global coordinate system

σ_{n f} Non-fibrillar matrix stress σt Total stress within tissue

~τ Moment applied to a body

~

ω Angular velocity k.k Euclidean distance

⊗ Tensor product

(20)

(21)

1 INTRODUCTION

The knee joint plays a crucial role in the kinematic chain of the human lower extremity, acting as a pivot between the longest bones of the body (i.e., tibia and femur), with the body’s strongest muscle groups acting across this joint. The knee joint complex primarily facilitates locomotion, provides stability, force transmission, and shock absorption during different body movements. However, the complexity of the joint mechanical linkage and high loading make the knee joint potentially vulnerable to injuries and diseases. Knee injuries are reported as the second most frequent musculoskeletal (MS) disease [1]. Several studies have revealed that altered knee joint kinematics and kinetics, and consequently, the mechanical responses (e.g., stress and strain) within the knee load-bearing tissue are key factors in the onset and progression of MS disorders such as knee osteoarthritis (KOA) [2–9]. Also, the suc- cess of knee rehabilitation protocols and the post-surgery restoration of joint function is often affected by knee joint loading [1, 10–16]. Hence, a thorough knowledge of the joint kinematics, kinetics, and soft tissue mechanical responses is essential to investigate the causes of MS disorders and restoration of knee functionality.

Several experimental studies have been conducted to evaluate the relationship between kinematics, kinetics, and tissue mechanical responses of the knee joint. In vivo andin situ knee joint contact forces (JCF), contact area, and contact pressure have been measured in several activities [17–22]. Although these experimental studies have revealed fundamental information on the knee joint mechanics, they are either limited to specific subjects (e.g., those with instrumented implants) or require highly invasive procedures. More importantly, experimental approaches cannot measure the key biomechanical parameters such as stress, strain, or fluid flow of the tissue either as a bulk medium or within the tissue’s constituents.

Alternatively, numerical simulations such as MS and finite element (FE) models have been developed to estimate detailed joint loading and tissue-level mechanical responses [23–28]. Using kinematic and kinetic data of a subject, MS models provide joint-level estimates of muscle forces and JCFs. To improve generic MS models, different subject-specific aspects, such as muscle activation patterns, joint geometries, and ligaments’ interactions, can be incorporated into the MS models while estimat- ing muscle forces [24, 27–30]. Yet, currently developed MS models cannot assess tissue-level mechanical responses and quantities that are important factors involved in the articular cartilage degradation, e.g., stress, strain, and fluid flow. FE models, in lower spatial scales than MS models, have been widely used to investigate tissue-level mechanical responses of the knee joint cartilages and menisci [6, 31–33].

Despite extensive efforts in the development of MS and FE models, only a few studies have introduced multiscale musculoskeletal finite element (MSFE) models [26, 34–38] to include the interplay between the joint- and tissue-level joint mechanics, i.e., considering the structure and function of the joint tissue in different spatial scales. Moreover, previously developed MSFE models are subjected to ma- jor limitations. First, none of those multiscale workflows have considered subject- specific muscle activation patterns in JCF estimations, although it has been shown that the muscle activation pattern is altered in subjects with MS disorders compared

(22)

to healthy individuals [24, 27, 39–45]. Second, none of the previously developed multiscale MSFE models have been assessed for their abilities to analyze tissue-level joint mechanics during activities other than gait. Third, the current multiscale MSFE models are not driven by the subject’s kinematics and kinetics [26, 34] and/or the coupling between the MS and FE models have been poorly defined [6, 35–38, 46, 47], especially in relation to the joint moments. Those MSFE models with a nested coupling between the MS and FE models (i.e., concurrent models) [24, 27, 30, 45] have not accounted for subject-specific muscle activation patterns, are not capable of es- timating tissue-level mechanical responses, have excluded crucial joint tissues, i.e., menisci, or have utilized simple soft tissue material models (i.e., isotropic elastic).

Few concurrent MSFE models [34, 35] have used a more complex material model of knee cartilage and menisci. However, these models rely on inputs that are difficult to measure in practice, and in addition, they can only analyze discrete time-instances (i.e., at steady-state) and not the transient response of the tissue [34, 35]. More importantly, none of the previously introduced multiscale MSFE models have been included as part of toolboxes or pipelines for rapid and clinically feasible assessments.

