, David B. Burr2, Neil A. Sharkey3 and David P. Fyhrie4
(1)
Orthopaedic Surgery, UC Davis Medical Center, Olga, WA, USA
(2)
Department of Anatomy and Cell Biology, Indiana University School of Medicine, Indianapolis, IN, USA
(3)
Kinesiology, Orthopedics and Rehabilitation, Pennsylvania State University, University Park, PA, USA
(4)
Orthopaedic Surgery, Biomedical Engineering, UC Davis Medical Center, Sacramento, CA, USA
Abstract
This chapter describes how engineering principles can be used to estimate joint forces. Principles of static and dynamic analysis are reviewed, with examples of static analysis applied to the hip and elbow joints and to the analysis of joint forces in human ancestors. Applications to indeterminant problems of joint mechanics are presented and utilized to analyze equine fetlock joints.
Keywords
JointsForcesStaticsDynamicsKinematicKinetics…mechanical science is of all the noblest and most useful, seeing that by means of this all animate bodies which have movement perform all their actions…
Leonardo Da Vinci (1452–1519)
1.1 Introduction
The skeleton is first and foremost a mechanical organ. Its primary functions are to transmit forces from one part of the body to another and to protect certain other organs (e.g., the brain) from mechanical forces that could damage them. Therefore, the principal biologic role of skeletal tissues is to bear loads with limited amounts of deformation. To appreciate the mechanical attributes that these tissues must have to perform this role, it is necessary to learn something about the forces which whole bones normally carry. In most cases, these forces result from loads being passed from the part of the body in contact with a more or less rigid environmental surface (e.g., the heel on the ground when walking) through one or more bones to the applied or supported load (e.g., the torso). In addition to the forces transmitted in bone-to-bone contact, muscle and ligament forces act on the bones, and these forces (especially the muscle forces) are large and important.
Most muscle, ligament, and bone-to-bone forces act in or near the body’s major diarthrodial joints. The purpose of this chapter is to explain how conventional engineering analysis may be used to estimate joint forces, and to provide some practical experience in solving such problems.
1.2 An Illustrative Experimental Model
A major advantage of conventional static and dynamic analyses is that the forces and moments imposed on bones can be estimated with reasonable precision from non-invasive external measurements. Actual physical measurement of internal loading is more challenging, for both experimental and ethical reasons. Although physical measurements of skeletal loading have been made in live subjects (we will explore an example later in this chapter) these types of studies are relatively rare and always limited in scope. One way around the ethical dilemma posed by invasive measurement is to apply life-like loading environments to cadaver specimens donated for scientific research, thereby enabling direct, detailed measurement of internal kinetics and its physical consequences within the tissues themselves. Orthopaedic research has a long and rich history using this approach and much has been learned about normal skeletal loading and the biomechanical consequences of pathology, surgical reconstruction, and joint replacement. Before tackling classic static analyses, we will examine some data sets taken from a series of cadaver experiments focused on the lower extremity, an exercise that will lend further appreciation of the large forces routinely imposed on our skeletons.
The data presented in Fig. 1.1 were gathered using an apparatus that reproduces the kinematics and kinetics of normal gait in cadaver lower extremities (Sharkey and Hamel 1998). One set of computer controlled actuations reproduces the motions of the proximal tibia as recorded in a standard gait lab while a second set simultaneously reproduces the actions of six major extrinsic muscle groups of the foot and ankle. These muscles reside in the shank and include the triceps surae (gastrocnemius and soleus), tibialis posterior, tibialis anterior, peronei (peroneus longus and brevis) flexor halluces longus and flexor digitorum longus. The vertical axis of the plot is force, given in units of body weight (BW) for easy comparison, and the horizontal axis represents increments of time when the foot is in contact with the ground during normal walking. The vertical force imposed by the ground is given by the heavy dark double-peaked plot and reflects the classic force-time curve for gait whereby the first peak is a component of the braking force occurring shortly after heel-strike and the second peak a component of the push-off force to propel the body forward. The shape and magnitude of this plot confirms life-like loading equivalent to that routinely measured in standard gait labs.
