The application of forces to the musculoskeletal system generates moments around joints, thereby facilitating locomotion. It is important for the orthopaedic surgeon to have an understanding of the various applications of musculoskeletal biomechanics and related concepts including implant design considerations, wear mechanisms, and rigid body mechanics.

A force is defined as a quantity that changes the velocity and/or direction of an object of the musculoskeletal system. Its magnitude is given by the product of the mass of the object and its acceleration. The unit of the force is given in Newtons (N = kg·m/s²). When a force acts on the musculoskeletal system over a distance from the joint center, the force creates a torque around the joint. A moment is defined as a quantity that changes the angular velocity of an object of the musculoskeletal system. The magnitude of the moment is defined by the product of its moment of inertia and its angular acceleration. The unit is the Newton-meter (N·m).

When an external force acts on a segment of the musculoskeletal system, for example, when a weight is being held with an outstretched arm, the moment created around the shoulder joint is equal to the product of the force acting on the body segment as well as the perpendicular distance between the line of action of the force and the joint center of rotation (defined as moment arm). Therefore, the moment acting on the shoulder joint as shown in Figure 1 is 80 N m. To resist the external load and keep the outstretched arm in an equilibrium position, internal muscle forces must be generated through the contraction of shoulder muscles including the deltoid to counteract the moment created by the external load.

The static equilibrium refers to the state of the musculoskeletal system, in which there is no acceleration of body segments. This means that the body segments are either at rest or moving with a constant velocity.

The concept of static equilibrium can be used to calculate the force required by the biceps muscle to keep the arm, as shown in Figure 2, in an equilibrium position. Without any acceleration of body segments, the
sum of the forces and moments around the elbow joint must be zero. If the weight in the hand and the biceps force are considered the only two forces generating a moment around the elbow joint, then the extension moment caused by the weight must be offset by the flexion moment created through the contraction of the biceps. As the moment arm of the biceps muscle around the elbow is an order of magnitude smaller than the moment arm of the weight (4 cm versus 30 cm), the biceps must generate a 7.5 times greater force to create the same magnitude moment around the elbow joint as was generated by the weight.

Free-body diagrams are used to identify all forces and moments acting on a segment of the musculoskeletal system to correctly apply the equilibrium equations. For that purpose, the musculoskeletal system is isolated from its surroundings and forces as well as moments are used to replace the effect of the surrounding environment. Figure 3 shows a forearm holding a weight. Point O describes the fixed axis of rotation for the elbow. Point A quantifies the biceps muscle attachment with the radius, point B describes the forearm center of gravity, and point C describes the center of gravity of the weight. W_O is the mass of the object and W_F is the total weight of the forearm. F_B connotates the biceps force exerted on the radius, whereas R_F connotates the elbow joint reaction force. The line of action for the muscle forces as well as the gravitational forces is assumed to be vertical.

The elbow joint reaction force (R_F) as well as the muscle force (F_B) can be computed through consideration of the equilibrium conditions at the forearm. Analyzing the rotational equilibrium conditions of the forearm around the elbow joint will lead to:

Analyzing the translation forearm equilibrium along the y-direction leads to:

These rotational and translational equilibrium equations can be solved for R_F and F_B for any given set of parameters. For example, if a = 4 cm, b = 15 cm, c = 35 cm, W_O = 80 N, and W_F = 20 N, this would yield a force of 775 N for F_B and a force of 675 N for R_F.

Free-body diagrams can also be used to calculate external intersegmental forces and moments at different joints during locomotion. In Figure 4, A, the subscripts f, s, and t denote foot, shank, and thigh, respectively. I denotes the moment of inertia, whereas α stands for the angular acceleration. The gravitational acceleration is denoted by g, whereas m and a stand for the segmental mass and linear acceleration, respectively. In Figure 4, B, the subscripts g, k, a, and h denote the ground, knee, ankle, and hip. The force is denoted by F, whereas T denotes the resultant torque. Following the creation of the free-body diagrams, equilibrium equations serve to calculate the unknown proximal forces. The calculations start from the distal end and proceed to the proximal end of the body segment, starting with the foot. Both the torque as well as the force at the distal part of the shank are equal and opposite to the torque and force of the ankle. With the knowledge of distal force and torque of the shank, the proximal force and torque of the shank can then be calculated.