During a dynamic short-term loading in which the fluid does not flow out from the tissue, a simple material model such as isotropic elastic (that is employed in the nested MSFE models) [24,27,45] may suffice to estimate tissue responses, i.e., contact pressure at the lowest spatial scale [48, 49]. However, poro(visco)elasticity is shown essential to precisely replicate the time-dependent mechanical response of a fluid- impregnated porous tissue such as cartilage and meniscus [50]. In addition, the fluid pressurization within the tissue carries up to 90% of a dynamic load [51,52]. Further- more, several studies [49, 53] have reported that a fibril-reinforced material model is essential to simultaneously estimate both the joint-level and the tissue-level mechanical responses of a fibrillar tissue such as knee cartilage and menisci. These char- acteristics of the knee soft tissue emphasize the requirements of a fibril-reinforced poroviscoelastic (FRPVE) material model, which can potentially provide the simulations with a more accurate estimation of tissue-level mechanical responses, especially if cartilage degradation and remodeling is the focus of interest [6, 33, 54].

Apart from the above limitations, modeling assumptions and uncertainties may also affect the simulation results. Changes in the estimated joint kinematics and kinetics due to alterations in conventional aspects of the MS models, such as em- ploying generic or subject-specific joint geometry, the number of active degrees of freedom (DoFs) of the knee joint, and informing the MS model with muscle activation patterns, have been extensively explored [27, 29, 30, 39–45, 55–58]. In the interest of attenuating the effect of uncertainties within an FE model, subject-specific knee geometries have been employed using magnetic resonance images (MRIs) [59, 60].

There have also been attempts to incorporate subject-specific material properties of cartilages into FE models [61–63]. Nonetheless, ligament properties, i.e., ligament pre-strains, are challenging to be approximated [64], which introduces uncertainties into modeling [65]. Yet more, no studies have investigated whether and how different higher-spatial scale (i.e., MS) modeling approaches affect the tissue-level mechanical responses estimated by the lower-spatial scale (i.e., the FE) model.

Hence, this thesis was conducted with two overarching goals to overcome the above-mentioned limitations concerning multiscale modeling of the knee joint. The first overarching goal (i.e., studies I and II) was to develop and validate differ- ent sequential and concurrent multiscale MSFE modeling pipelines. To this end, a workflow was developed and validated (studyI) comprised of an EMG-assisted MS

(23)

model sequentially linked with a muscle-force driven FE model of the knee joint with a focus on improving the integrity of the MS and FE models. Then the developed muscle-force driven MSFE workflow (established in studyI) was employed to create and assess different sequential and concurrent MSFE models of the knee joint (i.e., studyII).

The second overarching goal of this thesis was to establish an MSFE workflow with the potential to investigate the joint mechanics in functional activities of healthy individuals as well as in subjects with knee KOA not only for research purposes but also to help in clinical assessments. In this, first, a rapid EMG-assisted muscle-force driven FRPVE FE modeling and analysis pipeline was developed and examined (i.e., study III) for investigating joint-level and tissue-level knee mechanics. The focus was to break through the barrier to incorporating the state-of-the-art MSFE analyses of the knee joint in clinical assessments or research purposes with large cohorts. Finally, the feasibility of the pipeline developed in study III was showcased for clinical assessments and subject-specific rehabilitation design (i.e., study IV) considering knee joint tissue remodeling and degeneration responses.

(24)

(25)

2 KNEE JOINT STRUCTURE AND FUNCTION

The knee joint is vital in the lower extremity kinematic chain of bipedal creatures.

This synovial joint pivots the longest bones of the human body, i.e., tibia and femur. In addition, the body’s strongest muscle groups, i.e., quadriceps, hamstrings, and calf muscles, act across the knee joint. The knee joint itself consists of two different joints: the tibiofemoral and the patellofemoral joints (Figure 2.1). The tibiofemoral joint primarily provides the kinematic chain of the lower extremity with flexion/extension DoF (in the sagittal plane). The patellofemoral articulation, on the other hand, is a saddle joint between the femoral trochlea and patella with a primary role in the knee extensor mechanism, i.e., increasing the moment arm of the knee extensor muscles. In conjunction with the synovial fluid, the articular cartilages within the knee provide nearly frictionless contacts between the joint surfaces, bear the JCF, and partly absorb the mechanical shocks. More details on the knee joint’s structure and function are discussed in the following sections of this chapter.

2.1 MACROSCALE STRUCTURE AND FUNCTION OF THE KNEE JOINT 2.1.1 The tibiofemoral joint

The tibiofemoral joint, as the largest joint in the body, is comprised of the distal femur and proximal tibia articulations. The medial condyle of the femur is larger than the lateral femur, in both anteroposterior and proximodistal directions contributing

Figure 2.1: Frontal (left) and transverse (right) views of the knee joint.

(26)

to the valgus alignment of the knee. Also, the tibia has ~3 degrees lateral inclination relative to the joint line and ~9 degrees posterior slope [66]. As a result, the knee typically has ~10-12 degrees of valgus [66]. The menisci, situated between the femur and the tibia, effectually provide the tibiofemoral joint with a continued contact area and stress distribution across the joint, shock absorption, and joint lubrication [67].

