Development of a virtual model of musculoskeletal soft tissue: application to knee menisci
Abstract
L'articulació del genoll, una de les més grans i complexes del cos, té un paper crucial en la locomoció i l'estabilitat de les extremitats inferiors. La cavitat sinovial i el fluid dins de l'articulació del genoll contribueixen a la seva funció general facilitant el moviment, la lubricació de l'articulació, l'absorció d'impactes i el subministrament de nutrients, entre altres. Aquest estudi presenta un enfocament computacional utilitzant el programari obert OpenFOAM per simular el comportament del líquid sinovial en una quarta part de la cavitat del genoll. L'objectiu principal d'aquesta investigació és millorar la geometria, aconseguir una simulació estable i observar el comportament del fluid dins de la regió del genoll. Inicialment, la geometria de la cavitat del genoll es dissenya i millora amb SALOME, una plataforma de codi obert per a la dinàmica de fluids computacional, per representar amb més precisió les característiques anatòmiques. Posteriorment, es defineixen les condicions de contorn apropiades per simular el flux del fluid dins del model virtual a través d'un cicle de marxa. Mitjançant una anàlisi meticulosa dels paràmetres de simulació, s'aconsegueix una simulació estable. Els resultats de la simulació es processen posteriorment amb ParaView, una eina de visualització de codi obert. Es realitzen diversos anàlisis, incloent l'extracció de perfils de velocitat i l'avaluació de les variacions de volum dins de la cavitat simulada del genoll. Els resultats obtinguts proporcionen informació sobre la dinàmica del fluid dins de la cavitat del genoll, contribuint a una comprensió més profunda del comportament del fluid. Els perfils de velocitat revelen els patrons i les característiques del flux, mentre que l'anàlisi de les variacions de volum proporciona informació sobre la deformació i la redistribució del fluid dins de la cavitat del genoll. El model virtual desenvolupat serveix com a eina rellevant per a futures investigacions.
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FINAL DEGREE THESIS Bachelor’s Degree in Mechanical Engineering Bachelor’s Degree in Materials Engineering DEVELOPMENT OF A VIRTUAL MODEL OF MUSCULOSKELETAL SOFT TISSUE: APPLICATION TO KNEE MENISCI Report Author: Judit Llop Borràs Director: Gil Serrancolí Masferrer Co-director: Lluís Jofre Cruanyes Department: EM – Mechanical Engineering Call: June 2023
Development of a Virtual Model of Musculoskeletal Soft Tissue: Application to Knee Menisci i Abstract The knee joint, one of the largest and most complex joints in the body, plays a crucial role in supporting locomotion and providing stability to the lower extremities. The synovial cavity and fluid within the knee joint contribute to its overall function by facilitating movement, lubricating the joint, absorbing shock, and supplying nutrients, among others. This study presents a computational approach using OpenFOAM open-source software to simulate the behaviour of the synovial fluid in one-quarter of the knee cavity. The primary objective of this research is to enhance the geometry representation, achieve a stable simulation and observe the fluid's behaviour within the knee region. Initially, the geometry of the knee cavity is designed and improved in SALOME, an open-source platform for computational fluid dynamics, to represent the anatomical features more accurately. Subsequently, appropriate boundary conditions are defined to simulate the fluid flow within the virtual model through a gait cycle. Through meticulous analysis of the simulation parameters, a stable simulation is achieved. The simulation results are then post-processed using ParaView, an open-source visualization tool. Various analyses are performed, including the extraction of velocity profiles and the assessment of volume variations within the simulated knee cavity. The obtained results provide valuable insights into the fluid dynamics within the knee cavity, contributing to a deeper understanding of fluid behaviour. The velocity profiles reveal flow patterns and characteristics, while the analysis of volume variations provides information on deformation and fluid redistribution within the knee cavity. The developed virtual model serves as a valuable tool for further research in understanding and analysing the biomechanics of the knee joint and going further with the main project, which aims to develop a multi-scale model that can combine concepts in the field of biomechanics and fluid biodynamics.
Report ii Resum L'articulació del genoll, una de les més grans i complexes del cos, té un paper crucial en la locomoció i l'estabilitat de les extremitats inferiors. La cavitat sinovial i el fluid dins de l'articulació del genoll contribueixen a la seva funció general facilitant el moviment, la lubricació de l'articulació, l'absorció d'impactes i el subministrament de nutrients, entre altres. Aquest estudi presenta un enfocament computacional utilitzant el programari obert OpenFOAM per simular el comportament del líquid sinovial en una quarta part de la cavitat del genoll. L'objectiu principal d'aquesta investigació és millorar la geometria, aconseguir una simulació estable i observar el comportament del fluid dins de la regió del genoll. Inicialment, la geometria de la cavitat del genoll es dissenya i millora amb SALOME, una plataforma de codi obert per a la dinàmica de fluids computacional, per representar amb més precisió les característiques anatòmiques. Posteriorment, es defineixen les condicions de contorn apropiades per simular el flux del fluid dins del model virtual a través d'un cicle de marxa. Mitjançant una anàlisi meticulosa dels paràmetres de simulació, s'aconsegueix una simulació estable. Els resultats de la simulació es processen posteriorment amb ParaView, una eina de visualització de codi obert. Es realitzen diversos anàlisis, incloent l'extracció de perfils de velocitat i l'avaluació de les variacions de volum dins de la cavitat simulada del genoll. Els resultats obtinguts proporcionen informació sobre la dinàmica del fluid dins de la cavitat del genoll, contribuint a una comprensió més profunda del comportament del fluid. Els perfils de velocitat revelen els patrons i les característiques del flux, mentre que l'anàlisi de les variacions de volum proporciona informació sobre la deformació i la redistribució del fluid dins de la cavitat del genoll. El model virtual desenvolupat serveix com a eina rellevant per a futures investigacions en la comprensió i l'anàlisi de la biomecànica de l'articulació del genoll, i per continuar amb el projecte principal que té com a objectiu desenvolupar un model multi-escala que pugui combinar conceptes en el camp de la biomecànica i la biodinàmica de fluids.
Development of a Virtual Model of Musculoskeletal Soft Tissue: Application to Knee Menisci iii Resumen La articulación de la rodilla, una de las más grandes y complejas del cuerpo, desempeña un papel crucial en la locomoción y la estabilidad de las extremidades inferiores. La cavidad sinovial y el fluido dentro de la articulación de la rodilla contribuyen a su función general al facilitar el movimiento, la lubricación de la articulación, la absorción de impactos y el suministro de nutrientes, entre otros. Este estudio presenta un enfoque computacional utilizando el software abierto OpenFOAM para simular el comportamiento del líquido sinovial en una cuarta parte de la cavidad de la rodilla. El objetivo principal de esta investigación es mejorar la geometría, lograr una simulación estable y observar el comportamiento del fluido dentro de la región de la rodilla. Inicialmente, la geometría de la cavidad de la rodilla se diseña y mejora en SALOME, una plataforma de código abierto para la dinámica de fluidos computacional, para representar con más precisión las características anatómicas. Posteriormente, se definen las condiciones de contorno apropiadas para simular el flujo del fluido dentro del modelo virtual a través de un ciclo de marcha. Mediante un análisis meticuloso de los parámetros de simulación, se consigue una simulación estable. Los resultados de la simulación se procesan posteriormente con ParaView, una herramienta de visualización de código abierto. Se realizan varios análisis, incluida la extracción de perfiles de velocidad y la evaluación de las variaciones de volumen dentro de la cavidad simulada de la rodilla. Los resultados obtenidos proporcionan información sobre la dinámica del fluido dentro de la cavidad de la rodilla, contribuyendo a una comprensión más profunda del comportamiento del fluido. Los perfiles de velocidad revelan los patrones y las características del flujo, mientras que el análisis de las variaciones de volumen proporciona información sobre la deformación y la redistribución del fluido dentro de la cavidad de la rodilla. El modelo virtual desarrollado sirve como una relevante herramienta para futuras investigaciones en la comprensión y el análisis de la biomecánica de la articulación de la rodilla, y para continuar con el proyecto principal que tiene como objetivo desarrollar un modelo multiescala que puede combinar conceptos en el campo de la biomecánica y la biodinámica de fluidos.
Report iv Acknowledgments Firstly, I would like to thank my directors of the project, Gil and Lluís, for giving me the opportunity to take part of this ambitious project, for all their support, patience, and kindly help, because they have always been accessible and ready to give some guidance. I am also immensely grateful for all the help and support I have received from my family and friends, who have always supported and encouraged me throughout my studies and new challenges. Thanks to you all, today I hand over this project.
