Mathematical and Computational Methods and Algorithms in Biomechanics: Human Skeletal Systems

By Jirí Nedoma and Jiri Stehlik

Cutting-edge solutions to current problems in orthopedics, supported by modeling and numerical analysis

Despite the current successful methods and achievements of good joint implantations, it is essential to further optimize the shape of implants so they may better resist extreme long-term mechanical demands. This book provides the orthopedic, biomechanical, and mathematical basis for the simulation of surgical techniques in orthopedics. It focuses on the numerical modeling of total human joint replacements and simulation of their functions, along with the rigorous biomechanics of human joints and other skeletal parts. The book includes:

• An introduction to the anatomy and biomechanics of the human skeleton, biomaterials, and problems of alloarthroplasty

• The definition of selected simulated orthopedic problems

• Constructions of mathematical model problems of the biomechanics of the human skeleton and its parts

• Replacement parts of the human skeleton and corresponding mathematical model problems

• Detailed mathematical analyses of mathematical models based on functional analysis and finite element methods

• Biomechanical analyses of particular parts of the human skeleton, joints, and corresponding replacements

• A discussion of the problems of data processing from nuclear magnetic resonance imaging and computer tomography

This timely book offers a wealth of information on the current research in this field. The theories presented are applied to specific problems of orthopedics. Numerical results are presented and discussed from both biomechanical and orthopedic points of view and treatment methods are also briefly addressed. Emphasis is placed on the variational approach to the investigated model problems while preserving the orthopedic nature of the investigated problems. The book also presents a study of algorithmic procedures based on these simulation models.

This is a highly useful tool for designers, researchers, and manufacturers of joint implants who require the results of suggested experiments to improve existing shapes or to design new shapes. It also benefits graduate students in orthopedics, biomechanics, and applied mathematics.

• Language: English
• Publisher: Wiley-Interscience
• Release date: Jun 9, 2011
• ISBN: 9781118006467

Book preview

PART I

ANATOMY, BIOMECHANICS, AND ALLOARTHROPLASTY OF HUMAN JOINTS

CHAPTER 1

BIOMECHANICS OF THE HUMAN SKELETON AND THE PROBLEM OF ALLOARTHROPLASTY

1.1 INTRODUCTION TO HISTORY OF BIOMECHANICS AND ALLOARTHROPLASTY

Scientific branches were historically divided into five separate branches—physics, mechanics, chemistry, geology, and biology. Until the first half of the nineteenth century these branches were mutually isolated and independent. From the second half of the nineteenth century, and mainly during the twentieth century, these scientific branches started to cooperate. As a result new scientific branches, such as biophysics, biomechanics, biochemistry, and geophysics, have developed.

Modern physics creates theoretical background for our understanding of nature and its regularities. The principles and laws were found in physics, namely in classical mechanics, relativity, and quantum physics. Quantum mechanics facilitates an understanding of the structure of matter and the chemical coupling as well as further properties of mass. Statistical mechanics facilitates analyzing separate chemical reactions, therefore, facilitating the study of the fundamental basis of chemical reaction theory. Knowledge of the physical principles and regularities of the physics of nonliving mass was then applied to the physics of living mass, which creates a new branch—biomechanics. Understanding a living organism, its functions, its reproduction, as well as its further evolution on the higher qualitative level, yet realizing its dependence on previous evolutionary states, represents the assignment of biology. Biology as a scientific branch is divided into such fields as biophysics, biomechanics, and biochemistry.

Mechanics is the part of physics that is concerned with motion and deformation of bodies, which are or are not in mutual contact and which are loaded by external and internal forces. Mechanics is a very old branch of physics, dating back to the time of Aristotle (384–322 bc) and Archimedes (287–212 bc) and later to the time of Galileo Galilei (1564–1642) and Isaac Newton (1642–1727). The word mechanics was first used by Galileo in his book Discorsi e dimostrazioni matematiche intorno a due nuove scienze, attenenti alla mecanica & i movimenti locali (1638), where he describes force, motion, and strength of materials. Galileo made the first fundamental analyses in dynamics and mechanical experiments, while Newton formulated the laws of motion and gravity. As the study of mechanics evolved, it was shown that with greater experimental experience in the field of elastic mass properties our conceptions changed, but the fundamental idea that mass is continuously spread and that this mass spreading is due to the force actions, including thermal loading, was changed. Then we spoke of the continuum and about a branch called continuum mechanics.

Classical continuum mechanics of nonliving nature studies mass properties by considering the thermodynamics of closed systems. This idea was based on the fact that these systems do not need any further energy or any change of the mass with their neighborhood. Therefore, a change of entropy with the neighborhood is always positive in the closed system. Owing to dissipative processes the degree of the system organization decreases continuously.

Systems having properties of regularization, reproduction, and possibility of conservation of information represent open systems, for example, systems having an ability to change mass as well as energy with the surroundings. Such systems have dynamic equilibrium and their invariability is ensured by means of a mass exchange. Such systems describe the biological systems. The living systems receive mass by means of meals. The most important assignment of the control system is the conservation of its energetic foundation with the aim to conserve a certain level and stability of the function of the biosystem, also under the changed conditions of an outer medium. It is evident that the law of conservation of the living mass is a part of other conservation laws, known from classical and relativistic mechanics. The processes of acquisition, cumulation, transfer, and employment of energy in biosystems ensure both the growth of a living mass and conservation of the structure and realization of the function of the biological system, which is realized in the cooperation with receiving, processing, protection, and employment of the information.