The intrinsic material properties of the meniscus such as low permeability and low compressive stiffness (compared to its stiffness in the knee transverse plane) account for the above-mentioned functions of the tissue [67, 68].

It is thought that the muscles acting across the knee stabilize and control the knee flexion/extension DoF. Apart from the flexion/extension DoF, ligaments within the tibiofemoral joint (i.e., anterior and posterior cruciate ligaments and lateral and medial collateral ligaments) primarily restrain the relative translation and rotation of the femur and tibia in sagittal, frontal, and transverse planes.

The anterior cruciate ligament (ACL) is attached to the medial surface of the lateral femoral condyle in one end. On its other end, it is attached to the front and lateral edge of the medial tibial plateau (Figure 2.1). The ACL primarily restrains posterior translation of the femur relative to tibia whilst it does not resist anterior translation of the femur relative to the tibia [69–71]. The ACL secondarily restrains adduction rotation, external rotation, and hyperextension of the femur [69–71].

The femoral attachment site of the posterior cruciate ligament (PCL) is the lateral surface of the medial femoral condyle. The PCL tibial attachment is located in a hallow between the medial and lateral tibia plateaus and extends to the posterior surface of the tibia (Figure 2.1). The PCL primarily restrains the anterior translation of the femur relative to the tibia at all angles of knee flexion, with no resistance in the backward translation of the femur relative to the tibia [70–72]. The PCL, secondarily, restrains the abduction rotation and internal rotation of the femur [70–72].

The medial collateral ligament (MCL) can be divided into superficial and deep bundles, both originating from the medial epicondyle of the femur (immediately below the adductor tubercle). The superficial MCL is subdivided into anterior and posterior portions. The anterior superficial MCL stretches downward to the medial tibia, whereas the posterior superficial MCL passes obliquely backward and inserts into the medial meniscus [73, 74]. The deep MCL bundle is subdivided into meniscotibial and meniscofemoral ligaments. The meniscotibial portion originates from the lateral and posterior medial tibial plateau (inferior to the tibial cartilage) and inserts at the outer surface of the medial meniscus. The meniscofemoral ligament originates from the medial epicondyle of the femur and inserts at the posterior horn of the lateral menisci. The MCL, and mostly its superficial bundle, is an important restraint to abduction rotation, external rotation, and mediolateral translation of the femur relative to the tibia [66, 73, 75].

The lateral collateral ligament (LCL) stretches from the lateral epicondyle of the femur to the head of the fibula. In contrast to the MCL, the LCL is not connected with other tissues such as menisci. LCL primarily resists knee adduction rotation and secondarily restrains internal rotation and anterior translation of the femur relative to tibia [66, 75, 76].

2.1.2 The patellofemoral joint

The patellofemoral articulation is a saddle joint (cellar joint) between the patella and the femoral trochlea. The movement and alignment of the patella are controlled by muscle forces (i.e., quadriceps and hamstrings), contact forces (between

(27)

the patella and trochlear groove), and restraints of ligaments [66]. The ligaments which are attached to the patella consist of the patellar ligament, medial and lateral patellofemoral ligaments (MPFL and LPFL), medial patellotibial ligament, medial patellomeniscal ligament, medial retinaculum, and lateral retinaculum.

The patella provides the quadriceps muscles with a greater moment arm around the center of rotation of the knee joint, decreasing up to 30% of the quadriceps force required to extend the knee [66]. Interestingly, the moment arm of the patellar ligament is slightly longer than the moment arm of the quadriceps tendon [77, 78].

As a result, the ratio of the quadriceps force to the patellar ligament force varies at different knee flexion angles [79]. This feature of the patellofemoral joint is of interest, especially in the rehabilitation of the quadriceps muscle-tendon complex [79]. The patella also acts as a bony shield to protect the trochlea and distal femur, e.g., when the knee is flexed.

2.1.3 Functions of the muscles

Approximately 660 skeletal muscles, ~40% of body mass, provide the human body with coordination, movement, and stability under the control of the central ner- vous system (CNS) [80]. At each joint of the body, e.g., the knee joint, a certain magnitude of moments (on each joint’s DoF) is required to perform either static or dynamic tasks. These moments are provided actively via muscles or passively via, e.g., ligaments. In other words, the force generated by a muscle crossing a joint may have a twofold effect on the joint kinetics. First, the muscle force applies a moment around the center of rotation of the joint, which is equal to the muscle force multi- plied by its moment arm. The moment arm of each muscle varies (dominantly) as a function of the body kinematics. Consequently, the required moment at the joint is controlled by the CNS through adjusting the magnitude of the generated muscle forces. The second effect of the muscle force on the joint kinetic is a force equal to the force generated by the muscle. As a result, the total joint contact force (JCF) may exceed multiples of the bodyweight due to the interaction of several muscles simultaneously acting on the joint.