Development of a Virtual Model of Musculoskeletal Soft Tissue: Application to Knee Menisci v Index ABSTRACT ____________________________________________________________ I RESUM _____________________________________________________________ II RESUMEN ___________________________________________________________ III ACKNOWLEDGMENTS ________________________________________________ IV 1. PREFACE _______________________________________________________ 11 1.1. Background ............................................................................................................ 11 1.2. Motivation .............................................................................................................. 12 2. INTRODUCTION _________________________________________________ 13 2.1. Objective ................................................................................................................ 13 2.2. State of the Art ....................................................................................................... 14 2.3. Scope of the Project ............................................................................................... 16 3. THEORETICAL FRAMEWORK _______________________________________ 18 3.1. Knee Joint Anatomy and Biomechanics ................................................................ 18 3.1.1. Tissues of Knee Joint ............................................................................................... 19 3.1.2. Synovial Joints ......................................................................................................... 20 3.1.3. Biomechanics of the Knee Joint ............................................................................. 20 3.2. Synovial Cavity and Fluid ....................................................................................... 22 4. METHODOLOGY_________________________________________________ 24 4.1. Geometry ............................................................................................................... 24 4.1.1. Geometry Improvement ......................................................................................... 24 4.1.2. Geometry Characterisation .................................................................................... 26 4.2. Mesh generation .................................................................................................... 27 4.2.1. Mesh Hypothesis ..................................................................................................... 27 4.2.2. Sub-meshes Construction ....................................................................................... 28 4.2.3. Patches Creation ..................................................................................................... 30 4.2.4. Mesh Refinement .................................................................................................... 31 4.2.5. Mesh Exportation .................................................................................................... 31 4.3. Movement and Boundary Conditions ................................................................... 32 4.3.1. Boundary Conditions .............................................................................................. 32 4.3.2. Velocity and Pressure ............................................................................................. 34 4.3.3. Velocity Calculation ................................................................................................. 37
Report vi 4.3.4. Simulation Parameter Definition ........................................................................... 42 4.4. Solver ..................................................................................................................... 45 4.4.1. Navier-Stokes Equations ......................................................................................... 45 4.4.1.1. Mass Conservation ......................................................................................... 46 4.4.1.2. Momentum Conservation ............................................................................. 47 4.4.1.3. Energy Conservation ...................................................................................... 47 4.4.2. PimpleFoam ............................................................................................................. 48 4.5. Post-Processing ...................................................................................................... 49 4.5.1. ParaView .................................................................................................................. 49 5. RESULTS AND ANALYSIS __________________________________________ 50 5.1. Effect of Viscosity on Pressure Results .................................................................. 50 5.2. Volume Results ...................................................................................................... 53 5.3. Velocity Results ...................................................................................................... 54 5.3.1. Magnitude of Velocity ............................................................................................ 55 5.3.2. X-direction Velocity (Ux) ......................................................................................... 60 5.3.3. Y-direction Velocity (Uy) ......................................................................................... 66 5.3.4. Velocity Profile ........................................................................................................ 71 5.3.5. Velocity Glyphs ........................................................................................................ 73 5.4. Surface Flow ........................................................................................................... 76 6. ENVIRONMENTAL ANALYSIS _______________________________________ 78 7. CONCLUSIONS __________________________________________________ 79 ECONOMIC ANALYSIS _________________________________________________ 81 REFERENCES ________________________________________________________ 82
Development of a Virtual Model of Musculoskeletal Soft Tissue: Application to Knee Menisci vii Index of Figures Figure 1: Human knee joint [15]. ___________________________________________________ 18 Figure 2: Six different types of synovial joints [19]. _____________________________________ 20 Figure 3: The six degrees of freedom of the TF joint [22]. ________________________________ 21 Figure 4: Synovial cavity of the knee joint [24]. ________________________________________ 22 Figure 5: Lower part of the femur [30]. ______________________________________________ 25 Figure 6: RMI of the knee and the cavity's geometry on top [31]. _________________________ 25 Figure 7: Dimensions of the quarter of the synovial cavity. ______________________________ 26 Figure 8: Resulting improved geometry in SALOME. ____________________________________ 27 Figure 9: Resulting quadratic mesh from the improved geometry in SALOME. _______________ 29 Figure 10: Resulting mesh with corresponding patches. _________________________________ 30 Figure 11: Unit system in OpenFOAM [33]. ___________________________________________ 37 Figure 12: Velocity values for one cycle of 1s. _________________________________________ 38 Figure 13: Eulerian and Lagrangian representation of fluid flow equations [44]. _____________ 45 Figure 14: Pressure map for nu 1 at 0,36 s. ___________________________________________ 51 Figure 15: Pressure map for nu 2 at 0,36 s. ___________________________________________ 51 Figure 16: Pressure map for nu 3 at 0,36 s. ___________________________________________ 52 Figure 17: Variation of volume during a gait cycle. _____________________________________ 53 Figure 18: Geometry with the lines to plot the velocity. _________________________________ 54 Figure 19: Magnitude of velocity at 0.36 s. ___________________________________________ 55 Figure 20: Coloured map of magnitude of velocity at 0.36 s. _____________________________ 56 Figure 21: Magnitude of velocity at 0.53 s. ___________________________________________ 56 Figure 22: Coloured map of magnitude of velocity at 0.53 s. _____________________________ 57 Figure 23: Magnitude of velocity at 0.62 s. ___________________________________________ 57 Figure 24: Coloured map of magnitude of velocity at 0.62 s. _____________________________ 58
Report 14 2.2. State of the Art Computational Fluid Dynamics (CFD) simulations have been used extensively to study the fluid dynamics in the knee joint, which is a complex biomechanical system. In recent years, there have been significant advancements in knee modelling techniques and the accuracy of CFD simulations. However, a limitation in most of the studies is that they do not consider the cartilage and meniscus as biphasic material, as it has both solid and fluid phase or that they do not simulate the synovial fluid inside the cavity nor in a geometry similar to the cavity. Here are some surveys and review papers that study the modelling of synovial fluid in the knee joint: 1. "Mathematical Modelling of Synovial Fluids Flow" by P.Pustejovská [3]. This study models the synovial fluid flow, describing the fluid as an incompressible, viscous and non-Newtonian fluid, inside a two-dimensional rectangular domain with a focus on biochemical and rheological properties of the synovial fluid. The researcher propose two different viscosity models, which are derived from a combination of experimental data and numerical solutions. In contrast to the current study, the main difference lies in the fluid type being considered. While this study treats the synovial fluid as a non-Newtonian fluid, the current study investigates a Newtonian fluid. However, both studies share the common characteristics of incompressibility and being modelled within a two-dimensional cavity. It is important to note that the non-rectangular approach adopted by the current study is more appropriate for accurately representing the synovial knee cavity, unlike the rectangular approach used in this study. 2. “Computational Model for the Generalised Dispersion of Synovial Fluid” by M. Alshehri and S. K. Sharma [4]. In this study the theory of the convective diffusion mechanism occurring in the knee is studied, using a configuration of bearing model for analysis of two rectangular plates of infinite length. The partial differential equations are solved for the Newtonian fluid and an analytical solution for the unsteady dispersion problem has been obtained, which allows to understand better the nutritional transport phenomena of the synovial joint. The synovial fluid is represented as a viscous Newtonian fluid located between two rigid plane surfaces that are approaching each other. To represent the cartilage, a porous boundary is taken into account. While the study primarily focuses on the knee cavity bearing rather than fluid motion, a twodimensional rectangular plate is used as the domain, which is reasonable but not ideal since it does not simulate the actual geometry of the synovial cavity. The key finding of the study is the distribution of nutrients through the synovial fluid, which is influenced by the concentration of hyaluronic acid molecules and other proteins, using a mathematical model.