Biomechanics is a scientific branch that combines the field of applied (e.g., engineering) mechanics with the fields of biology and physiology and it is concerned with a human body. In biomechanics, the principles of mechanics are applied to the conception, design, development, and analysis of the equipment and systems in biology and medicine. One of the main goals of biomechanics is to study responses of living tissues on an external energetic function from the physiological point of view, where we assume that the living tissue is a composite material with controlled properties. A biological material is a strongly organized material with the ability of self-evolution, reproduction, and a possibility of conforming to the surroundings. The modern development of biomechanics started in the fifteenth century, based on the studies of Leonardo da Vinci (1452–1519), though the concepts of biomechanics were probably given in Greek (Aristotle 384–322 bc) and Chinese writings. Later the contributors to biomechanics were Galileo Galilei (1564–1642), René Descartes (1596–1650), Giovanni Alfonso Borelli (1608–1679), Robert Boyle (1627–1691), Robert Hooke (1635–1703), Isaac Newton (1642–1727), Leonard Euler (1707–1783), Thomas Young (1773–1829), Jean Poiseuille (1797–1869), Hermann von Helmholtz (1821–1894), and the others. The development of the biomechanics as a separate branch has improved our understanding of the mechanics of human joints. In the last several decades this understanding has grown to include the total replacements of joints as well as implants, the mechanics of a blood flow, the mechanics of an airflow in the lungs, the mechanics of soft tissues, and the mechanics of the growth and a form of joints. Moreover, biomechanics has contributed to the development of medical diagnostic and treatment procedures as well as to the development of designing and manufacturing medical instruments and devices for the handicapped. It has also contributed to the development of sport and forensic medicine.

Materials of nonliving nature, because of their low organization, are only slightly accepted or not accepted by highly organized living systems because they cannot be quickly regenerated and renovated. Another disadvantage is the insufficient adaptability of contacts between living and nonliving systems. This contributes to the problems that artificial replacements for biological organs and their parts experience. One of the main aims of biomechanics is the detailed research of composite materials suitable for the development of artificial replacements for human organs and joints. To apply materials of nonliving nature for the development of artificial replacements for human organs the following fundamental criteria and properties must be satisfied:

1. Unconditional adaptability to the surrounding materials of the living systems

2. Sufficient range of elastic deformations with reasonable nonlinearity and with useful elastic modules

3. Useful orientation of deformable properties with regard to the type and direction of force actions

4. Useful nonreturned deformations rendering an adaptation possible without great time and space changes of properties and evoked damages, but rendering a stress relaxation and precluding microdamage origins possible

5. Limitation of a total deformation during the growth of stresses with reinsurance of an elastic behavior with high local solidity and with a minimal requirement of a further delivery of energy

6. High biocorrosion resistance

7. High solidity against cyclical loading with high initial damping

8. High quality of the surface design, ensuring biocompatibility and a decrease in the possibility of biocorrosion

9. Having an ability of certain regeneration in connection with a neighboring living mass–tissue, as a higher form of biocompatibility

At present a most vigorous development of biomechanics is associated with orthopedics because the most frequent use of surgical intervention is with patients with musculoskeletal problems. In orthopedics, results of biomechanics have become everyday clinical tools. Therefore, the most urgent problems in biomechanics are problems connected with static and dynamic loading of human joint systems and their artificial joint replacements. Then fundamental research has included not only surgery, prostheses, implantable materials, and artificial limbs but also cellular and molecular aspects of healing in relation to stress and strain and, moreover, tissue engineering of a cartilage, tendon, muscle, and bone. Thus, rheology of biological tissues, transfer of a synovial fluid between both parts of joint systems, mass transfer through membranes, microcirculation, and interfacial phenomena must be investigated. We see that biomechanics represents a strongly interdisciplinary branch.

The first replacement of a human joint was made by Carnochan in 1840 in New York. The first replaced human joint was the temporomandibular joint. This attempt was unsuccessful because the implanted material was a wood. The first successful attempts of the hip prosthesis were made at the turn of the twentieth century. A successful replacement was made by Jones. His attempt was based on a gold plate used as an insert into the hip joint. This replacement functioned for 21 years, that is, until the patient’s death. In the twentieth century Smith-Petersen accomplished marked successes by putting a cap on the femur head. Next attempts are connected with Delbet, Hey-Growes, and Judet. Delbet and Hey-Groves were the first to implant an artificial replacement of the whole head of the femur (in the twentieth century). Later, Judet used a type of replacement that was further developed by Thompson, Zanoli, Townley, Movin, Güntz, Merle d’Aubigue, Lange, Neff Marine-Zuco, and Gosett. Moore implanted a new type of a whole-metal femoral component—an artificial head of the femur with the stem fixed in the medullar cavity. This type was modified by Thompson, Eichler, Lipmann, McKee, Reiley, and others. In 1951 Haboush introduced a bone cement for fixation of the stem into the marrow channel. This technique was also applied by McKee and Charnley. In 1950 Urist and in 1960 McBride implanted in addition to the stem also the acetabular cup. In this way they were able to do a total hip joint replacements. From this time on we speak about total hip replacements. The modern arthroplasty of the hip joint and then of the knee and other joint replacements started with the Charnley shape of the hip joint replacement and mainly with his low-friction arthroplasty (Charnley, 1979). After a transient failure in which Teflon was used like a material for an artificial acetabulum, Charnley established the ultra-high molecular polyethylene—UHMWPE—into the construction. His replacement was solved as a metal femoral component with the metal head, whose stem was fixed by a bone cement. The artificial acetabular cup made of UHMWPE was also fixed with bone cement, and a mutual motion between the head of the femur and the cup of the acetabulum was realized by pairing metal–polyethylene. This low-friction arthroplasty was a model for many other authors and it has been used in most up to the present time. A change in the shape of the femoral stem, made by Müller in the late 1960s, was important and it enabled a component implantation without the necessity to apply a large femur trochanter. Because of this all new types of artificial joints were introduced. Ceramics were the preferred material for the production of the femur head (pairing ceramics–polyethylene), although some replacements were made without contact with polyethylene (pairing metal–metal or ceramics–ceramics). The problem of fixation of individual components in a bone without using bone cement was solved by developing cementless types of joints.