Muscles acting on a joint can be classified (typically) into four categories during every specific task. The prime mover (agonist) muscles are those producing most of the moment required to perform the task. For instance, hamstring muscles are the agonists during knee flexion. A muscle that aids the agonist is termed as a synergistic muscle. For example, medial and lateral gastrocnemius are considered as synergistic muscles during knee flexion. The third group of muscles, which are termed antagonist, are the muscles that oppose the agonist muscles. Antagonist muscles provide more control over the joint and prevent excessive movement, in- appropriate action, or joint injury. When flexing the knee, the quadriceps act as antagonists. The fourth group of the muscles are called fixators since they improve the stability of the joint such as relative translations of bones when other muscles drive the joint. The recruitment coordination of these four muscle functional roles when performing an action is termed muscle synergy or the muscle activation pattern [81]. Muscle synergy may be altered in individuals with different MS disorders despite the comparable body kinematics [39–45, 82].

(28)

2.2 MICROSCALE COMPOSITION AND STRUCTURE OF THE KNEE SOFT TISSUE

2.2.1 Knee joint cartilage

The articular cartilage can be described as a hydrated, swollen tissue with an extra- cellular matrix (ECM) composed of organized collagen fibers and negatively charged proteoglycans (PGs) [83]. In healthy and fully hydrated cartilage, the fluid content is ~70% of the total mass at the bone-cartilage interface. The degree of hydration increases from the deep zone towards the surface of the cartilage linearly, up to

~85% of the cartilage mass at the cartilage surface (Figure 2.2). Nonetheless, the fluid content may differ with age, joint pathology (e.g., KOA), and location in the joint [84, 85]. The fluid flow within and across the cartilage is essential for the mechanical response of the tissue, lubrication of the joint, and providing the tissue with nutrition [86, 87].

In general, the fluid flow through a fluid-impregnated porous tissue (such as cartilage) may happen due to two different mechanisms [87]. First, the interstitial fluid may flow out due to a direct (and pure) fluid pressure gradient across the tissue. In this mechanism, the fluid flow can be represented by Darcy’s law as a function of the permeability of the tissue [87]. The permeability of the cartilage varies throughout the cartilage depth, which is attributed to the different densities of the collagen fiber network within the tissue [87, 88]. Moreover, the permeability may become altered in degraded cartilage [84, 89].

The second fluid transport mode within and across the cartilage may happen due to the deformation (i.e., strain) of the cartilage matrix. The volume of the solid matrix of the tissue changes due to the strain, and consequently, the fluid flows out. This is known as the flow due to consolidation [87], in which the fluid flow depends on both stiffness and the permeability of the medium. It is believed that the fluid flow of normal cartilage during physiological loading of the knee happens due to both pressure gradient and consolidation [87]. Accordingly, the cartilage permeability is shown to be both depth-dependent and strain-dependent [90, 91].

The PG content is about 20-30% of the dry weight of the cartilage [92]. In healthy tissue, the PG concentration is typically lowest in the superficial zone of the cartilage and it increases with the depth (Figure 2.2). PGs are negatively charged and trapped within the ECM collagen network [92]. Consequently, positively charged ions (e.g., Na+) diffuse into the ECM. As a result, swelling happens due to osmotic pressure gradient and also repulsive forces between PGs [86]. When the cartilage is com- pressed, fluid flows out from the tissue, and the PG density increases. Thereby, both the internal osmotic pressure and the expansion due to repulsive forces increase.

This mechanism accounts for the contribution of PGs’ to the compression stiffness of the cartilage in long-term loading (i.e., at equilibrium) [93].

About two-thirds of the dry weight of the cartilage is composed of the collagen network, predominantly collagen type II (~80-95%) [94], providing the cartilage with higher resistance against tensile stress and fluid-independent viscoelastic response of the cartilage [86, 95]. It has been shown that collagen fibers (along with the fluid pressurization within the tissue) effectually resist compressive (fast) dynamic- loading of the tissue [96]. Each collagen fiber is made up of multiple collagen fibrils arranged along the collagen fiber length [97]. The collagen fibers’ orientation varies through the depth of the cartilage, and accordingly, the cartilage is divided into three zones: superficial zone, middle zone, and deep zone (Figure 2.2). The superficial

(29)

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

zone is composed of tightly-woven collagen fibers, preferably oriented parallel to the surface. The collagen fibers are randomly oriented and fairly homogeneously dis- persed within the middle zone. Within the deep zone, collagen fibers are arranged perpendicular to the surface and partially anastomosed, forming longer fibers. The collagen fibers insert into the calcified cartilage zone anchoring the soft tissue to the bony structure [86] (Figure 2.2). It has been reported that the mentioned arrange- ment of the collagen fibers provides an ideal architecture to minimize the mechanical stress within the cartilage [86].