Development of a Virtual Model of Musculoskeletal Soft Tissue: Application to Knee Menisci 15 3. “A Model of Synovial Fluid Lubricant Composition in Normal and Injured Joints” by M. Blewis, G. Nugent-Derfus, T. Schmidt, B. Schumacher, and R. Sah [5]. The objective of this study was to create a quantitative intercompartmental model that could predict the concentration of synovial fluid lubricants in the knee joint. The model consists of a synovial fluid compartment surrounded by articular cartilage and a semi-permeable synovial membrane, and was designed to integrate various biophysical processes, enabling the identification of environmental and clinical factors that impact the composition of synovial fluid. An advantageous aspect of this study was the integration of multiple layers, including the inclusion of cartilage and the synovial membrane. While this approach is commendable, it is important to note that the model primarily focused on biological aspects. 4. “Computed Synovial Fluid Flow in a Simple Knee Joint Model” by K. Pekkan, R. Nalim and H. Yokota [6]. The study focuses on predicting shear stress induced by fluid flow on knee joint cells, represented in a sinusoidal oscillating flat plate. It examines oscillatory flow of a Newtonian synovial fluid in two-dimensional joint geometries using an experimental model. Additionally, the effect of articular cartilage curvature is investigated in a model with and without menisci. Like the current study, the synovial fluid is considered Newtonian and the domain in this research is also two-dimensional. A notable aspect is the investigation of fluid motion within curved cartilage, and investigating how the inclusion of the menisci in the model affects the shear stresses. On the other hand, a sinusoidal oscillating flat plate typically represents a simplified mechanical system but comparing a knee synovial cavity to it may not be the most appropriate approach. While both involve oscillatory motion, the nature and behaviour of a knee synovial cavity and a sinusoidal flat plate are fundamentally different, a knee synovial cavity is a complex structure with fluid-filled compartments, cartilage, and other anatomical features. 5. “Numerical Modeling of the Knee Synovial Joint Under Deformation Using the Finite Volume Method” by N. Martínez-Gutiérrez, L. A. Ibarra-Bracamontes, S. R. Galván-González, A. AguilarCorona & G. Viramontes-Gamboa [7]. In this study, a two-dimensional model of a knee synovial joint is presented, focusing on the angular pressure distribution along the synovial fluid-cartilage interface under simulated loads. The model incorporates a mobile circular wall for the condyle, a larger circular wall for the condylar concave cavity, porous medium for the articular cartilage, and high viscosity Newtonian fluid for the synovial fluid. This study incorporates the loads experienced by the knee, which is expected to yield favourable outcomes. It also includes an analysis of the interaction between the fluid and cartilage at the interface, while defining the synovial fluid as Newtonian, akin to the current study. Although this study may seem the most advanced and bears similarities with the current study, primarily
Report 16 focuses on pressure rather than fluid motion and utilizes a quarter-circle geometry as an approximation, the geometry should aim for greater accuracy. A positive aspect of this study is the consideration of porous cartilage in the analysis of angular pressure distribution and the inclusion of compressive movement in the model. However, the study does not further investigate fluid motion. 6. “On the Modelling of Synovial Fluid” by J. Hron, J. Málek, P. Pustějovská, and K. Rajagopal [8]. In this study, researchers focused on synovial fluid, describing it as a polymeric liquid with viscoelastic properties due to the presence of hyaluronan (HA) molecules. Depending on the concentration of HA the synovial fluid was having a Newtonian or non-Newtonian behaviour. The authors examined the behaviour of synovial fluid when it behaves as a viscous fluid. They developed a new generalized power-law fluid model that incorporates the concentration of hyaluronan as a factor in determining the power-law exponent. The researchers aimed to describe the flow of synovial fluid inside a rectangular cavity under certain conditions, assuming that it is not subjected to immediate changes. The model included constraints of incompressibility, the balance of linear momentum, and a convection-diffusion equation to account for the concentration of hyaluronan. To validate their models, the researchers compared their results with predictions from a previously used model for synovial fluid. One model had a very good approach to studying the fluid behaviour, but in the end, the simulation was not conducted inside a real-shaped cavity but rather a rectangular one. 2.3. Scope of the Project This study is part of an ongoing project aimed at developing a virtual model that accurately replicates the knee. The global objective of this project is to provide a dynamic virtual representation of the knee allowing for monitoring, analysis, and optimization. This actual study builds upon previous final degree [9] or master projects [10], using and refining existing data while also incorporating additional improvements to enhance accuracy. In previous studies, an introduction to OpenFOAM software was conducted, which provided beneficial in gaining familiarity with the software before embarking on the actual project. Additionally, a preliminary simulation approach from the quarter of the synovial cavity of the knee was obtained from previous studies, which helped understanding how to address the problem of study. In those previous studies, a first geometry was designed and a simulation with a vertical movement was performed. Building upon this foundation and once acquainted with the software, the next step involved an
Development of a Virtual Model of Musculoskeletal Soft Tissue: Application to Knee Menisci 17 improvement of the synovial cavity geometry to attain a more accurate representation, ensuring at the same time a stable dynamic mesh and readjusting the boundary conditions for accurate simulation. The primary limitation of the project is the time allocated for a Bachelor thesis, in this case a double Bachelor thesis. While there may be a desire to investigate deeper into the subject matter, it is crucial to adapt the research to accommodate the time constraint.
Report 18 3. Theoretical Framework 3.1. Knee Joint Anatomy and Biomechanics The knee joint is essentially made up of three bones: the thigh bone (femur), the shinbone (tibia) and the kneecap (patella). The joint head of the femur has two strong bony projections, the condyles, which are covered with articular cartilage. The articular surface of the tibia is called the tibia plateau [11]. The human knee is the largest joint in the musculoskeletal system, which supports the body weight and facilitates locomotion. The knee consists of two distinct articulations, the tibiofemoral and the patellofemoral joints [12]. The tibiofemoral joint is one of the most complex articulations of the human body and its main tissues are the femur, tibia, fibula, articular cartilages, menisci, and ligaments. The tibiofemoral joint enables the relative motion of the femur and tibia, which is facilitated through mechanical contacts between the cartilages and menisci [13]. Another articulation is the patellofemoral. The function of the patella is multifaceted. Its primary purpose is to serve as a mechanical pulley for the quadriceps as the patella changes the direction of the extension force throughout knee range of motion [14]. A human knee join can be seen in Figure 1. Figure 1: Human knee joint [15].
Development of a Virtual Model of Musculoskeletal Soft Tissue: Application to Knee Menisci 19 3.1.1. Tissues of Knee Joint In the Knee Join apart from the hard tissues, the bones described above, soft tissues can be found. There are different types of soft tissue: - Cartilage: slippery and flexible, hyaline (articular) cartilage within the knee joint allows, has less friction than two pieces of glass placed together. This allows the joint to move with minimal friction in a healthy knee. There are two primary types of cartilage in the knee: Articular (Hyaline), which covers the bones where they meet at the knee joint, and Meniscus (Fibrocartilage), two C-shaped cartilaginous structures within the knee [16]. - Tendons: are flexible tissues that attached muscle to bone. In the knee 3 different can be found: o The hamstring, there are three hamstring tendons that cross the knee joint on the back of the knee. Two are on the inside (medial) part of the knee attaching to the shin bone (Semimembranosus and Semitendinosus) and one is on the outside (lateral) part of the knee, attaching to the fibula (Biceps femoris). o The quadriceps, this tendon is composed of contributions from the four quadriceps muscles (the vastus lateralis, vastus intermedius, vastus medialis, and rectus femoris). It attaches the powerful quadriceps muscles to the top part of the patella. o The patellar, which attaches the bottom part of the patella to the top part of the tibia. - Ligaments: fibrous bands of tissue that connect bones to other bones. In the knee 4 can be found: o the Anterior Cruciate Ligament (ACL), that connects the tibia to the femur and functions to prevent the tibia from sliding forward on the femur. o The Posterior Cruciate Ligament (PCL). The PCL also connects the tibia to the femur. It functions to prevent the tibia from sliding backward on the femur. The PCL works with the ACL for stabilization of the knee. o The Lateral Collateral Ligament (LCL). The LCL, which is also known as the fibular collateral ligament, is located on the outside (lateral side) of the knee. It connects the outside, bottom edge of the femur to the outside, top edge of the fibula. The LCL helps stabilize the knee joint by limiting outward (varus) force across the knee. o The Medial Collateral Ligament (MCL). The MCL is located on the inside (medial side) of the knee, connecting the inside, bottom edge of the femur with the inside, top edge of the tibia. The MCL helps to stabilize the knee by limiting inward (valgus) force across the knee. The MCL works with the LCL to prevent unwanted side-to-side motion. - Membrane: the synovial membrane surrounds all the joints in the knee and holds the synovial fluid inside. This fluid will provide lubrication and necessary nutrients to the knee joint [16].