Subsequently primal and revised types of implants were developed. Both types of implants were available as cemented or cementless. At present, replacements for temporomandibular joints have been developed and are available for use.

1.2 BIOMECHANICS OF HUMAN JOINTS AND TISSUES

Biomechanics as a modern science combines the developments in engineering mechanics with developments in biology and physiology. Biomechanics is concerned with the human body; thus it is a natural science concerned with living systems mechanics. It is the study of mechanical movements, their function in the whole biosystem as well as its individual parts. Modeling biosystems and the simulation of their functions developed into a significant understanding of the construction and function of living mass (Valenta, 1985, 1993). In biomechanics, the principles of classical mechanics are applied to the conception, design, development, and an analysis of equipment and systems in medicine.

Modeling of biosystems can be divided into two categories of models. The first one introduces mathematical models that model structures, functions, and quantities of investigated biosystems. Into this group we can add mathematical-mechanical models, which model specific biomechanical systems and simulate their functions. Models of physical-mechanical properties, thermodynamic models, structural models, and function models also belong here. Theoretical biomechanics is concerned with these problems. Real experimental models, modeling of specified biomechanical actions, or biomechanical objects belong in the second category. These models’ task is to verify theoretical possibilities concerning their structure and function or why a biomechanical object such as a unit or its parts can verify the rightness, exactness, and accuracy of a solution of abstract mathematical models, to investigate biomechanical problems such as stress states, deformation ability, and the like. Experimental biomechanics is concerned with this topic. For that reason when artificial joint replacements have to be reliable and function long term, their construction must follow the principles of biomechanics and biomechanical relationships in an appropriate joint sector of the movement system. The study of the forces that have an effect on the joint system, natural or artificial, concerns fields such as biostatics, biodynamics, biokinematics, biokinetics, and tribology. In biomechanics we are concerned with the study of external and internal forces, which are summarized and transferred by joints. As a consequence of external and internal forces, we study the distributions of deformation and stresses in the movement apparatus, their character and relationship in the course of movement through time.

Arrangements and shapes of human joints establish their kinematic and dynamic characteristics. A kinematic characteristic is given by a joint geometry, by a shape of contact areas, and by their cartilaginous surface. Ligaments satisfy a function of mechanical stops or leading and stabilizing elements. From analyses of movements of other living creatures it is evident that their movement organs are constructed on the principle of the lever system with an alternating movement. Shapes and forms of individual structural elements of joint connection are so various that it is not possible to construct a universal artificial replacement for whatever human joint to fully comprehend its real functional properties in a human organism. It is the main goal of orthopedic specialists and design engineers to approach this ideal state. In that case mathematical modeling and mathematical simulation of the function of a joint and of its optimal replacements can be also helpful.

CHAPTER 2

INTRODUCTION TO THE ANATOMY OF THE SKELETAL SYSTEM

2.1 ANATOMY OF THE SKELETAL SYSTEM

From a biomechanical point of view in the human locomotor apparatus we can speak about tissues, biological structures, and passive and active formations. Passive elements are tissues that transfer originated and acting forces. Simultaneously, they must resist acting forces and they must satisfy certain conditions of strength and elasticity. These tissues would orient acting forces in different ways, regulate them, and change their ordering. Bones and their parts, including joint surfaces covered by a hyaline cartilage and further ligaments, tendons, and fasciae, belong among passive elements. Generally, bones are nonhomogenous and anisotropic materials.

Among active elements we can include muscles, which are able to change the energy of biomechanical reactions into work and develop some power for achieving movement. Mechanical qualities of active and passive elements of a human locomotive organ have been changing during its life.

A bone tissue is a connective type of tissue whose solid composition enhances its supportive and protective roles. It consists of cells and an organic extracellular matrix of fibers and a ground substance produced by the cells. Moreover, bones contain a high content of inorganic materials in the form of mineral salts, which are combined intimately with the organic matrix. These inorganic components make bone tissues hard and rigid; organic components give a bone its flexibility and elasticity. These inorganic components consist of calcium and phosphate in the form of small crystals of Ca10(PO4)6(OH)2. Bone minerals are embedded in variously oriented fibers of the protein collagen, in the inorganic matrix. Water is fairly abundant in a living bone, ∼ up to 25% of its total weight. From a macroscopic point of view bones are divided into two types of osseous tissue—that is, cortical or compact and cancellous or trabecular or spongy bones. Cortical bones form the outer cover of the bone and they have dense structures, while a cancellous bone within the cortical shell is composed of thin plates, or trabeculae, in loose mesh structures, where the interstices between the trabeculae are filled with a red marrow. The cancellous bone tissue is arranged in concentric lacunae-containing lamellae. Bone tissues behave like composite (bio)materials.