Figure 1.1
Top: Image of an experimental apparatus enabling detailed measurements of biomechanical phenomena in skeletal tissues under normal or pathologic conditions. Bottom: Plots of the forces carried in the select skeletal tissues during normal walking.
The remaining plots in the graph represent the internal tissue forces present during normal walking. Peak tension in the Achilles tendon (a function of triceps surae contraction needed to propel the body forward) is approximately 1.8× BW while the peak tensile force carried in the plantar fascia, a tough band of ligamentous tissue that spans the sole of the foot is roughly equivalent to body weight (Erdemir et al. 2004). Perhaps most striking are the plots of joint reaction force measured in the tibiotalar (ankle) and first metatarsophalangeal (MTP; at the base of the great toe) joints. The routine load transmitted through the ankle is over 4.0× BW (Sharkey and Hamel 1998) while that carried through a very small digital joint in the first ray of the foot is almost equal to BW (Kirane et al. 2008). This is a remarkable feat (no pun intended). With each step we take day in and day out, both our hard and soft skeletal tissues are exposed to tensile and compressive forces considerably greater than body weight and this is only for walking; the forces are much larger during more strenuous activities like running and jumping. The mechanical explanation for these high forces is examined in detail within the next sections of this chapter, while future chapters examine the mechanisms enabling such an incredibly strong and durable system.
1.3 Static Analysis of Forces in Joints
1.3.1 Forces in the Elbow Joint
To extend this subject, let us consider the human elbow joint. We choose this joint because it works more or less as a simple hinge, and because it is familiar to us, conforming to the image that we acquired in grade school of the skeleton as a system of levers. Although we use this simplicity in solving for the forces, we should spend a few moments examining the anatomy to appreciate the simplifying assumptions that we are making.
A lateral view of the bones of the elbow is shown in Fig. 1.2a. The olecranon of the ulna wraps around the distal end of the humerus to form the major part of the elbow’s hinge joint. The proximal joint surface of the radius articulates with the distal joint surface of the humerus and also with a cartilage-covered notch (the radial notch) on the lateral aspect of the ulna; this means there are three articulations in the elbow joint. These articulations allow for flexion and extension of this joint as well as pronation and supination1 of the forearm. The radius and ulna are bound together by the interosseous membrane along the central part of their length and by several ligaments at their proximal and distal ends (not shown). We assume that these structures cause the two forearm bones to act as one structural unit. We also assume that the wrist joint is stabilized by its musculature so that the hand and forearm flex about the elbow as a simple hinge joint.
Figure 1.2
(a–d) Anatomical drawings of the elbow joint. (a) Overall view of the flexed joint and the biceps inserting primarily on the radius. (b, c, d) Approximate locations of the biceps, brachialis, and brachioradialis muscles.
When the elbow is flexed with the palm upward, two muscles are primarily responsible: the biceps and the brachialis (Fig. 1.2b, c). The biceps has two heads (i.e., points of origin). The short head originates on the coracoid process of the scapula, while the long head runs through the shoulder joint to attach to the superior lip of the glenoid fossa (the scapular part of the shoulder joint). The biceps divides distally into a tendon that inserts on the proximal radius and an aponeurosis (tough band of connective tissue) which blends with other muscles in the proximal forearm (Fig. 1.2b). The brachialis originates along the anterior surface of the central humerus and inserts on the proximal ulna (Fig. 1.2c). When the forearm is flexed rapidly, or a large force must be applied during flexion, the brachioradialis also acts between the distal humerus and the distal radius (Fig. 1.2d). Other muscles also cross the elbow and participate in flexion. However, to simplify matters, and because the present problem is for a static situation, we assume that the biceps and brachialis muscles act as one in holding the elbow in a flexed position against the action of a weight held in the hand. We want to find the force required in the “biceps-brachialis muscle” to support the flexed forearm and the total force exerted on the distal end of the humerus by the radius and ulna. These forces are expressed as multiples of the weight in the hand.