Report 20 3.1.2. Synovial Joints The classification of joints is based on the amount of movement possible. All diarthroses, or freely movable joints, are synovial joints because they share similarities of structure [17]. Because of the presence of a synovial membrane, diarthrodial joints are often referred to as synovial joints [18]. Seven different categories of diarthrodial (synovial) joints exist, each with unique functional abilities; however, all synovial joints contain the seven common elements: synovial fluid, articular cartilage, articular capsule, synovial membrane, capsular ligaments, blood vessels and sensory nerves[17]. In Figure 2, six different types of synovial joints are shown, being A: Plane, B: Hinge, C: Pivot, D: Ball-andsocket, E: Condyloid and F: Saddle type. Figure 2: Six different types of synovial joints [19]. In the case of study, the knee joint, it is a bi-condylar type of synovial joint, which mainly allows for flexion and extension (and a small degree of medial and lateral rotation). Condyloid joints are composed of the articulation between a large, rounded, convex member and a relatively shallow concave member. Most often, these joints allow 2 degrees of freedom; ligaments as well as the bony structure of the joint typically prevent motion from occurring in a third plane [17]. 3.1.3. Biomechanics of the Knee Joint The knee joint comprises two distinctly separate joints/articulations; the tibiofemoral (TF) joint - the articulation of the femur over the tibia - and the patellofemoral (PF) joint - the articulation of the patella over the femur. The main roles of the knee joint complex are to allow locomotion with minimum energy requirements from the muscles and stability, accommodating for different terrains; and to
Development of a Virtual Model of Musculoskeletal Soft Tissue: Application to Knee Menisci 21 transmit, absorb and redistribute forces caused during the activities of daily life[20]. The movement of the knee joint involves several key actions, primarily flexion and extension, but also including slight rotation and gliding movements. The primary motion of the TF joint is rotation in the sagittal plane (flexion-extension); therefore, a simplified description is that it acts as a hinge joint in the sagittal plane. The complex three-dimensional motion of the TF joint can be described through six degrees of freedom (three rotations and three translations [21]. The six degrees of freedom of TF joint motion are visible in Figure 3. Mediolateral and flexion-extension movements take place along the epicondylar femoral axis, while joint distraction and internal-external rotation occur along the tibial long axis. Anterior-posterior translation and varusvalgus rotation occur along a floating axis that is perpendicular to both the femoral epicondylar and tibial long axes. Figure 3: The six degrees of freedom of the TF joint [22]. At full extension, usually defined when the long axes of the tibia and femur are aligned in the sagittal plane, the femur is internally rotated with respect to the tibia and is located anterior to the mid-point of the tibial plateau [20], [21]. As the knee flexes, the lateral condyle of the femur consistently translates posteriorly on the lateral tibial plateau with increasing flexion. Active knee flexion is possible primarily through hamstring contraction and usually reaches 130º, whereas passive flexion can reach 160º [21]. The medial condyle on the other hand experiences a less consistent motion pattern that includes posterior translation and anterior slide during knee flexion. This leads to a reduced overall posterior translation of the medial condyle on the medial tibial plateau from full extension to maximum knee flexion. The larger magnitude of the lateral femoral translation, when coupled with a lesser
Report 22 translation of the medial femoral translation, leads to the femur externally rotating with respect to the tibia as the knee flexes [20]. This complex mechanism of knee joint motion means that the geometry itself is not adequate to maintain stability, requiring input from passive soft tissue (e.g., ligaments) and muscle tensions. It also means that large forces acting on small articulating areas generate high articular stresses, commonly called joint contact pressure [21]. 3.2. Synovial Cavity and Fluid The synovial cavity of the knee, also known as the knee joint cavity, is a space within the knee joint that is filled with synovial fluid, a representation is shown in Figure 4. The purpose of this fluid is to lubricate the cartilage of the bone joint and provide nourishment through diffusion. It is made from a ultrafiltrate of blood plasma and is regulated by the synovium, a specialized connective soft-tissue membrane that lines the inner surface of synovial joint capsules [23]. The synovial fluid within the synovial cavity helps reduce friction, nourishes the joint structures, and provides shock absorption during movements of the knee joint. Figure 4: Synovial cavity of the knee joint [24]. On the joint surface of each bone is the articular cartilage, which provides a smooth surface. The joint capsule, made of fibrous connective tissue, encloses the joint in a strong sheath, like a sleeve. Lining the joint capsule is the synovial membrane, which secretes synovial fluid into the joint cavity. Synovial
Development of a Virtual Model of Musculoskeletal Soft Tissue: Application to Knee Menisci 23 fluid is thick and slippery and prevents friction as the bones move. Many synovial joints also have bursae (or bursas), which are small sacs of synovial fluid between the joint and the tendons that cross over the joint. Bursae permit the tendons to slide easily as the bones are moved. If a joint is used excessively, the bursae may become inflamed and painful; this condition is called bursitis [18]. The lubrication for the natural joint is provided by a small amount of viscous fluid consisting of mainly hyaluronic acid, proteins, and other factors such as lubricin and surface-active phospholipid (SAPL) that allow for effective gliding of the joints [25]. As with the tissue, the characteristics and properties of the fluid vary considerably as the result of joint health, injury status and other host factors such as inflammation and hypersensitivity. Injury to the joint may result in an abundance of fluid (a joint effusion), distension of the tissues and interruption of the normal functions of the synovial tissue [26]. Synovial fluid composition and visual characteristic are strongly dependent on the state of health of the patient, without any disorders the synovial fluid has egg white like consistency and colour [3]. In hemarthroses it will be red or orange and in inflammatory arthropathies may appear cream or white. Normal synovial fluid has a thick mucoid consistency because of complex interactions between proteins and proteoglycans. These interactions are fundamental to the lubricating properties of synovial fluid [27]. The synovial fluid was assumed to be a homogeneous and incompressible Newtonian fluid with a density of 1010 kg/m3 and a dynamic viscosity of 0.008 kg/(m·s) [28].
Report 30 4.2.3. Patches Creation To be able to assign different properties to the different cells of the geometry, patches should be defined. Patches can be understood like each face on the surface that forms the surface geometry. In 4.3.1. Boundary Conditions section, basic types of patches will be explained. The user can create these patches using SALOME feature: groups of sub-shapes of the geometry. A group can be formed by different types of elements such as nodes, faces or volumes among others. In the case of study, the groups will be created using faces as the element types and this groups can consist of one or more faces. All the groups can be named and by doing this, the user is later able to establish different boundary conditions, pressure, and velocity to each group in OpenFOAM software. In Figure 10 the different names for each group or patches can be seen as follows. Figure 10: Resulting mesh with corresponding patches. The cavity is divided into 7 main patches: left, movingWall, farFieldWall, right, fixedWall, front and back. Some of these main patches have been divided in cells (or other patches) to later be able to assign different velocities to each cell or patch. For example, MovingWall has been divided into 30 groups,
Development of a Virtual Model of Musculoskeletal Soft Tissue: Application to Knee Menisci 31 from MovingWall1 to MovingWall30, so it is formed by 30 patches. Left and right have been divided into 15 groups or faces, and the others, fixedWall, front, back and farFieldWall, all boundary faces form one group or one patch. From now on in this study, when mentioning this groups, faces or cells, only the name of patches will be used. 4.2.4. Mesh Refinement Mesh refinement is the process of improving the quality of a mesh by adding more vertices or adjusting the position of existing vertices. To do so, the user can use the Local Refinement option to refine specific areas of the mesh, or the Global Refinement option to refine the entire mesh. In this study, the mesh refinement was in the global geometry and the process involved increasing the number of segments per edge to achieve smaller cell sizes, resulting in a higher cell count. The main goal was to get a smooth mesh with sufficiently small cells in the cavity of the femoral condyle, as the mesh refinement can increase the accuracy of the simulation by reducing numerical errors. Once the mesh is done, the user can use an extra function in OpenFOAM to check the mesh quality. The following function has been used: checkMesh, which checks the validity of the mesh to ensure that the refined mesh has good topology and is free of errors. 4.2.5. Mesh Exportation Last step is to export the mesh. SALOME offers various formats like UNV, MED and CGND, for use in numerical simulations. In this study the UNV format has been used and will be converted to Foam format, to be able to read it in OpenFOAM. In Algorithm 1, the command used can be seen, being Mesh_1 the name of the geometry’s file in SALOME. ideasUnvToFoam Mesh_1.unv Algorithm 1: Command to export the mesh to UNV format. Running this command, a new folder named constant will be created. Inside this folder, another folder named polyMesh can be found, where all the information about the geometry can be obtained, including the boundary file where the user can see all the boundary patches and later edit the boundary conditions. This will be further explained in chapter 4.3.1. Boundary conditions.
Report 32 4.3. Movement and Boundary Conditions 4.3.1. Boundary Conditions Setting appropriate boundary conditions is vital for a successful simulation. Ill-posed boundary conditions will lead to physically incorrect predictions, and in many cases solver failure. Users must specify the boundary conditions for each solved field. We first need to consider that, for the purpose of applying boundary conditions, a boundary is generally broken up into a set of patches. One patch may include one or more enclosed areas of the boundary surface which do not necessarily need to be physically connected [32]. Basic patch types are: patch The basic patch type for a patch condition that contains no geometric or topological information about the mesh (with the exception of wall), e.g. an inlet or an outlet [32]. wall There are instances where a patch that coincides with a wall needs to be identifiable as such, particularly where specialist modelling is applied at wall boundaries. A good example is wall turbulence modelling where a wall must be specified with a wall patch type, so that the distance from the wall of the cell centres next to the wall are stored as part of the patch [32]. symmetry This boundary condition enforces a symmetry constraint. For any (non-planar) patch which uses the symmetry plane (slip) condition [32]. empty While OpenFOAM always generates geometries in 3 dimensions, it can be instructed to solve in 2 (or 1) dimensions by specifying a special empty condition on each patch whose plane is normal to the 3rd (and 2nd) dimension for which no solution is required. For solutions in in 2 (or 1) dimensions (2D/1D), the type used on each patch whose plane is normal to the 3rd (and 2nd) dimension for which no solution is required [32]. In the case of study, a generic patch, patch, will be used as the condition for movingWall, right, and farFieldWall; slip condition for left; symmetry condition, symmetry, will be used for fixedWall; and
Development of a Virtual Model of Musculoskeletal Soft Tissue: Application to Knee Menisci 33 empty boundary condition for front and back patches. An example of the boundary file can be seen below in Setting 1Error! Reference source not found.. ( front { type empty; nFaces 450; startFace 855; } back { type empty; nFaces 450; startFace 1305; } fixedWall { type symmetry; nFaces 15; startFace 1755; } farFieldWall { type patch; nFaces 15; startFace 1770; } left1 { type slip; nFaces 1; startFace 1785; } ... movingWall1 { type patch; nFaces 1; startFace 1800; } ... right1 { type patch; nFaces 1; startFace 1830; } ... ) Setting 1: Part of the boundary file in PolyMesh folder.