The role of the ligaments and joint capsules, which connect a bone with a bone, is to augment the mechanical stability of the joints, to guide joint motion, and to prevent excessive motion. Ligaments and joint capsules act as static restraints. The muscle–tendon units are formed by tendons and muscles and they act as dynamic restraints. Moreover, the tendons also enable an optimal distance between a muscle and a joint. Tendons and ligaments are dense connective tissues, that is, parallel-fibered collagenous tissues. The great mechanical stability of collagen gives the tendons and ligaments their characteristic strength and flexibility. Like other connective tissues tendons and ligaments also consist of relatively few cells, the so-called fibroblasts, and an abundant extracellular matrix.

The human body consists of three muscle types—the cardiac muscles (which compose the heart), smooth muscles (which line hollow internal organs), and skeletal (striated or voluntary) muscles (which are attached to the skeleton via tendons). Skeletal muscles create ∼40–45% of the total weight of the human body. In the human body there can be found more than 430 skeletal muscles; they are in pairs on the right and left sides of the body. The most vigorous movements are produced by fewer than 80 pairs. The muscles provide strength and protection to the skeleton by distributing loads and absorbing shock. Moreover, they enable the bones to move at the joints. The skeletal muscles perform both static and dynamic work, which permits locomotion.

The human body has three types of joints—fibrous, cartilaginous, and synovial. The synovial joints are the main joints of the skeletal system as they allow large degrees of motions. The specific property of synovial joints, with which the articulating bone ends, are covered by a thin (1–6 mm), dense, translucent, white connective tissue layer, known as the articular cartilage. This cartilage is a highly specialized tissue that enables one to transform and transmit the highly loaded joint environment without any failure during a relatively long lifetime. Its cellular density is less than that of any other tissues. Its function is to distribute joint loads over a relatively wide area and to allow a relative movement of the opposing joint surfaces with minimal values of friction and wear. Its properties are incompressibility, immiscibility, and distinct phases—an interstitial fluid phase and a porous-permeable solid phase. The water contributes to its mechanical properties, so that the articular cartilage can be considered as a fluid-filled porous-permeable biphasic medium. Due to the articulation cartilage the coefficient of friction is very low. From the point of view of its nature the articular cartilage can be assumed as a bioviscoelastic material. Like bone, the articular cartilage is also anisotropic and its material properties depend on the direction of loading and it allows also a creep. For more on the compositions and properties of bones, muscles, cartilages, tendons, and ligaments see Fung (1981), Nordin and Frankel (2001), and Covin (2001).

We can summarize components of the locomotive apparatus from a biomechanical point of view into the following systems:

1. System of Skeletal Muscles Its main function is the production of active mechanical movements.

2. System of Body Segments and Skeleton Elements Its function is the passive transmission and facilitation of active forces onto the surroundings; it creates movable and carrying bases for muscle fixing, ligaments, and fasciae.

3. System of Intermediate Elements It connects body segments by ligaments, cartilage, bone, joints; it connects muscles with tendon segments; its main function is a mechanical connection between the first and second systems and among themselves.

4. Informative System It is created by receptive elements in muscles, joints, tendons, and proprioceptors; then ear, eye, dermatic detectors, the exteroceptors; visceroceptors and afferent neural tracks. Its function is mechanical reception and transfer of information.

5. Innervation System It is formed by motor neurons, myoneural junctions, and efferent neuron tracks, synaptic transfers, and the like; its function is activation of motor muscle units.

6. Central Neural System It is created by the spinal cord, inter- and intrasegmental connections, subcortical and cortical mobile subsystems; its function is a collection, selection, saving, and an information analysis, representation of mechanical properties of an environment, a reflective activity, determination, and the like.

The human skeleton bones are mutually connected. These connections can be continuous and movable. The movable connection is a joint. We can distinguish the following formations:

1. Articular contact surfaces.

2. Articular cartilages, which cover joint contact surfaces.

3. An articular capsule, anticular ligaments. The capsule is created by two layers—a surface external layer and a synovial internal layer secreting into an articular cavity a viscous synovial liquid, which nourishes a cartilage and increases articular surface adhesion.

4. An articular cavity, which is a slot space between small-sized contact surfaces and it is closed by an articular capsule and filled by a microscopic amount of the viscous liquid, the so-called synovia.

Mobility of the joints is a combination of movements around three axes:

1. Around a horizontal axis in the sagittal plane, the flexion and extension

2. Around a horizontal axis in the frontal plane, the adduction and abduction

3. Around a longitudinal axis of the bone, the rotation

With regard to the human skeleton construction and a way of human walking in the upright position, these are the most exercised loading joints of lower limbs, that is, the hip and knee joints as well as the ankle joint.

2.2 HUMAN JOINTS AND THEIR FUNCTIONS

The synovial joints are subjected to an enormous range of loading conditions. Moreover, because of the very good properties of the articular cartilage, the cartilage surface sustains little wear. The minimal wear of the cartilage, associated with loads indicates the high quality of cartilage materials as well as the possibility of lubrication based on the high quality of the synovial liquid. There are two types of lubrication. The first is boundary lubrication, which involves a single monolayer of lubricant molecules adsorbed on each bearing surface. The second is fluid-film lubrication in which a thin fluid-film layer provides surface-to-surface separation. The coefficient of friction has an extremely low value of 0.02 and, therefore, the low rates of wear as well.

Developing artificial joint replacements must be based on knowledge of basic functions of natural joints. We must try to simulate them mathematically because only comparing mathematical simulation of basic functions of natural joints and the results of simulation of natural joint functions replaced by total replacements allow us to decide which surgical method to use for joint implantation. Osteointegration of the artificial joint represents, similar to a natural joint, a balance system, where a shape corresponds to a function. Violation of this balance in the natural joint enables arthritic changes with an ensuing destruction of the bone joint structures. The artificial joint leads to infringement of this balance and to the wear and tear of individual components of the replacement joint and afterward to their mechanical breakdown and untimely loosening. Elimination of consequences of violation of this balance is a goal of the mathematical simulation of the function of natural and artificial joints.