Chondrocytes are the only cells present (and metabolically active) in the cartilage, insulated within the ECM (Figure 2.2). Chondrocytes are surrounded by a nar- row region called the pericellular matrix (PCM). The whole complex together, com- prising the cell and the surrounding PCM, has been termed as the chondron [98].

The main function of the chondrocytes in cartilage is to synthesize and maintain the ECM [99]. Since the ECM is avascular, chondrocytes rely on diffusion for the exchange of nutrition and waste [98, 99]. Hence, proper loading of chondrocytes is crucial for maintaining normal mechanotransduction and cartilage health. Nonethe- less, these cells are well-adapted to the cartilage environment, i.e., uniquely oper- ating with a very low demand for oxygen [99]. Chondrocytes differ in shape and size throughout the cartilage’s depth. In the superficial layer of the cartilage, chondrocytes have an ellipsoidal shape with their longer axis being in line with collagen fibers whilst they are more spherical within the middle zone. In the deep zone, chondrocytes form column-like structures with multiple cells within a single chondron [98] (Figure 2.2).

(30)

2.2.2 Meniscus

About 70% of the meniscus is comprised of water, with the remaining (~30%) comprised of PGs (~1-2%), collagen (~15-22%), elastin (< 1%), and cells [68,100,101]. The menisci provide up to 20% capacity of the knee shock absorption [102]. This function is attributed to the low permeability and viscous property of the menisci [102]. Un- like the cartilage, the meniscus is (region-specific) vascular and innervated. Hence, the meniscus is suggested to have a proprioceptive role due to the existence of the mechanoreceptors within its anterior and posterior horns [103–105]. In addition, a system of micro-canals close to the blood vessels within the meniscus commu- nicates with the synovial cavity. It has been reported that these micro-canals may provide fluid transport for lubrication and nutrition of the knee joint tissue, i.e., cartilages [106].

Meniscus collagen content and collagen type differ region-wise. Collagen type I is dominant in vascular regions (~80% of the dry weight), while collagen type II is prevalent (~40% of dry weight) in avascular/aneural regions, followed by collagen type I (~30% of dry weight) [107]. The collagen fibers within the superficial zone of the meniscus are randomly oriented, conferring a multi-directional resistance against mechanical loading [108, 109]. Within the deep zone, collagen fibers are circumferentially oriented [109]. The wedge shape of the meniscus and its horn attachments converts the compressive force on the meniscus to horizontal hoop (circumferential) stress. Therefore, the aforementioned collagen architecture of the menisci seems to be appropriately adapted to withstand the stress within the tissue [108, 110, 111].

2.2.3 Ligaments and tendons

Ligaments and tendons, primarily known as connectors between bones and muscles, are composed of water (50-72%), collagen (primarily type I, 55-87% of dry weight), elastin (2-11% of dry weight), PGs (<2.5% of dry weight), cells and other proteins [112–116]. Densely packed collagen fibers run parallel to the axis of loading of the tissue and are considered as the main load-bearing constituent in ligaments and tendons. These collagen fibers have a crimpy pattern in an unloaded ligament [117], which disappears as the tissue undergoes tensile loading. This gradual recruitment of fibers within the ligament results in a nonlinear stress-strain behavior (i.e., toe region), especially for the longitudinal strains less than ~6% [118–120].

Elastin, within the ligaments and tendons, has been known to provide the tissue with reversible elasticity [115, 121] and resists transverse stress and shear deformation [121]. The PG content of the ligaments and tendons is known to facilitates the sliding of the collagen fibers within the densely packed fiber bundles [122, 123].

Cells within the ligaments synthesize the components of the ECM.

2.2.4 Muscle structure and the mechanism of force production

Skeletal muscles are made up of fiber bundles, nerves, blood vessels, cells, and other tissues. The muscle fibers within the skeletal muscles can be divided into intrafusal muscle fibers (providing proprioception) and extrafusal muscle fibers (providing contraction) [124]. The intrafusal muscle fibers, also known as muscle spindle, lie within and alongside the extrafusal muscle fibers [124] (Figure 2.3).

(31)

Figure 2.3: Structure of a typical skeletal muscle.

The extrafusal muscle fibers are organized in bundles, called fascicles (Figure 2.3). These muscle fibers vary in length from a few millimeters up to ~30 cm and in diameter from 10 to 500 µm [80]. They also have different arrangements in muscles, such as parallel to the muscle line of action, pennated, convergent, or fusiform, influencing some of the mechanical properties of muscles. For instance, muscles designed for speed tend to have parallel fibers, whereas the muscles intended for strength are typically pennated [125]. Each fiber within the fascicles is made up of myofibrils. Myofibrils, wrapped in the sarcoplasmic reticulum, are composed of sarcomeres arranged mechanically in series along the myofibril length (Figure 2.3).