Report 34 The patch types are specified in the mesh and field files of a OpenFOAM case. More precisely, the base type is specified under the type keyword for each patch in the boundary file, located in the constant/polyMesh directory [32]. In boundary file the base type is specified, this will be discussed in the following section. 4.3.2. Velocity and Pressure In the time directory folder, a folder that contains the output files generated during a time-dependent OpenFOAM simulation, the user need to introduce all input data at the initial time step. Usually "0” will be defined as the initial time step. This input data will mainly be velocity and pressure definition but could contain other variables. Boundary conditions in OpenFOAM are used to specify the values of variables of the computational domain such as velocity and pressure. To define parameters at the boundaries, the user should apply specific boundary condition types to each boundary face. Some basic boundary conditions to define those parameters are: calculated This boundary condition is not designed to be evaluated; it is assmued that the value is assigned via field assignment, and not via a call to e.g. updateCoeffs or evaluate [32]. fixedValue This boundary condition supplies a fixed value constraint, and is the base class for a number of other boundary conditions [32]. zeroGradient This boundary condition applies a zero-gradient condition from the patch internal field onto the patch faces. That means the gradient of respective quantity is zero, meaning that the actual value is constant [32]. totalPressure This boundary condition provides a total pressure condition, is uniform total pressure applied everywhere in the inlet patch. Total pressure value is equal to p0 [32]. movingWallVelocity
Development of a Virtual Model of Musculoskeletal Soft Tissue: Application to Knee Menisci 35 This boundary condition provides a velocity condition for cases with moving walls [32]. For the case of study, the boundary conditions used for velocity and pressure are shown in Table 1. Table 1. Boundary conditions for each patch Patch U (velocity) p (pressure) left slip slip movingWall movingWallVelocity totalPressure farFieldWall zeroGradient zeroGradient right movingWallVelocity zeroGradient Back empty empty front empty empty fixedWall symmetry symmetry In Setting 2Error! Reference source not found., part of the p file can be seen, where the user can add all boundary conditions regarding the pressure. dimensions [0 2 -2 0 0 0 0]; internalField uniform 1000; boundaryField { movingWall1 { type totalPressure; p0 $internalField; } ... fixedWall { type symmetry; } farFieldWall { type zeroGradient; } left1 { type slip; } ...
Report 36 right1 { type zeroGradient; } ... back { type empty; } front { type empty; } } Setting 2: Part of the pressure file in 0 folder. In Setting 3, part of the U file is shown, where the user can describe the boundary conditions related to the velocity. dimensions [0 1 -1 0 0 0 0]; internalField uniform (0 0 0); boundaryField { movingWall1 { type movingWallVelocity; value $internalField; } ... farFieldWall { type zeroGradient; } fixedWall { type symmetry; } left1 { type slip; } ...
Development of a Virtual Model of Musculoskeletal Soft Tissue: Application to Knee Menisci 37 right1 { type movingWallVelocity; value $internalField; } ... back { type empty; } front { type empty; } } Setting 3: Part of the velocity file in 0 folder. In U file, dimensions entry specifies the dimensions of the field. In Figure 11, it is shown how the system does work in OpenFOAM. For each position inside the bracket of dimensions a property is related and accordingly the unit. For example, in U file, dimensions [0 1 -1 0 0 0 0]; equals to m·s−1 that is the same as saying m/s, as there is a 1 in the second position (length) and -1 in third position (time). Figure 11: Unit system in OpenFOAM [33]. 4.3.3. Velocity Calculation In the study, the movement done by the knee when a human is walking is simulated. The walking gait can be divided into two main phases: stance (about 0-65% of gait) and swing phases (about 65-100% of gait) [34], [35]. The stance phase consists of three subphases: initial (heel strike to foot flat), middle (foot flat to opposite heel strike), and terminal stance (opposite heel strike to toe off) [35], [36]. The knee joint in the stance phase is regarded as a shock damping mechanism to accept the body weight
Report 38 [37]. The swing phase consists of two subphases: initial (toe off to knee maximum flexion) and terminal swing (knee maximum flexion to heel strike) [35], [36]. The main function of the knee in the swing phase is assisting flexion-extension for toe clearance, foot placement, and taking over the load in the next step [38], [39]. To obtain the real movement of the knee the “Influence of musculoskeletal model parameter values on prediction of accurate knee contact forces during walking” [40] study has been used. This study uses a musculoskeletal model to predict knee contact forces during walking, and then vary the values of selected parameters to evaluate the sensitivity of the model to parameter changes. Throughout fluorescence experimentation and an algorithm, this study could obtain the translation between the femur and knee coordinate systems of a gait step, from heel strike to the next heel strike. In the case of study, only vertical movement will be applied, and the horizontal translation will be simplified, considering that in the support phase, where the foot is in contact with the ground and supporting the body weight, there is no movement. Another simplification is that all the movements will be considered without acceleration, so all velocities will be lineal velocities. The total time of one gait cycle will be set to 1s, that is why the data taken from the study will be adjusted accordingly. The mesh will perform a cyclic movement to simulate the movement of the knee with a gait step, so the start point will be the same as the end point. In the preceding study, the linear velocities were obtained from a velocity versus time graph constructed using data from the "Influence of musculoskeletal model parameter values on prediction of accurate knee contact forces during walking" study. The velocities were determined by calculating the slopes of the lines on the graph. In Figure 12, the velocities obtained at different times can be seen. 𝑣= { 0 mm/s −24,1065 mm/s 11,7577mm/s − 7,4775 mm/s 12,1923 mm/s 0 mm/s for 0 s≤𝑡<0,36 s for 0,36 s≤𝑡<0,53 s for 0,53 s≤𝑡<0,66 s for 0,66 s≤𝑡<0,74 s for 0,74 s≤𝑡<1 s for 𝑡=1 s Figure 12: Velocity values for one cycle of 1s. At the beginning of the simulation, there is no movement until t=0.36 s, where the velocity reaches a negative value of 24.1 mm/s. From t=0.36 s to t=0.53 s, the velocity increases to 11.7 mm/s. Then, at t=0.66 s, it changes to a negative velocity of 7.5 mm/s, which persists until t=0.74 s. From t=0.74 s to t=1 s, the velocity increases again to 12.2 mm/s. Finally, at t=1 s, the velocity is set to 0 to close the cycle and prepare for the next iteration, ensuring that the start and end points match to maintain a cyclic pattern. To prevent excessive compression or deformation of the bottom cells, which would not be compatible with our mesh, we will distribute the velocities across the patches. This will involve assigning higher
Development of a Virtual Model of Musculoskeletal Soft Tissue: Application to Knee Menisci 39 velocities to the upper patches and gradually decreasing them. For instance, in the MovingWall patches, the first patch, MovingWall1 (on the top side), will have higher velocity compared to MovingWall2, and MovingWall2 will have higher velocity compared to MovingWall3, and so on. Consequently, the lowest velocity of MovingWall patches will be found on MovingWall30, which is the last patch of the MovingWall. Regarding the left patches, left1 will initially have the same velocity as the neighbour patch, MovingWall30, and the velocities will progressively decrease until left15, where the velocity becomes 0. This is because the fixedWall has no velocity, hence its name. The same procedure will be applied to all the right patches. As for the farFieldWall, all the patches will have the same velocity as their neighbouring patches, MovingWall30 on the left side and right1 on the right side. These velocities will be added to the pointMotionUy file, as the movement is happening in y-direction, inside of the initial time step (“0”) in the time directory folder. The velocities profile will be set as in Setting 4. dimensions [0 1 -1 0 0 0 0]; internalField uniform 0; boundaryField { movingWall1 { type uniformFixedValue; uniformValue table ( (0.0 0.0) (0.359 0.0) (0.36 -0.0241) (0.529 -0.0241) (0.53 0.0117) (0.659 0.0117) (0.66 -0.0075) (0.739 -0.0075) (0.74 0.0122) (0.999 0.0122) (1.0 0.0) ); } ... movingWall30 { type uniformFixedValue; uniformValue table ( (0.0 0.0) (0.359 0.0) (0.36 -0.0098) (0.529 -0.0098) (0.53 0.0048) (0.659 0.0048)