Investigating human joints leads to studies from a kinematic, static, dynamic, and tribilogy points of view. Results of an analysis of articular kinematics of the locomotive apparatus lead to the explanation of relative movements in the joint, not only from a quantitative but also from a qualitative point of view. Kinematic articular connection characteristics, for example, degrees of freedom and transference ratio, are determined by contiguous shapes of articular surfaces, a bone epiphysis shape and a cartilage surface shape. Roentgen kinematography or stroboscopic photography are technical devices that help us to obtain necessary information. A reliable human joint function is influenced not only by its skeleton shape and its cartilage covering but also by the effectiveness of different muscle groups, tissues, and viscous elastic characteristics of the synovial liquid. Individual articular connections are strained during ordinary limb movements in different ways. Basically, we can characterize forces affecting an articular connection of the upper limb, such as a tension force, and a force affecting the upper limb joints, such as a compressive force.

During a tension force, articular contact surfaces are pulled away from each other, and the effect of operating force is absorbed by a tissue apparatus of an individual joint. In this case a joint functions as a movable connection, allowing relative movements of skeleton-connected parts.

By a pressure character force contact surfaces are pressed against one another; thus, the relative movements in connected skeleton parts occur mainly in conditions of a close contact with sliding articular surfaces under pressure. In this case the joint represents a flexible pressure connection. These joints are strained much more than upper limb joints, and that is also the reason why degenerative changes happen more often. A pressure force value, acting, for example, on the femur head, can be determined by a force parallelogram, that is, from a body weight and a resultant of muscle forces retaining the balance. During a static situation, forces acting in the hip joint, the knee joint, and the ankle joint become an anatomical structure whose influence is bigger than the body’s weight.

Dynamics in the sense of a human movement puts high demands on the locomotive apparatus, then specially on joints and their contact surfaces. During walking, lower limb joints are subjected to dynamic forces, which with regard to their long-term cyclic operation, leads often to the destruction of contact cartilage surfaces.

The hip, knee, and ankle joints have contact surfaces adapted in such a way as to absorb dynamical forces corresponding to the human weight. For example, the diameter of the adult human femur head is 38–56 mm. When walking, the pelvic acetabulum is in contact with the femur head spreading on an area of about 80% of a half-circle, which is the same as the head diameter. Thanks to the elasticity of the cartilage covering the articular surfaces, this contact surface, which takes in the hip joint a value of ∼430–920 mm², increases during overloading a value depending on the cartilage size, its elasticity, and an operating pressure force. For example, in the knee joint case it was shown that this contact surface enlargement can be up to 50%, during flexion the contact surface size is decreasing proportionately with the knee flexion increasing. The knee joint adapts to the strain to which it is exposed most of the time.

In consequence of straining lower limb joints by considerable forces and pressure tension concentration on a relatively small area, lubrication has a substantial significance for a right joint function; in human joints this function is represented by a synovial liquid. The synovial liquid is secreted into the articular cavity by an internal layer of the articular capsule. It gives elasticity to cartilage coverings of articular surfaces and, because of its viscoelastic properties, it is able to absorb a certain value of pressures. Human joints are constituted by components from living tissues, which have a regeneration ability. It means that microcracks created as a consequence of overloading do not lead by a consequence of tissue regeneration straight to the joint destruction.

According to shapes of articular contact surfaces, human anatomy displays the following types of joints:

1. Spherically shaped (ball-and-socket shaped)—contact surfaces are created from parts of spherical areas.

2. Bounded shaped (walnut shaped, enarthrosis)—articular surfaces are larger than half of the spherical surfaces or of rotational ellipsoidal surfaces, for example, the hip joint.

3. Free shaped (arthrodis)—articular surfaces are smaller than half of spherical surfaces, for example, the shoulder joint.

4. Cylindrically shaped—articulatio radiohumeralis.

5. Trochlear shaped (gynglimus)—articular surfaces have a leading edge and groove.

6. Pivot shaped (wheel shaped)—a movement around an axis parallel with a longitudinal bone axis.

7. Egg shaped—contact surfaces are parts of an ellipsoid.

8. Saddle shaped—contact surfaces have a horse saddle shape.

9. Flat shaped—contact surfaces are of a plane type, for example, joints of the cervical spine.

10. Amphiarthrosis—irregular contact surfaces; they have minimal movements.

11. Combinated shaped—an anatomically separated joint, it is functionally connected with another joint and movements act simultaneously.

2.3 TRIBOLOGY OF HUMAN JOINTS

Tribology is a theory concerned with the friction mechanism and wear of rigid bodies. Friction is a term expressing the resistance against a movement that originates between two bodies being in mutual contacts in the area of their surface contact in a transverse direction toward them. Between bodies, which are in contact, a certain medium can be present—the so-called frictional interlayer, which can be liquid, solid, or gaseous. Wear is an unfavorable change of a surface or surfaces of rigid bodies, which are in mutual close contact, caused by a mutual activity of functional surfaces or of a functional surface and a medium, respectively, that evokes the wear. Wear is not a material property but a property of a body system, including an interlayer. Between the friction and the wear there is no straight connection. Friction in the Coulombian sense depends on the absolute value of a normal component of an acting force, with an increasing normal force the friction and wear increase, too.