The main function of sarcoplasmic reticulum is to store the calcium ions involved in muscle contraction. The sarcomere, as the basic contractile unit of the muscle, is composed of two contractile proteins: myosin and actin (Figure 2.3).

It is thought that the muscle contraction happens when an actin filament slides along the myosin filament, known as the cross-bridge theory [126]. When a motor unit activates the muscle, action potential travels through the muscle fiber causing sarcoplasmic reticulum to release calcium ions. This increase in the calcium ion concentration initiates the cross-bridge formation between the actin and myosin fil- aments leading to muscle contraction. The relaxation of the muscle, on the other hand, happens due to the re-uptake of calcium ions into the sarcoplasmic reticulum.

Both the release and re-uptake of the calcium ions are not instantaneous, causing a delay (~5-50 milliseconds) between muscle excitation (i.e., firing of the motor unit) and muscle activation (i.e., concentration of calcium ions). The relationship between the excitation of a muscle and its activation is known as activation dynamics.

In addition to the activation dynamics, the force generation capacity of the skeletal muscles is also limited to the current length and velocity of the muscle, termed as muscle contraction dynamics [127]. The total force generated by the muscle consists of an active force (due to excitation of the muscle) and a passive force (due to resistance of the muscle against elongation) [128]. Both the active and passive components vary nonlinearly with the muscle length, represented by the force-length curve (Figure 2.4-A). Moreover, the force generated by the muscle varies nonlinearly with the rate of changes of the muscle’s length, represented by the force-velocity curve (Figure 2.4-B). Tendons, as the connector elements between the muscles and

(32)

Figure 2.4: Schematic force-length-velocity relationship of a muscle and force- length relationship of a tendon. (A): Active (in blue) and passive (in orange) force- length relationship, (B): force-velocity relationship and (C): tendon force-length relationship of a skeletal muscle (i.e., contraction dynamics). Forces are normalized to the maximum isometric force of the muscle [127–133].

bones, also display a nonlinear force-length relationship (Figure 2.4-C). The elongation in the tendon alters the whole muscle-tendon’s length that can potentially affect the muscle force-generation capacity, i.e., contraction dynamics [127].

The CNS controls the level of muscle activation, and consequently, the active muscle forces by altering the frequency and magnitude of stimulation passed to motor neurons [80]. Three groups of motor neurons (α,β, andγ) are distinguished by the targeted fiber type, diameter, and propagation velocity. The extrafusal muscle fibers are innervated at neuromuscular junctions along their length by branches ofα (the longer and faster) orβmotor neurons. The muscle spindle, on the other hand, is innervated byβorγmotor neurons for sensing muscle length and force [80,124,134].

Although the mechanism underpinning muscle structure and force production is relatively well-established, the muscle recruitment strategy controlled by CNS is not fully understood [135]. Hence, electromyography (EMG) is conventionally used to measure the train of action potentials along the muscles as a way of exploring the muscle activation patterns and CNS muscle requirement strategy [136].

(33)

3 KNEE JOINT MODELING AND SIMULATION

As briefly explained in chapter 2, the mechanical responses of the knee joint load- bearing tissues (i.e., cartilages and menisci) are attributed to the intricate combination of macroscale and microscale biomechanical and biological factors. Sev- eral studies have provided convincing evidence that altered joint kinematics and kinetics, and consequently, altered tissue-level mechanical responses can effectually lead to undesirable remodeling in the structure, composition, and metabolism of joint load-bearing tissue, and as a result, initiate MS disorders such as KOA [2–5,7,9,31,137–139]. Hence, joint realignment strategies (either non-invasive or sur- gical based), as well as rehabilitation protocols, have been developed to favorably re-distribute and manipulate joint kinematics and kinetics in the interest of prevent- ing or decelerating KOA and increasing the joint’s functionality [1, 10, 16, 140, 141].

Yet, reliable assessment of these interventions requires a thorough knowledge of associated joint-level and tissue-level mechanical responses.