Report 46 𝜕𝜌 𝜕𝑡+𝛻·(𝜌·𝑈 )=0 (3) 𝜕(𝜌·𝑈 ) 𝜕𝑡 +𝛻·(𝜌·𝑈·𝑈)=−𝛻·𝑃+𝛻·𝜏 (4) 𝜕(𝜌·𝐸) 𝜕𝑡 +𝛻·(𝜌·𝑈 ·𝐸)=−𝛻·𝑞𝑠 −𝛻·(𝜌·𝑈 )+𝛻·(𝜏·𝑈 )+𝑓𝑏·𝑈 +𝑞𝑉 (5) where: 𝜌 is the density of the fluid 𝑈 is the velocity vector of the fluid 𝑃 is the pressure of the fluid 𝜏 is the stress tensor of the fluid 𝐸 is the total energy of the fluid 𝑞𝑠 is the rate of heat transfer per unit area across the surface area of the material element 𝑞𝑉 is represents the rate of heat source within the material volume per unit volume 𝑇 is the temperature of the fluid 𝑓𝑏 is the body forces. 4.4.1.1. Mass Conservation The mass in the control volume can be neither created nor destroyed, it can only be transported from one region to another. The conservation of the mass states that the mass flow difference throughout the system is zero, that can be seen in Eq. (3): 𝜕𝜌 𝜕𝑡+𝛻·(𝜌·𝑈 )=0 (3) The first term on the left-hand side represents the time rate of change of density, while the second term represents the divergence of the mass flux vector. The gradient operator 𝛻 is described in Eq. (6): 𝛻 =𝑖 𝜕 𝜕𝑥+𝑗 𝜕 𝜕𝑦+𝑘 𝜕 𝜕𝑧 (6) Assuming the fluid is incompressible, where the density remains constant, the continuity equation can be simplified to represent a steady-state process, as shown in Eq. (7):
Development of a Virtual Model of Musculoskeletal Soft Tissue: Application to Knee Menisci 47 𝜕𝜌 𝜕𝑡 =0→𝛻·𝑈 =𝜕𝑢 𝜕𝑥+𝜕𝑣 𝜕𝑦+𝜕𝑤 𝜕𝑧 (7) 4.4.1.2. Momentum Conservation Momentum describes a mass in motion and is measured as the product between an object’s mass and velocity. As the momentum in control volume remains constant, the conservation of momentum implies momentum is neither created nor destroyed. It can only change through the action of forces based on Newton’s law [45]. 𝜕(𝜌·𝑈 ) 𝜕𝑡 +𝛻·(𝜌·𝑈·𝑈)=−𝛻·𝑃+𝛻·𝜏 (4) Eq. (4) describes the conservation of momentum of the fluid. The first term on the left-hand side represents the time rate of change of momentum, while the second term represents the convection of momentum. The first term on the right-hand side represents the pressure gradient force, while the second term represents the viscous forces acting on the fluid, which for some cases of study can be omitted. The description is set up in accordance with the Eq. (8), the expression of Newton’s Second Law of Motion: 𝐹=𝑚·𝑎 (8) where: 𝐹 is the net force applied to the object 𝑚 is the mass of th eobject 𝑎 is the resulting acceleration of th eobject 4.4.1.3. Energy Conservation Conservation of energy is the first law of thermodynamics which states that the sum of the work and heat added to the system will result in an increase in the total energy of the system [45], this is described In Eq. (5): 𝜕(𝜌·𝐸) 𝜕𝑡 +𝛻·(𝜌·𝑈 ·𝐸)=−𝛻·𝑞𝑠 −𝛻·(𝜌·𝑈 )+𝛻·(𝜏·𝑈 )+𝑓𝑏·𝑈 +𝑞𝑉 (5)
Report 48 The first term on the left-hand side represents the time rate of change of total energy of the fluid, the second term represents the convective transport of energy. The first term on the right-hand side represents the conductive heat transfer within the fluid and last term represents any heat generation of absorption per unit of volume. The first law of thermodynamic equation, Eq. (9), can be described as follows: 𝑑𝐸𝑡=𝑑𝑄+𝑑𝑊 (9) where 𝑑𝑄 is the heat added to the system, 𝑑𝑊 is the net work on the system, and 𝑑𝐸𝑡 is the increment in the total energy of the system. 4.4.2. PimpleFoam The PIMPLE algorithm is a numerical algorithm for solving the Navier-Stokes equations in OpenFOAM. PIMPLE stands for “Pressure-Implicit with Splitting of Operators" and is a combination of the SIMPLE (Semi-Implicit Method for Pressure-Linked Equations) and PISO (Pressure-Implicit with Splitting of Operators) algorithms. Both algorithms, the pressure-implicit split-operator (PISO) or semi-implicit method for pressure-linked equations (SIMPLE) are iterative procedures for solving equations for velocity and pressure, PISO being used for transient problems and SIMPLE for steady-state [46]. Both algorithms are based on evaluating initial solutions and subsequently correcting them. However, the algorithms differ in the number of corrections they perform. In the SIMPLE algorithm a pressure correction term is used while the velocity corrections are neglected because they are unknown. This results in rather slow convergence. The PISO algorithm also neglects the velocity correction in the first step, but then performs one in a later stage, which leads to additional corrections for the pressure [47]. The PIMPLE algorithm uses a segregated approach to solve for the pressure and velocity fields in a time-marching manner. It iteratively solves the momentum equations (using the PISO algorithm) and the pressure-correction equation (using the SIMPLE algorithm) until convergence is achieved. The PIMPLE algorithm also allows for the use of adaptive time-stepping and under-relaxation factors, which can improve the stability and convergence of the solution. This algorithm is suitable for simulating incompressible and transient fluids, as well as handling turbulence and finite volume options. The input requirements for this algorithm include the kinematic pressure and the initial velocity.
Development of a Virtual Model of Musculoskeletal Soft Tissue: Application to Knee Menisci 49 4.5. Post-Processing OpenFOAM is supplied with a post-processing utility paraFoam that uses ParaView, an open-source visualisation application. ParaView operates a tree-based structure in which data can be filtered from the top-level case module to create sets of sub-modules. The strength of ParaView is that the user can create a number of sub-modules and display whichever ones they feel to create the desired image or animation [48]. 4.5.1. ParaView In the post-processing step, the simulation results can be visualized and analysed using ParaView, an open-source visualization tool. ParaView offers various options for visualizing data in different formats and provides a wide range of tools for analysis and exploration. Users can generate plots, graphs, and other visual representations to gain insights into the simulation results and understand the behaviour of the system being simulated.
Report 50 5. Results and Analysis To analyse the results as mentioned before, the post-processing tool of OpenFOAM, ParaView, has been used. The primary focus of analysis for this project will be on velocity profiles, as they provide valuable insights into the flow patterns and characteristics within the simulated knee cavity. In addition to velocity profiles, other relevant parameters will also be analysed to further support the correctness of the simulation and enhance our understanding of the fluid dynamics in the synovial cavity. By conducting a comprehensive analysis of these parameters, we aim to demonstrate the fidelity and reliability of the simulation results. The findings will be presented and discussed in this section and plots, graphs, and visualizations will be presented to illustrate the key findings. 5.1. Effect of Viscosity on Pressure Results To see the relationship between the pressure and the kinematic viscosity in our model, three different kinematic viscosities (nu) have been taken and introduced into the same model: • nu1=4,95·10−5 𝑚2/𝑠 • nu2=1·10−3 𝑚2/𝑠 • nu3=1,5·10−3 𝑚2/𝑠 As described in the methodology section, the estimated value of the synovial fluid viscosity was adopted based on the literature. Then the viscosity was increased to see its impact on the physical properties. In this case, the effect of the viscosity variations on the pressure within the system was studied. Following, the pressure maps for the three different kinematic viscosity cases are presented. These color-coded maps illustrate the distribution of pressure throughout the geometry. Extracted from the model at time 0,36𝑠 of the gait cycle, this specific time corresponds to the transition from the stance phase to the swing phase. It represents the compression of the synovial cavity due to the femur movement towards the cavity. The image displays the pressure ranges for each case on the right side. It is important to note that OpenFOAM calculates pressure relative to density, which means that the observed values should be multiplied by the fluid density (1010 kg/m3). However, it is worth emphasizing that the focus is not solely on the absolute value but rather on the range of values obtained. The results obtained are the following:
Development of a Virtual Model of Musculoskeletal Soft Tissue: Application to Knee Menisci 51 For nu1=4,95·10−5 𝑚2/𝑠 the range of pressures obtain can be observed in Figure 14. Figure 14: Pressure map for nu 1 at 0,36 s. For nu2=1·10−3 𝑚2/𝑠 the range of pressures obtain can be seen in Figure 15. Figure 15: Pressure map for nu 2 at 0,36 s.