We speak about friction and sliding pairs. Friction pairs are two bodies in mutual contact, inside which friction appears like a demanding effect (e.g., brakes), sliding pairs are two bodies in mutual contact, inside which friction is an unfavorable effect (e.g., arthroplasty). From tribology of human joints the viscoelastic properties of the synovial liquid are sufficiently known. It seems that according to experimental results of many authors (Walker, Erkman, Weightman, Duff-Barclay, and Spillman), the synovial liquid after an artificial joint application is secreted into the joint in the same chemical constitution as in the natural joint.

A mathematical simulation of the joint function based on a mathematical theory of contact problems of the Signorini type allows us to study stress and force conditions on the contact surfaces, magnitude of deformation of contact surfaces as well as the transfer of loading forces, transferred from the acetabulum onto the head of the femur. A comparison of the results of a mathematical simulation of natural and artificial joints allows one to judge whether the function of the hip joint arthroplasty will be able to satisfy fully functional demands, and whether it can provide determination about what kind of clearance in the joint can be optimal for the full functional ability of the total hip arthroplasty (THA).

2.4 BIOMECHANICS OF THE SKELETAL SYSTEM

Generalized continuum mechanics of nonliving nature is used when studing mass properties of the thermodynamics of closed systems, and these systems do not need further energy or metabolism from their surroundings to maintain a balanced state. That is why entropy change in the surroundings of a closed system has been always positive. As a result of a dissipative process the degree of system organization decreases all the time. Systems that have a regulation property, a reproduction property, and possibilities to keep the information are open systems, that is, systems, that are able to change the mass and energy with the surroundings. For systems with a dynamic balance constancy is supplied by metabolism. These systems describe biological systems. Living systems increase their mass by an intake of energy, which enables them to make new structures, that is, to eliminate and correct mistakes in their organization structure. Thus biosystems are autoregulating and directing systems at the same time. The most important assignment of the control system is to conserve its energetic background with the aim to conserve the level and constancy of biosystem functioning, mainly by changing the conditions of the exterior surroundings. It is evident that the law of conservation of living mass is used here as a part of other conservation laws known from classical and relativistic mechanics. Processes of obtaining, accumulating, transforming, and fully utilizing the energy in biosystems ensure that a living mass grows as a conservation of the structure and realization of a biological systems function as well. What is realized in cooperation is the receiving, elaborating, protection, and a full use of the information. Informative control mechanisms influence the quality energetic processes of biosystems and the speed of all structural and functional changes.

Biosystems are characterized by (i) a high level of complexity, characterized by control systems that provide an information exchange, their processing for a need to provide the function and structure, (ii) a collective production principle, (iii) mechanisms of selection, (iv) significant stochasticity of structures, providing biosystem evolution, and (v) stability and lability, where high stability provides vitality and liveness and lability provides further evolution.

Biosystems are studied by biological, biochemical, biophysical, biomechanical, and medical sciences as well as combinations of these fields because living tissues must be studied as the coupled problems. Hereafter we will concentrate on biomechanical problems of living tissues only.

Biomechanics is based on applications of mechanical and biological laws to biological and medical sciences. One of the main goals of biomechanics is to study living tissue responses to exterior energetic actions from a physiological point of view, whereas we suppose that the living tissue is basically a composite material with controlling properties. A nonliving natural material is relatively little organized (e.g., a structure and a construction of crystals), and it is not able to do self-evolution—sense and function is added by humans only. On the contrary, biological materials are highly organized with a self-evolution possibility, reproduction, and adaptation possibility. The system function then provides its existence. Nonliving materials, thanks to their low organization, are very neither accepted nor nonaccepted by highly organized living systems because it is not possible to regenerate and recover them sufficiently fast. Their disadvantage is also their insufficient adaptation to the dividing line between living and nonliving systems. This is why there are problems with artificial replacements of biological organs and their parts. One goal of biomechanics is the detailed research of composite materials suitable for use in the development of artificial replacements of human organs. In order to use these materials in artificial human organs, they must have certain fundamental properties:

1. Unconditional adaptation to the surrounding materials of the living systems

2. Sufficient range of elastic deformation with a reasonable nonlinearity and a needed elastic modulus

3. Suitable orientation of deforming properties with regard to a type and direction of a force operation

4. Suitable irreversible deformations providing adaptation without greater time and space property changes and evoked damaging, which allow relaxation of stresses and prevention of microfailure formations

5. Limitation of deformation because of increasing stresses while providing an elastic behavior of the biomaterials with high strength and a minimum need of more energy delivery

6. High immunity against biocorrosion

7. High strength against cyclic loading with high beginning damping

8. High-quality surface treatment that will provide biocompatibility and not allow biocorrosion

9. Ability of certain regeneration in relation to surrounding living mass—a tissue—as a higher level of biocompatibility.

An increase of further biochemical knowledge will bring new possibilities of tissue and organ replacements in connection with new methods of living tissue cloning. For that reason future research is going to be focused on mechanical properties research of the muscle–skeletal system based on macro- and microscopic, that is, cellular, living tissue structures. This research will be focused not only on an experimental study of living tissue structure but also on mathematical modeling and mathematical simulation of processes in living tissues and organisms. Presently, the most urgent and most accessible seems to be a study of biomechanical and biodynamical processes of the locomotive human system and its relation to surrounding tissues and its influences on the human locomotive apparatus. This research will be the focus of research in fields such as bioengineering, medical engineering, and in the medicine disciplines and applied mathematics. In biomechanics the main focus in medicine will be the skeletal and muscular apparatus with consideration of the analysis and synthesis of stiffness and dynamics of the human skeleton together with considering the influences of applied muscle forces, rheological properties of synovial liquids, research of muscle utilitization and temperature regulation as well as their influences on human skeleton pathological appearances such as osteophytes and other pathological deformations of the human skeleton and its parts. Another task of biomechanics is its utilization in the elimination of congenital and obtained defects of the locomotive apparatus.