To solve this problem and others like it, three steps are necessary:
1.
Draw a free-body diagram.
2.
Write the equations of static equilibrium.
3.
Solve these equations simultaneously to obtain the unknown forces.
We simplify the analysis by assuming that it is two dimensional and that the forces all act in a sagittal plane containing the humerus and the forearm. The free-body diagram is constructed by isolating the structure being analyzed such that the internal forces to be determined are exposed and replacing all the forces acting on the structure by vectors. The free-body diagram for the elbow problem is shown in Fig. 1.3. The forearm is made a free body by “amputating” through the elbow joint, whose internal forces we wish to reveal. The forearm lies in the x–y plane. The weight in the hand, the muscle force, and the joint reaction force are represented by the vectors W, B, and J, respectively.2 The joint force is assumed to pass through a fixed center of rotation for the joint, shown by a dot. The insertion point of the muscle is b meters (m) along the forearm from the joint center, and the center of gravity of the weight in the hand is w meters from this point. The equations for static equilibrium in two dimensions are
Figure 1.3
Free-body diagram for the elbow force calculation. J is the joint reaction force, W is the weight in the hand, and B is the biceps force.
(1.1)
where M stands for moments about an arbitrary point, and F X and F Y are force components in the x– and y-directions. Because, in two dimensions, there are three such equations, one may solve for three unknown force components. Alternatively, as usually happens, one may solve for one force vector and the magnitude of another force of known direction.
For the elbow problem, taking moments about the joint center, the equilibrium equations are
(Here, any moment of the x-component of B about the joint center is ignored.) Solving these equations yields
If θ = 75°, w = 0.35 m, and b = 0.04 m, then B = 9.1 W, J X = 2.3 W, and J Y = 7.8 W. The magnitude of the joint reaction force is 
and its orientation is arctan(J y /J x ) = 74° with respect to the x-axis. Thus, the muscle and joint reaction forces are eight to nine times greater than the weight held in the hand. This result is typical of virtually all the joints in the body in that the skeleton works at a mechanical disadvantage in terms of force, and as a consequence the forces acting on bones are high relative to the forces applied by (or to) the environment.
(1.2)
(1.3)
(1.4)
(1.5)
(1.6)
(1.7)
and its orientation is arctan(J y /J x ) = 74° with respect to the x-axis. Thus, the muscle and joint reaction forces are eight to nine times greater than the weight held in the hand. This result is typical of virtually all the joints in the body in that the skeleton works at a mechanical disadvantage in terms of force, and as a consequence the forces acting on bones are high relative to the forces applied by (or to) the environment.There is, of course, a good reason for this. Muscles can only contract about 30 % of their length. In the case of the biceps, for example, the overall length is about 25 cm, so the maximum contraction is 7–8 cm. The lever action of the forearm magnifies this distance by the ratio w/b in Fig. 1.3, enabling the hand to move much further (and also much faster). Of equal importance, the muscles insert proximally on the radius and ulna and do not create a “web” of flesh across the front of the elbow. Therefore, magnification of motion enables larger movements to be accomplished with a compact structure. The price that is paid for this compactness and magnification of motion is high forces in the muscles, across the joint surfaces, and within the bones.
Box 1.1 Giovanni Borelli on the Movement of Animals
Giovanni Alfonso Borelli (1608–1679) was the greatest of early biomechanicians. He held the chair in mathematics at Pisa in Italy, where he was a close friend of Malpighi, who was the professor of theoretical medicine. Together, they did much to persuade seventeenth-century physicians of the importance of physics in understanding medicine and physiology. Borelli’s treatise, De Motu Animalium (On the Movement of Animals) (Borelli 1989), has been translated into English and is a marvelous testament to his genius and ability to explain musculoskeletal mechanics clearly and simply. In the preface to his translation, Maquet points out that later biomechanicians unwittingly duplicated several of Borelli’s important discoveries. While some of his results contain errors, it must be remembered that Newton’s laws were not published until 1687. Thus, although Borelli understood very well the principle of balancing moments about a fulcrum, he could not have been expected to fully understand static equilibrium as we do, based on Newton’s laws. One of Borelli’s greatest achievements was the discovery that animals’ joints work at a mechanical disadvantage in terms of force. He discusses this in the following excerpt (Fig. 1.4).