Report 52 For nu3=1,5·10−3 𝑚2/𝑠 the range of pressures obtain can be seen in Figure 16. Figure 16: Pressure map for nu 3 at 0,36 s. In Figure 14, the maximum pressure relative to density is observed to be 0,038 Pa·(kg/m3) for the case of nu1=4,95·10−5 𝑚2/𝑠. The highest pressure value is located just below the medial condyle of the femur, where the synovial fluid is being compressed due to the bone movement. When the kinematic viscosity is increased to nu2=1·10−3 𝑚2/𝑠, the corresponding effect on pressure relative to the density of the fluid is also an increase, reaching 0,24 Pa·(kg/m3), as depicted in Figure 15. Finally, further increase on the kinematic viscosity, nu3=1,5·10−3 𝑚2/𝑠, lead to higher pressure relative to density, as shown in Figure 16, where the relative pressure reaches a value of 0,27 Pa·kg·m−3. The results were as expected, as the effect of viscosity on pressure can be understood through the concept of resistance to flow. When the viscosity of the fluid increases, it becomes more resistant to flow. This increased resistance hinders the movement of the fluid particles, resulting in higher internal friction within the fluid. As a result, the pressure within the system increases. The influence of viscosity on pressure can be attributed to the higher intermolecular forces of attraction associated with increased fluid viscosity. Consequently, the available free volume within the cavity decreases due to compression and the molecules experience reduced freedom of movement, resulting in higher internal friction forces.
Development of a Virtual Model of Musculoskeletal Soft Tissue: Application to Knee Menisci 53 5.2. Volume Results The volume can be calculated using the “Integrate Variable” in ParaView and selecting the desired variable to integrate. For volume calculation, "1" can be chosen as the integration variable. This corresponds to integrating a constant value of 1 over the domain, effectively calculating the volume. The volume of the fully expanded cavity reached a value of 2,9215·10−6 m3, whereas the maximum compression, occurring at 0.53 s, resulted in a volume of 2,5166·10−6 m3. Figure 17 displays a plot illustrating the variation in percentage in the volume of the cavity. Figure 17: Variation of volume during a gait cycle. As anticipated, the volume of the synovial cavity undergoes changes throughout the gait cycle, specifically during the initial swing phase. At 0.36 s, the cavity experiences the first compression due to the bone movement, resulting in a 14% decrease in the volume, as the femur moves downward in the y-axis direction. This compression continues until 0.53 s, when the femur starts moving upward, causing the volume of the cavity to expand again. In comparison with the non-compressed cavity, more than 8% of volume loss is observed. The last contraction of the cavity occurs at 0.66 s, corresponding to the terminal swing phase. During this phase, the volume decreases by 11% compared to the initial state. At 0.74 s, the volume expands once again. The gait cycle concludes at 1 s, with the cavity fully expanded, restoring the volume to its initial state.
Report 54 5.3. Velocity Results To analyse the velocity results, ParaView plot over a line function will be used. This feature enables the user to plot data values probed along a line through the dataset. In this case, six vertical lines will be defined at various positions all over the geometry, as seen in Figure 18. Figure 18: Geometry with the lines to plot the velocity. These lines have been distributed along the x-axis, with the x position per each line listed below. The lines cross the entire geometry vertically, blue lines on Figure 18, and they are positioned in the midpoint of the z-axis (0.25 cm). Exact position for the lines can be seen as follows: • Position 1 in x-axis: 0.2 cm • Position 4 in x-axis: 2.48 cm • Position 2 in x-axis: 1.43 cm • Position 5 in x-axis: 2.7 cm • Position 3 in x-axis: 2.035 cm • Position 6 in x-axis: 2.98 cm To facilitate the identification of these lines on the plots, the legend will display the x-axis position, and each case will be assigned a consistent colour across all velocity results plots.
Development of a Virtual Model of Musculoskeletal Soft Tissue: Application to Knee Menisci 55 5.3.1. Magnitude of Velocity The magnitude of velocity has been studied for 5 different time steps: 0.36 s, 0.53 s, 0.62 s, 0.74 s and 0.9 s. In Figure 19, the magnitude of velocity at 0.36 s can be seen for the different lines going across the geometry. Figure 19: Magnitude of velocity at 0.36 s. As the cavity is compressed due to the movement of the femur at 0.36 s, the liquid is expelled towards the lateral part of the cavity. This motion is depicted in the first three curves, which represent the lower part of the cavity. In these curves, the velocity of the liquid reaches a maximum of 1 to 15 mm/s. In the purple curve, specifically at x=2.48 cm, the liquid remains still until 2.5 cm, after which it starts accelerating, reaching an approximate velocity of 10 mm/s. In the other curves, a gradual increase in velocity is observed, indicating that the fluid is not significantly affected during the initial compression of the cavity. Regarding the behaviour of the fluid in the lateral region of the cavity, the fluid is slightly accelerated at positions x=2.7 cm and x=2.98 cm. However, at x=2.48 cm, which is the closest line to the lateral part of the femur, the fluid is more accelerated, reaching speeds of up to 9 mm/s. Figure 20 will display the colour map representing the magnitude of velocity for a time step of 0.36 s. To facilitate visual comparison of the coloured maps across different time steps in the simulation, a consistent scale range will be applied to all colour maps.
Report 62 Comparing it to the previous timeframe, the Ux velocity values at 0.53 s are almost 10 times higher (Figure 31). This significant increase in velocity is attributed to the maximum compression of the cavity, resulting in the expulsion of fluid in the positive x-direction. Figure 31: X-axis velocity at 0.53 s. Maximum velocities in the bottom side of the cavity, in the femur medial condyle, where the fluid is being expelled towards the lateral cavity are observed (Figure 32). Figure 32: Coloured map of x-axis velocity at 0.53 s.
Development of a Virtual Model of Musculoskeletal Soft Tissue: Application to Knee Menisci 63 At time step 0.62 s, the cavity experiences expansion, leading to a decrease in fluid velocity. The flow direction changes near the medial condyle, moving towards the left boundary. In the lateral cavity, the highest velocities are observed at the movingWall boundary (Figure 33). Figure 33: X-axis velocity at 0.62 s. Figure 34 displays the coloured map, providing a visual representation of the phenomena described. Figure 34: Coloured map of x-axis velocity at 0.62 s.
Report 64 At 0.74 s, the bone is moving downwards again. In Figure 35, higher velocities are observed at the cavity’s bottom, particularly just below the femur medial condyle. In the lateral cavity, the fluid motion slightly reduces the velocity, at x=2.48 cm and x=2.7 cm until a change in motion direction occurs. Figure 35: X-axis velocity at 0.74 s. Figure 36 shows the coloured map for the horizontal component of velocity at 0.74 s. Figure 36: Coloured map of x-axis velocity at 0.74 s.
Development of a Virtual Model of Musculoskeletal Soft Tissue: Application to Knee Menisci 65 Finally, at 0.90 s the gait cycle is about to end. A change in the flow motion direction can be observed, where at the bottom of the cavity, the fluid is moving towards the left boundary, but as it gets closer to the movingWall boundary, it changes direction towards the right as in the lateral side (Figure 37). Figure 37: X-axis velocity at 0.90 s. The highest negative velocities are observed to be in the bottom side of the cavity whereas the highest positives are in the top and in the movingWall boundary, close to the medial epicondyle (Figure 38). Figure 38: Coloured map of x-axis velocity at 0.90 s.
Report 66 Generally, it is apparent that there is significant fluid motion in the x-direction at the bottom of the synovial cavity, resulting from the compression of the cavity, due to the approach of the femur and tibia, and the rapid expulsion of fluid towards the lateral cavity. Conversely, in the lateral cavity, the fluid motion in the x-direction is less prominent, as the fluid tends to follow the geometry of the cavity and exhibit greater movement in the y-direction. 5.3.3. Y-direction Velocity (Uy) In this section, we will present the findings regarding the y-direction component of velocity. This component represents the vertical movement of the synovial fluid. The velocity, Uy, has been analysed at same five time steps as before. Given the significant variations in velocities, the scale of the coloured maps has been adjusted at each time step to provide a clearer visual representation of the data. At 0.36 s, the femur initiates a vertical movement, approaching the tibia and leading to the compression of the cavity. Near the movingWall boundary, fluid velocity in y-direction reach up to -15 mm/s in the case of x=2.035 cm and x=1.43cm. For x=2.48 cm, the fluid gains velocity just below the medial epicondyle whereas in the lateral side a velocity around -9 mm/s is achieved (Figure 39). Figure 39: Y-axis velocity at 0.36 s. At 0.36 s time step, the maximum negative velocities in y-direction are found in the movingWall, and in the upper part of the right boundary (Figure 40).
Development of a Virtual Model of Musculoskeletal Soft Tissue: Application to Knee Menisci 67 Figure 40: Coloured map of y-axis velocity at 0.36 s. The maximum compression of the cavity, at 0.53 s, in the bottom part, near the movingWall an increase in Uy velocity is observed, reaching values around 6 mm/s. In the middle of the lateral cavity, the fluid moves upwards, forming a parabolic shape in the velocity profile and reaching a positive velocity of more than 20 mm/s (Figure 41). Figure 41: Y-axis velocity at 0.53 s. The coloured map, in Figure 42, highlights the highest positive velocities within the lateral cavity, as the compression expels the fluid from the bottom and propels it towards the farFieldWall boundary.