In vivo and in situ measurements of knee joint mechanics, such as JCF, contact area, and contact pressure [4, 17–20, 22, 142, 143], are either limited to specific subjects (e.g., those with instrumented implants) or require highly invasive procedures. More importantly, experimental approaches are unable to measure crucial mechanical quantities such as stress, strain, or fluid flow within the tissue. Alterna- tively, computational methods have become a tool of choice to investigate multiscale biomechanics of the joint tissue. MS models, on the basis of rigid-body dynamics, have been developed and used to predict joint-level quantities such as muscle forces, JCF, and contact pressure. At lower spatial scales (e.g., tissue or cell level scales), continuum mechanics along with numerical approaches (predominantly the FE method) have been extensively utilized to describe the mechanobiology of the tissue. Nonetheless, rather few multiscale models are developed to simulate both joint-level and tissue-level joint mechanics while taking into account different subject-specific aspects such as muscle synergy, joint geometry, kinematics, and kinetics. The most conventional modeling approaches with their potential advantages and limitations are discussed in this chapter.

3.1 MUSCULOSKELETAL MODELING

MS analyses provide multibody dynamic simulations on the basis of an integrated model of anatomy and the neuromusculoskeletal system. To this end, a multibody system (i.e., the MS model) composed of rigid or partly elastic bodies along with joint drivers (e.g., muscles), occasionally with passive resistances (e.g., ligaments), are defined according to the anatomical dimensions of the subject of interest. The MS model of a participant can be created by either scaling of a generic MS model [23], constructing a subject-specific MS model (e.g., using subjects MRIs) [30, 144], or a combination of both methods [24].

In the scaling approach, mass properties (i.e., mass and inertia tensor) and the body segments and length-dependent muscle properties of the MS model are linearly scaled [23]. The scaling is executed based on relative distances between pairs

(34)

of markers attached to the subject (e.g., acquired from the motion capture system) and the corresponding marker locations in the MS model [23]. Cautions (such as precise placement of the model’s markers consistent with those attached to the subject) are required when scaling an MS model to avoid possible scaling errors which may considerably influence the estimated kinematics and kinetics [145]. Although the scaling approach accommodates variations in subject size, subject-specific MS modeling (based on segmentation of the subject’s medical images) [30, 144,146] may be required to account for specific variations in the MS geometries, such as torsional deformities in individuals with cerebral palsy [146].

Nonetheless, MS models in general can provide two different analysis approaches: forward dynamics (FD) and inverse dynamics (ID). FD analyses are intended to estimate movements of the body and the forces applied from the body to the environment (e.g., from the foot to the ground) for a given set of muscle activations. Nevertheless, estimation of physiologically realistic muscle activation patterns in FD problems requires sophisticated predictive algorithms, and consequently, is computationally expensive [147–150]. Conventionally, FD simulations are utilized to explore the principles underlying the locomotion and muscle synergy [147–151]. It worth mentioning that the FD analysis was not used in this thesis, and the developed workflows (i.e., MS analyses) were based on the ID approach.

Hence, the FD approach is described briefly here.

In the second approach of the MS analyses, i.e., ID, the equations of motion are employed to describe the movement of the body segments, the external loads (i.e., forces and moments) applied on the joints, and finally, to estimate the muscle activations/forces required to counterbalance those external loads on the joint during a specific task. By using the equations of motion (i.e., Newton’s second law of motion), the force applied on a body (~F) is related to the rate of change in the momentum (~P) of the body over time (t). Hence, we can write:

~F= d~P

dt (3.1)

For an object with constant mass (m), Equation (3.1) can be re-stated as:

~F= d(m~v)

dt = md(~v)

dt (3.2)

where~vis the translational velocity of the object. Similarly, the net external moment on a body (~τ) is related to its moment of inertia relative to its center of mass (Ic) and its angular velocity (~ω):

~τ= d(Icω)~

dt (3.3)

Accordingly, ID is specifically used to determine forces and moments at each joint (i.e.,~Fand~τin Equations 3.2 and 3.3) responsible for a given body movements (i.e.,~v andω~ in Equations 3.2 and 3.3). Using inverse kinematics (IK), the body movement required by Equations 3.2 and 3.3 are obtained according to the experimental data (Figure 3.1). Several measurements techniques may be utilized to measure body motion, such as radiography based devices (e.g., dual fluoroscopy) [152]

or reflective markers [153]. Radiography based motion capture may provide precise motion data for a specific region of interest; however, reflective-marker based

(35)

Figure 3.1: (A): Inverse kinematics with which joint angles are determined while minimizing the error between the experimental markers (blue) and the corresponding model markers (pink). (B): Inverse dynamics with which the external moments (represented in local coordinate systems) on the joints are calculated.

approaches are more extensively used due to their lower costs, ease of use, and cap- turing a much wider field of view [154]. In this respect, a calibrated set of video cameras is used to capture the 3-dimensional (3D) location of the reflective markers attached to the body. The same number of markers are also defined on the MS model of the subject in the corresponding anatomical locations. In some cases, tracking of an enforced joint angle may be of interest in addition to following the measured marker trajectories.