Figure 1.4
Diagram of elbow force problem from Borelli’s On the Movement of Animals (Borelli 1989). With kind permission of Springer Science + Business Media.
Proposition VIII
“It is commonly thought that Nature raises considerable weights by using the machines of the muscles with a weak moving force.
The magnitude of the vital force of the muscles … sustains, raises, and moves not only an arm or a leg, but the whole animal machine, enabling it even to dance. Besides the mass of the animal, heavy enough by itself, this force carries, pulls and pushes considerable weights.
Aristotle…did not recognize the muscles but imagined spirits which pull and push the limbs. [He] remarked how difficult it would be for the huge mass of an elephant to be moved…by tenuous spirit or wind. He met the difficulty by saying that Nature moves the joints and limbs of the animal by using very small force…and said that the operation is carried out by way of a lever. Therefore, it is not surprising that huge weights can be moved…by a small force. Lucretius used the same example….Galen also says that a tendon is like a lever. He thinks that, consequently, a small force of the animal faculty can pull and move heavy weights.
This general opinion seems to be so likely that, to my knowledge, not surprisingly, it has been questioned by nobody. Who indeed would be stupid enough to look for a machine [in the body] to move a very light weight with a great force, i.e., not to save forces but rather to spend forces? And if this is rightly considered as stupid, how is it possible that wise Nature, everywhere looking for economy, simplicity and facility, builds with great efficiency in animals machines to move, not heavy weights with a small force, but on the contrary, light weights with almost boundless force? …I shall demonstrate that multiple and different machines actually are used in the motions of animals but that light weights are carried by large and strong force rather than heavy weights being supported by small force.”
1.3.2 Forces in the Hip Joint
The hip joint is the articulation between the femur and the acetabulum of the pelvis. The hip is one of the most important joints in the body from a medical perspective. Especially in aging individuals, overall health is promoted by the exercise that goes with walking. The ability to walk depends on having a healthy, painfree hip. Two kinds of diseases, both very common among the elderly, preferentially affect the hip. Osteoarthritis is the primary reason that about 200,000 total hip replacement procedures are performed each year in the United States, and osteoporosis in the femoral neck results in several hundred thousand hip fractures each year, virtually all of which require surgical treatment with a metal fixation device or hip replacement. To better understand the mechanics of the hip and the demands on the implants used to correct its problems, it is important to know the forces across the hip during walking and their determining factors.
When one is walking, the lower extremities and other parts of the body are moving (i.e., accelerating), so the conditions of static equilibrium are not satisfied. However, most people do not usually walk very vigorously, so the accelerations are small relative to the forces produced by muscle pulls and gravity. Therefore, the problem is usually solved as though the person were simply standing on one leg, assuming that this approximates the conditions during the “single leg stance phase” of gait, that is, when all the weight is being carried by one leg. In addition to this assumption, we assume that the problem is two dimensional, in the frontal plane, and that only one muscle is acting. If you try to stand on one leg, the force pulling your center of gravity downward tends to rotate your torso toward the medial side of the leg you are standing on. The muscles that resist this movement are the same ones which you use to abduct your thigh when lying down (i.e., move your lower limb away from the body’s center line). Again, several muscles act to do this, but we can lump them all together and simply call them the “abductor muscles.”