Report 68 Figure 42: Coloured map of y-axis velocity at 0.53 s. At 0.62 s, the cavity is being expanded and moving upwards. Consequently, the fluid on the upper side is propelled upwards, pushed by the bone, while the fluid on the bottom side of the lateral cavity moves towards the lower part in response to the cavity expansion (Figure 43). Figure 43: Y-axis velocity at 0.62 s. The coloured map in Figure 44 shows a visual representation of these phenomena described above.
Development of a Virtual Model of Musculoskeletal Soft Tissue: Application to Knee Menisci 69 Figure 44: Coloured map of y-axis velocity at 0.62 s. At 0.74 s, the second compression of the cavity occurs. The highest vertical velocities are located near the movingWall in the bottom side and in the middle of the lateral cavity, because of the fluid being expelled due to the compression (Figure 45), similar to previously described behaviours. Figure 45: Y-axis velocity at 0.74 s. Following, the coloured map for the vertical component of velocity can be seen in Figure 46.
Report 70 Figure 46: Coloured map of y-axis velocity at 0.74 s. At 0.90 s, the gait cycle is becoming to an end. As the cavity expands during this time step, a behaviour similar to that observed at 0.62 s is seen. However, since this last compression is less pronounced, the value of the vertical velocities as the cavity expands is also lower (Figure 47). Figure 47: Y-axis velocity at 0.90 s.
Development of a Virtual Model of Musculoskeletal Soft Tissue: Application to Knee Menisci 71 As the cavity expands at 0.90s, the lateral side exhibits maximum negative velocities, indicating downward fluid motion as a response to the expansion of the bottom side of the cavity while in close proximity to the left side of the movingWall boundary, the fluid is pushed upwards by the movement of the femur (Figure 48). Figure 48: Coloured map of y-axis velocity at 0.90 s. In general, as the movement of the bone has been simplified to be only vertical the fluid motion in the bottom side of the cavity consistently follows this vertical motion. This is evident in the vertical movement observed for x=0.2 cm, x=1.43 cm, and x=2.035 cm, as these lines are located in the bottom region of the cavity. The fluid is pushed downwards and consequently, the fluid in the bottom is also pushing upwards, as the fixedWall boundary prevents the fluid from penetrating. In the lateral cavity, significant velocity values are observed when the femur approaches the tibia, resulting in compression of the cavity and propulsion of the fluid towards the lateral cavity. The fluid follows the geometry of the cavity towards the farFieldWall boundary. Notably, some velocity profiles in the middle of the lateral cavity exhibit a parabolic shape. 5.3.4. Velocity Profile Further analysis will be conducted on the velocity profile along the line at x=2.7 cm, focusing on the magnitude of velocity. This will provide valuable insights into how the fluid motion varies in response to the movement of the femur.
Report 78 6. Environmental analysis The environmental impact of the project is minimal because it was conducted using a personal electronic device, to be exact an HP ENVY 16inch. The manufacturing process, distribution, use and end of life have been the only environmental impact. To be able to calculate it, data obtained from the manufacturer, HP, will be considered. The information provided here represents the lifecycle carbon footprint of an industry-average notebook computer with the specifications listed in Assumptions table [32], Table 3 According to Figure 63, the manufacturing phase contributes 70% to the overall carbon footprint. This phase includes the production of various components such as the motherboard, SSD, and display, which are responsible for a significant portion of the emissions, as indicated in Table 2. The use phase accounts for 23% of the emissions, while the distribution phase only contributes 7% of the greenhouse gas emissions. HP's environmental impact calculations are done in accordance with ISO 14040/44. All estimates of impact results are uncertain, resulting largely from data limitations and data quality [32]. To mitigate this uncertainty, HP has developed HPspecific tools that use a combination of HP processes and product data, as well as highquality lifecycle assessment data [32]. HP strives to provide the most accurate environmental impact results, but uncertainty will never be completely minimized and results should be considered accordingly [32]. The estimated impact considering these assumptions presented in Table 3, was of 290 kg CO2eq for 4 years of service life. The current project was made in 4 months, so based on the proportional calculation, the total carbon footprint of the project is estimated to be 24,17 kg CO2eq. Figure 63: Greenhouse gas emissions (GHG) of an HP Envy 16inch laptop. Table 3: Assumptions made to calculate of the GHG emissions. Table 2: Manufacturing GHG Emissions of an HP Envy 16inch Laptop.
Development of a Virtual Model of Musculoskeletal Soft Tissue: Application to Knee Menisci 79 7. Conclusions The utilization of OpenFOAM, an open-source software, facilitated the achievement of a stable simulation through the application of suitable meshing techniques and the provision of a well-defined problem description with appropriate boundary conditions. The simulation was focused on modelling an incompressible Newtonian fluid, the synovial fluid, inside the synovial cavity which was achieved with minimal computational costs while ensuring control over the Courant number. By incorporating realistic parameters and accurately designing the cavity geometry, significant findings could be extracted and thoroughly analysed. The study demonstrated that a higher fluid viscosity resulted in increased resistance to flow, leading to elevated pressure within the system. This phenomenon was attributed to reduced molecular mobility and increased internal friction caused by higher fluid viscosity, which restricts the free volume and impacts the fluid’s ability to flow freely. Additionally, the simulation revealed dynamic changes in the mesh geometry during the compression and expansion of the cavity, resulting in a reduction of total volume. When moving boundaries are included in the model, the dynamic mesh can change its geometry during the simulation. This behaviour was particularly observed in the compression of the mesh, leading to a reduction of total volume, which is later recovered during the expansion of the cavity. By incorporating realistic parameters, the study captured the fluid dynamics and highlighted the redistribution of fluid within a closed cavity during the gait cycle. When the synovial cavity gets compressed, the fluid must be redistributed to accommodate the reduced space, resulting in a significant increase in pressure. In terms of velocity motion, after compression in the bottom of the cavity, the fluid is expelled towards the lateral side. The fluid then follows the geometry of the cavity, moving laterally towards the upper boundaries. Furthermore, during the simulation it has been observed that due to the cavity’s geometry, the velocity patterns in the bottom, when the femur approaches the tibia and compression occurs, the x-direction velocity has a more significant value as the synovial fluid moves horizontally towards the lateral side. In contrast, in the lateral side of the cavity, the y-direction velocity played a more prominent role. It exhibited predominantly vertical motion in both upward and downward directions, with less horizontal motion observed. These findings, obtained through the use of a robust simulation framework with a short computation time and the inclusion of realistic parameters, greatly contribute to understanding the synovial fluid behaviour within the knee's synovial cavity.
Report 80 The study represents a continuation of an ambitious project with the objective of creating a virtualization of the knee. Improving the geometry, parameter simulation, and the implementation of appropriate boundary conditions was a significant step towards achieving this goal. For future research, the menisci should be considered and added to the model. Creating an even more realistic cavity by making it thinner on the lateral side could enhance the accuracy of the virtualization. However, such a modification may introduce computational risks, as the Courant number could increase significantly. Additionally, conducting a more detailed study on the pressure and forces exerted by the bone on the synovial fluid would provide valuable insights and further the understanding of the knee's biomechanics. Finally, to provide a more realistic representation, introducing a more accurate gait cycle motion into the model is advisable as the current approximation allowed for the analysis of fluid behaviour but could be further refined to enhance realism.
Development of a Virtual Model of Musculoskeletal Soft Tissue: Application to Knee Menisci 81 Economic analysis This section refers to the project’s development costs. The development costs for the project primarily include personal costs (hours invested by the engineers) and software costs. The personal costs are further broken down based on the hours invested by each engineer. Three engineers were involved in this project, the accounted hours are counting for 4 months that we have been working on the project, although this would be only for this specific study as the global project has been ongoing for longer. Two of the engineers, the supervisor, and the co-supervisor, invested their time in weekly meetings. The specific hours invested by the student are shown in Table 4: Labor costs. It is stated that 20% of the student's time was dedicated to familiarizing themselves with the software used for modelling the problem, 60% was spent directly on developing the model and extracting results, and 20% was allocated for working on the final report. Table 4: Labor costs. Hours Worked Cost per hour worked [€/hour] Final Cost [€] Engineer 1 (director) 30 30 900 Engineer 2 (co-director) 30 30 900 Engineer 3 (student) 750 30 22,500 TOTAL 24,300€ Regarding software costs, in Table 5, four different software packages have been used. Two of them, SALOME and OpenFOAM, are open-source and available for free. However, licenses are required for the other two software packages, Microsoft Word, and MATLAB. Table 5: Software costs. License Cost [€] Time of use [month] Time of active license [month] Final Cost [€] SALOME 0 4 - 0 OpenFOAM 0 4 - 0 Microsoft 365 69 4 12 23 MATLAB 69 4 12 23 TOTAL 46€ So that, the total cost of the project, including the labour and software costs, is 24.346,00 €.
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