2.4.1 The Hip Joint

The hip joint (see Fig. 2.1) has a unique importance in the human body. It is one of the largest and most stable joints. As a consequence of bipedal walking, from the mechanical point of view, it becomes the most exposed place, where the carrying free limb is connected with the solid pelvic girdle on which the spine system is connected. A right bipedal system is observed only in humans and in birds. The difference is in the location of a central point. Birds (and in the past dinosaurs too) have the central point bellow the hip joint level, so the bird body resembles a hanged pendulum. This system is very stable. It demands a minimal force for keeping an erect position, which was very important for dinosaurs with their enormous weight. Humans are a different example. Their central point is over the hip joint in the area of the Th 10-11 disks. Therefore, to keep the erect position the muscle apparatus keeps the balance still functional. The hip joint has an intrinsic stability provided by its relatively rigid balland-socket configuration. It has also a great deal of mobility, which allows normal locomotion in the performance of daily activities. For this posture the whole human skeleton is adapted, but especially the skeleton of the pelvis, which is flat with an open pelvis muscle apparatus. By this the origin of abductors shifted laterally, far from the center of hip joint rotation, which causes extension and enables these muscles to act as levers. Loading relations during standing and moving are very complicated. The hip joint has four degrees of freedom, which are rotation movements along the x, y, z axes and one displacement movement in the direction of the y axis (this movement occurs only during luxation). Thus, from the mechanical point of view, 6 muscles would be enough to move the hip. In the hip joint area there are 20 different muscles, whose combinative operation causes different phases of individual movements of the joint itself.

FIG. 2.1 Hip joint (X ray).

The hip joint is made up of the acetabulum of the pelvis and the head of the femur. This articulation has a loose joint capsule surrounded by large and strong muscles. The acetabulum is a concave component of a ball-and-socket configuration of the hip joint. The acetabular surface is covered with the articular cartilage. The cavity of the acetabulum faces obliquely forward, outward, and downward. Osseous acetabulum in the hip joint is deep and provides a substantial static stability to the hip (Nordin and Frankel, 2001). The femoral head is the convex component of the ball-and-socket configuration of the hip joint. The hip joint is created by approximately two-thirds of a sphere. The articular cartilage covers the femoral head; the variations in its thickness result in a different strength and stiffness in various regions of the femoral head.

Moreover, the hip joint is basically a spherically shaped joint with bounded dynamic mobility, with the shape of a rotational ellipsoid, whose lengths of axes are not that different. Joint cartilage covering the femur head and the acetabulum is very flexible hyaline cartilage, which because of its deformations allows movements corresponding to the ball-shaped joint, which is its main function from a biomechanical point of view. The femoral head is covered with hyaline cartilage from 2.2 to 3.7 mm thickness. Hyaline cartilage of the acetabulum covers the facies lunads a minimal force for keeping an erect position, which was very important for dinosaurs with their enormous weight. Humans are a different example. Their central point is over the hip joint in the area of the Th 10–11 disks. Therefore, to keep the erect position the muscle apparatus keeps the balance still functional. The hip joint has an intrinsic stability provided by its relatively rigid ball-and-socket configuration. It has also a great deal of mobility, which allows normal locomotion in the performance of daily activities. For this posture the whole human skeleton is adapted, but especially the skeleta acetabuli, which has a horseshoe shape and is of about 0.5–0.9,mm thick and places the horseshoe of ∼0.8–3.0,mm thick approximately in the convex part of the horseshoe and on the periphery of the facies lunatum acetabuli. A part of the facies lunatum acetabuli is created from cartilage, that is, the fossa acetabuli, which is filled with pulvinar acetabuli with the insertion of ligamentum capitis femoris. The synovial membrane covering the pulvinar acetabuli, whose background is a ligamentous stroma with an amount of fat cellular elements, has its function in mechanics and joint nourishment (an analogy with Hoff’s knee joint corpus). The hip joint has a unique status in the body because it connects a limb carrying the whole body weight with a relatively nonmovable pelvic round. It is substantially different, for example, from the shoulder joint, inside which the humerus is connected with a trunk by an interlink—scapula and the heterogeneous muscles. Moreover, the proximal limb does not have any carrying function. The hip joint skeleton architecture is therefore determined by the combination of statical loading and a dynamical component determined by the muscle traction and a tonus ligament. Both of the components get the same denominator factor in a period, when the hip joint begins to realize the biomechanical requirements that walking imposes. The joint capsule of the hip joint is very thick and together with ligaments, which strengthen it, creates an anatomical and functional unit. The joint capsule is strengthened on the front side by the strongest ligament in the body, the so-called ligamentum iliofemorale, which is stretched during standing with the hyperextended hip joint as the arresting mechanism. In order to reach an elastic bounded extension of the hip joint its normal structure is necessary. Two other ligaments, which strengthen the articular capsula, are the ligamenta ischiocapsulare and pubocapsulare, circulary rounded by sheafs connecting all the ligaments and unifying them functionally (zone orbicularis).

From the value of the compression force operating on the head of the femur due to classical biomechanics it is possible to obtain the force paralelogram, that is, from body weight and the resultant muscle forces that maintain balance. Movement of this muscle force comes from a great trochanter, where all the mentioned abductors are fixed, in the direction of the upper pelvic edge. The muscle moment is determined by many factors, for example, the length of the neck of the femur, the size of the trochanter, the size of the angle that contains the neck with the femur axis, and by the femur location against the pelvis. The total limb length of the lower limb has a certain influence on the absolute value of this muscle moment.