Finally, an optimization algorithm is utilized to estimate the joint coordinates (i.e., IK) of the MS model at which the experimentally measured joint angles and markers best match with the corresponding joint angles and markers on the MS model, separately at each time point of the motion. Although there are several optimization algorithms [23, 155], the weighted least square optimization method has been most widely used in which the following objective function (fobj) is minimized separately at each time point [23]:

fobj=

∑

M i=1

wi(~r^s_i −~r^m_i )²+

∑

N j=1

wj(θ^s_j−θ^m_j )² (3.4) where~r^s_i and~r^m_i are the position of thei^th marker respectively on the subject and on the model and θ^s_j andθ^m_j are the values for the j^th joint of the subject and the model, respectively. wi andwj are factors to favorably weight the markers and joint angles, separately.

Following the IK, joint velocities and accelerations required by ID (i.e., Equations 3.2 and 3.3), and consequently, the external moments applied to each joint’s DoF can be calculated (Figure 3.1). Next, these external moments should be resolved into the muscle forces/activations, as follows:

τ_i =

∑

N j=1

f_ir_j,i (3.5)

whereτ_i is the external moment applied to thei^thDoF of the model (i.e., obtained from ID, f_j is the force that the j^th muscle of the model should generate, r_(j,i) is

(36)

the moment arm of the j^th muscle around the rotation axis of the i^th DoF of the model, and N is the total number of muscles acting on the i^th DoF of the model.

The fj in Equation (3.5), which will be discussed later in this section, is generally a function of the muscle activation level, the maximum isometric force of the muscle, and current length, and contraction velocity of the muscle-tendon unit (replicating muscle contraction dynamics).

Muscle forces, and consequently, muscle activations, are the unknowns in the Equation (3.5). However, Equation (3.5) is underdetermined since almost all the DoFs of the body joints are driven by more than one muscle (e.g., agonist, antagonist, etc.) and, more importantly, the muscle recruitment strategy controlled by the CNS is far from being fully understood [135]. Accordingly, Equation (3.5) should be solved by considering two aspects. First, to resolve the external moments into the muscle forces, and second, to respect the muscle activation dynamics and contraction dynamics in order to estimate physiologically feasible muscle forces. As a result, the calculation of the muscle forces/activations (Equation 3.5) relies on optimization-based methods, i.e., the assumed criterion of the recruitment strategy of the CNS [23, 24, 27, 135, 156] along with muscle activation/contraction dynamics.

The employed optimization algorithms may estimate muscle activations according to the states of the MS model (e.g., static-optimization-based MS models), or alternatively they might be assisted with extra experimental inputs such as measured EMGs (i.e., EMG-informed MS models). These two approaches will be discussed in more detail in the following sections.

3.1.1 The muscle activation and contraction dynamics

In addition to solving the equations of motion of the system (IK and ID), MS models provide mathematical representations of musculotendon dynamics, i.e., activation dynamics and contraction dynamics of the muscle-tendon structure (Figure 2.4). To this end, two classes of musculotendon models have been developed based on two different spatial levels. At a lower spatial scale, the cross-bridge models [157–159]

were devised on the basis of a knowledge of the fundamental structures of the muscle [158]. This category of the musculotendon models requires numerous (and difficult to measure) parameters with a high computational demand, especially in the MS models with many muscles. Alternatively, lumped-element Hill-type models (at a higher spatial level) provide computationally efficient and easy-to-implement muscle-tendon dynamics [127], and hence, have been widely used in MS analyses [23].

A general Hill-type muscle model can be represented with three components (Figure 3.2). An active contractile element represents the active force generation behavior of the muscle, a passive nonlinear-elastic element represents the passive resistance of the muscle against the elongation, and the tendon element represents the tendon with its force-length relationship (Figures 2.4 and 3.2). The pennation angle of the muscle fibers is considered asα_p(Figure 3.2).

Neglecting the mass of the muscle-tendon unit, the force equilibrium of the muscle-tendon unit results in the following equation:

f_T= f_Mcosαp (3.6)

where f_T and f_M are the force in the tendon and muscle components, respectively

Multiscale computational modeling of the knee joint: Development, validation, and application of musculoskeletal finite element modeling methods

Dissertations in Forestry and Natural Sciences

AMIR ESRAFILIAN

Multiscale computational modeling of the knee joint:

Amir Esrafilian

MULTISCALE COMPUTATIONAL MODELING OF THE KNEE JOINT:

DEVELOPMENT, VALIDATION, AND APPLICATION OF MUSCULOSKELETAL FINITE ELEMENT MODELING

METHODS

TABLE OF CONTENTS

1 INTRODUCTION

2 KNEE JOINT STRUCTURE AND FUNCTION

3 KNEE JOINT MODELING AND SIMULATION

∑

∑

∑