Again, the first step in calculating the forces in the joint is to draw a free-body diagram, “amputating” through the joint in question to “reveal” the force vectors acting there. In this case we delete the lower extremity and study the remaining portion of the body. Figure 1.5 shows this situation. It is important to understand that the subject was standing on the leg that has been “amputated,” and that the other leg, of which only a portion is shown, was not touching the ground. The abductor muscle force is represented by the force vector P. The joint reaction force acting on the middle of the acetabulum in the pelvis is F. The weight of the body, represented by the vector B, is actually the entire body weight, W, minus the weight of the leg supporting the body. Because each lower extremity weighs about 1/6 W, we let B = 5 W/6. This force acts downward slightly to the right of the centerline of the body (Fig. 1.5). Taking moments about the center of the acetabulum, we have
The lengths of the moment arms b and c have been estimated from antero-posterior pelvic radiographs. It was found that the b/c ratio ranges between 2 and 3.5. Following the lead of Frankel and Burstein (1973), we choose a conservative value of 2.4 and obtain the convenient result that P = 2 W. That is, the force required in the abductor muscles to balance the body on the head of the weight-bearing femur during the stance phase of gait is twofold body weight.
Figure 1.5
Free-body diagram for the calculation of the hip joint force while walking. The “amputated” leg is the one supporting the body. The leg in the free-body diagram is not in contact with the ground. B is the weight of the body (minus the amputated leg), P is the abductor muscle force, and F is the joint reaction force.
(1.8)
(1.9)
To solve the rest of the problem, we write the equations for force equilibrium, assuming that the x-direction is horizontal and the y-direction is vertical:
where Θ ≈ 30° is the angle that the abductor muscle line of action makes with the y-axis. Because sin 30° = 0.5, the components of the joint reaction force F are
The horizontal force of the femur on the pelvis is equal to body weight, and the vertical force is 2.5 times as much. The total joint reaction force is 
. This force acts at an angle to the horizontal θ = arctan(F y /F x ) = 68.7°.
(1.10)
(1.11)
(1.12)
. This force acts at an angle to the horizontal θ = arctan(F y /F x ) = 68.7°.We have seen that the magnitude of the forces in the hip joint depends critically on the ratio of the body weight moment arm to the abductor muscle moment arm. Thus, anything that increases the former or decreases the latter increases the abductor muscle force required for gait and consequently the force on the head of the femur as well. People with short femoral necks have higher hip forces, other things being equal. More significantly, people with a wide pelvis also have larger hip forces. This tendency means that women have larger hip forces than men because their pelves must accommodate a birth canal (Burr et al. 1977). This fact may be one reason that women have more hip fractures and hip replacements because of arthritis than men do. It is also conceivable that this places women at a biomechanical disadvantage with respect to some athletic activities, although studies do not always show gender differences in the biomechanics of running, particularly endurance running (Atwater 1990).
Box 1.2 Measurement of Hip Joint Forces In Vivo
To verify the estimates of hip joint forces made using free-body calculations, several groups of investigators have implanted devices that allowed measurement of hip joint forces into hip replacement patients (Rydell 1965, 1966; English and Kilvington 1979; Davy et al. 1988; Kotzar et al. 1991; Bergmann et al. 1988, 1993, 2001; Taylor et al. 1997). When humans are used as experimental subjects, ethical concerns demand that the subject’s health not be endangered in any way. This ethical mandate has extensive technical consequences for measurement of hip joint forces. To begin with, a healthy human subject cannot be used; experimentation must be restricted to a subject with a diseased or injured hip who will be undergoing replacement of the proximal femur with a prosthesis by way of treatment for this condition. This condition provides the opportunity for using an instrumented prosthesis (Fig. 1.6) that can measure joint forces instead of a standard prosthesis. However, the instrumented prosthesis must be as strong and as durable as a standard one, which imposes important constraints on the instrumentation design. Moreover, wires cannot be run from the prosthesis to the surface of the skin, where they could pose a risk of infection or other problems. Consequently, the instrument must be fully contained within the prosthesis, with a working space of a few cubic centimeters, and it must be capable of transmitting data to a receiver outside the body while the patient walks.
Figure 1.6
Left: Exploded view of telemetry electronics inside an instrumented hip prosthesis (Davy et al. 1988). Right: Three-dimensional plot of hip joint force vector during a gait cycle (with crutches). Orientations and lengths of lines emanating from the surface of the prosthesis head indicate force direction and magnitude, respectively (Davy et al. 1988).