To determine the value of pressure force functioning on the femur head we will assume that individual skeleton segments are perfectly solid and mutually connected bodies. Let point C be the center of the femoral head (Fig. 2.5). Then from a moment balance condition to this point we determine a reactive force Fs as a resultant of muscle forces keeping the balance. We have

2.1

where is the weight of the body G lowered by the weight of the supporting lower limb, a is an arm of the acting weight force G with respect to the point C, and b is an arm of the acting force Fs with respect to the point C.

For a rate of the moment arm from the radiographs we find 2 < a/b < 3.5. Hence and from (2.1) it follows that 1.7G < Fs < 2.9G.

The force of the abductors, Fs, acts on the angle β = 22°. For G = 800 N components Fsv and Fsh are equal to Fsv = Fscos β = 1237–2110 N, Fsh = Fssin β = 500–853 N (see Fig. 2.3). From the balance of forces acting on the pelvis it follows that , tg α = Fsh/(Fsv + G′), then α = 15–17° (see Fig. 2.3).

It means that when standing on one leg a force acts on the femur head that is 3.7,times greater than the weight of the human body. This force acts on the femur head in a direction diverted medially from 16° from the vertical. The resultant muscle forces Fs and the resulting pressure force P in this static situation act in one plane. This plane goes in a direction of an acting pressure force of the human body weight and the center of rotation of the femur head (Beznoska et al., 1987). During walking this tension plane is turned probably to a small angle against the frontal plane going through both centers of femur heads, so that the mathematical factor for a muscle force resultant will be ≈0.94. Probably the direction of the resulting pressure force P acting in the hip joint in the relation with the vertical is medially diverted in different phases of gait in the range of 2°–20°. The pressure force resultant acting on the femur head is possible with an advantage and with a better exactness to get an analysis of the biomechanical model problem based on the contact theory, which are going to be the main parts of this study concerning the hip joint. As a result we will not get only the value of the resulting force at the point C but also the distribution of the normal and tangential components of stresses on a contact surface between the femur and the acetabulum and also the distribution of the stress and strain fields in the whole articular system. Figure 2.2 shows the schematic cross section of the hip joint. Figures 2.3 and 2.4 present the hip joint models according to Pauwels and Nedoma. The Nedoma model is based on the contact theory in linear (visco-)elasticity and thermoelasticity, and it will be analyzed in detail from the mathematical point of view in Chapter 6.

FIG. 2.2 Schematic cross section of the hip joint [modified after Beznoska et al. (1987) and Nedoma et al. (2006)], where a = the pelvis, b = the femur, c = the contact surface, d = the cartilages, e = the joint capsule, and f = the joint cavity.

FIG. 2.3 The Pauwels model. [Modified after Bartoš (1998) and Nedoma et al. (2006).]

Fig. 2.4 The Nedoma model. [After Nedoma (1993b) and Nedoma et al. (2006).]

The hip joint is stabilized by the muscle system, which is important for biomechanical relationships in a joint, because it affects force relationships in a limb and its movements. Movements in the hip joint are controlled by muscles, which do not lay directly on the hip joint. Where muscles grip and the direction of their operation are very important from a biomechanical point of views. They are muscles, which provide flexion, extension, abduction, adduction, and internal and external rotation.

The distribution of muscles and their positions surrounding the hip and the forces they are developing will be necessary to derive the boundary conditions at the simulation of the hip joint function and tension and deformation relations inside the articular system.

For the hip joint arthroplasty following morphological data are important: (i) a colo-diaphysaire (CD) angle about 126° with a range of 115°–140°. This angle changes with age. (ii) An angle neck anteversion is quoted as having an average of 12° with a range of 4°–20°.

Fundamental morphological constitution of the upper femur end establishes biomechanical characteristics on which musculatures and other structures also establish. The bone architecture depends also on changes related with flows of forces in the neighborhood of the upper end of the femur. Two basic trajectory systems—tension and pressure, as it follows from experimental and theoretical biomechanics—are a morphological substrate of tensile and compression forces during loading of the end of the upper femur. Biomechanically less active, less exposed places do not have such a structure, and they are labeled like the so-called Ward triangle, seen on the frontal cut of the end of the upper femur. These anatomical experiences and basic knowledge are necessary to the mathematical simulation function of the hip joint and during the mathematical simulation as well as design of total replacement, but also during its construction and own implantation of an artificial joint (THA).

Many models try to explain the biomechanics of the hip joint. The basic ordinary model is the static Fischer hip joint model; see Fig 2.5 (Pauwels, 1973; Bombelli, 1983; Bartoš, 1998). The center of gravity is situated in the middle sagittal plane at the Th 10–11 disk area. Both joints carry the weight of the body, which is distributed between both limbs and the median halves vertically cut the line connecting the centers of both hip joints. The head is in contact with the acetabular surface and we refer to it as the load zone. The resultant of post-operating forces act on the head of the hip joint in a vertical direction. The Pauwels model (see Fig. 2.3) developed from the previous model of the so-called sixteenth period of the gait, where the gravity center S is situated in the vertical frontal plane led by the joint center O. Biomechanical relations in a joint correspond to a double-reversible lever with a center of rotation O in the center of the joint of the carrying limb, where on a longer arm of a lever system OC the resulting force corresponds to the body weight without a carrying limb, and this moment is eliminated by a force of muscles acting on a lever arm, corresponding to a distance from a joint center to where muscles are inserted in the great trochanter. According to Pauwels, the direction of adductoral muscle movement is