The solution to this technical problem has been to use strain gauges (3–12 in number) to measure strains on interior surfaces of the prosthesis, and microelectronic circuitry to process the strain signal, which is broadcast as an FM signal to a receiver held against the skin. The device must be fully calibrated in the laboratory before insertion into the subject, so that the measured strains can be converted to force components on the ball or head of the prosthesis. The power source in the prosthesis used by (Davy et al. 1988) Davy et al. was a battery located in the stem of the prosthesis and activated by a magnetic switch turned on by a magnet held outside the patient’s leg (Fig. 1.6). Because batteries carry a slight risk of releasing toxic components, and because they have a finite life, other investigators (Bergmann et al. 1993) used an inductive power source. A magnetic coil wrapped around the subject’s leg induced current in a receiver coil within the prosthesis.
One might question whether a patient who has a degenerated joint, has recently had surgery on it, and is in a laboratory environment, is capable of walking “normally”—but these are the best data available. Findings from all studies are consistent with one another and with the theoretical estimates. During the single leg stance phase of gait, hip joint forces between 2.5 and 3.3 times body weight were measured. The highest force recorded was in an individual who was attempting static single-leg balancing; during recovery from a momentary loss of balance, a force of 5.5 body weight was measured in the hip (Davy et al. 1988).
1.3.3 Clinical Significance of High Joint Forces
Because diarthrodial joints work at a mechanical force disadvantage so that limbs can move far and rapidly with short muscle contractions, high stresses are produced in the tissues of the bones and joints. Normally, these tissues carry their loads without causing pain, but various diseases and injuries can damage the tissues so that the deformations associated with loading are painful. To some degree, the pain is proportional to the amount of force carried by the tissues; in other words,
There are no nerves in cartilage, and the source of joint pain is poorly understood, but experience shows that reducing joint forces often alleviates pain. Often the physician is not able to do much about the disease, so the first consideration in controlling pain may be to reduce the forces in the joint. For example, the patient can lose weight, or walk with a cane, and thereby reduce the forces transmitted across the joint.
1.4 Hip Forces in Human Ancestors
Physical anthropologists have analyzed hip joint forces in skeletons of various hominids. Of particular interest is Australopithecus because of collateral evidence (footprints) that these individuals may have walked very much like modern humans. Figure 1.7 is a depiction of the famous Australopithecus skeleton known as Lucy. The differences between the anatomy of the femur and pelvis of Australopithecus and Homo sapiens include factors that seem to be very pertinent to hip joint force. For example, the neck of the Australopithecus femur; shaded in Fig. 1.8, overlaying the outline of a modern femur; was proportionately longer than ours, but the bone was smaller overall. The pelvis was also smaller overall, but the ilium (shown shaded in Fig. 1.9, on one side of a modern human pelvis of similar size) was more outwardly flared, moving the line of action of the abductor muscles away from the hip joint. These factors affect the moment arms of both the abductors and the body weight vector. The analysis of Lovejoy and co-workers (Lovejoy et al. 1973) indicates that the hip joint force in Australopithecus was about 2.5-fold body weight, a value somewhat less than that of modern humans. However, when they performed a similar analysis on the skeletons of Native Americans excavated at archeological sites (and thus more comparable in their condition to fragmented Australopithecus skeletons), they obtained a value of 2.5 for these as well. Therefore, Lovejoy et al. concluded that Australopithecus and modern humans experienced similar hip forces (Lovejoy et al. 1973). They then considered the pressure on the head of the femur in these two species. Considering the degree to which the head of the Australopithecus femur was smaller than ours, and the estimated difference in body weights, it was concluded that the pressure on the Australopithecus cartilage was about half that on our femoral heads. However, these estimates are quite tenuous because of the scarcity and fragmented condition of the Australopithecus skeletons. Later in their paper, Lovejoy et al. conceded that other observations suggest that the Australopithecus hip force may have been substantially less than that of Homo sapiens. Two of the exercises at the end of this chapter allow you to explore this application of skeletal biomechanics.