Bone represents the most abundant of the tissues in the musculoskeletal system. Bones contain 40% inorganic material (hydroxyapatite), 35% organic matrix (type I collagen), and 25% water.¹ When loaded, the inorganic material contributes to the compressive behavior and the type I collagen plays a significant role in the tensile properties of bones. At the micrometer level, bone is composed of mineralized collagen fibrils that are embedded within the organic extracellular matrix. The fibrils are arranged into lamellae with a thickness of 7 to 10 µm. Circumferential structures of numerous lamellae with a central haversian canal form osteon, a structure that is approximately 200 µm in diameter. Lamellae as well as osteons form cortical and cancellous bone structures at the highest level of scale (greater than 1 mm). The cortical bone is made up of longitudinally oriented lamellae and osteons, whereas cancellous bone (cortical bone) is characterized by an open cellular structure with a variable microstructure of trabeculae and struts.

Based on the hierarchical structure of bones, its material properties are anisotropic (vary in different directions) and viscoelastic (dependent on loading rate). Cortical bone loaded longitudinally is more resistant compared with loads applied in the transverse direction.¹ The cortical bone is loaded the greatest extent in the longitudinal direction during functional daily activities including walking and running. Additionally, cortical bone is stronger in compression, when compared with tension. Stiffness and strength of cortical bone are dependent on the loading rate, with a stiffer behavior and a decreased failure strain when loaded more quickly² (Figure 5). With regard to cancellous bone, its material properties are more challenging to generalize.
Its material properties are dependent on the apparent density, quantified through linear or exponential relationships.³ Cancellous bone, similar to cortical bone, is stronger in compression compared with tension.

The mechanical behavior of tendons and ligaments is similar, despite their different functions in the musculoskeletal system. Ligaments connect bone to bone to provide joint stability, whereas tendons connect muscle to bone to transfer muscle-generated force to the skeleton to facilitate motion. The tissue matrices of both tendons and ligaments are composed of type I collagen (75% to 80% dry weight) and proteoglycans (1% to 3% dry weight).⁴ They are both highly hydrated, with one major difference. Ligaments contain a greater percentage of elastin (10% to 15% dry weight) compared with tendons (less than 3% dry weight). With regard to tendons, the collagen fibers are aligned uniformly along the loading direction of the tendon. In contrast, the collagen fibril orientation in ligaments is less structural, reflecting the greater range of loading conditions experienced by ligaments.

The mechanical properties of tendons and ligaments are characterized by anisotropic behavior. The tensile stiffness of tendons and ligaments is smaller in comparison with bone but greater than that of cartilage. The nonlinear behavior of tendons and ligaments under tensile loading conditions is characterized by an initial toe region, where low loads lead to high deformations as fiber realignment occurs. The toe region is followed by a linear region where the aligned fibers bear the load and elongate until yield and failure occur.

The knee and hip joints are diarthrodial joints, which facilitate the relative movement of the articulating bones in the synovial cavity. The joint motions are commonly defined in three mutually perpendicular anatomic planes: coronal (frontal), sagittal (longitudinal), and transverse (horizontal). Rotations of the lower limb in the coronal plane are defined as abduction (leg is raised away from the center of the body) and adduction (leg moves back toward the center of the body). Rotations in the sagittal plane are defined as flexion (hip rotates forward) and extension (hip rotates back toward straightened position). With regard to the transverse plane, rotations are defined as internal rotation (rotate around its longitudinal axis toward the center of the body) and external rotation (rotate around its longitudinal axis outward).
The hip joint allows all three of these rotations, thus it has three degrees of freedom for rotation, in addition to the three degrees of freedom for translation, making the hip joint an articulation with six degrees of freedom. The knee joint allows the translation of the articulating surfaces in the three anatomic directions, in addition to the three rotations, thus the knee joint has six degrees of freedom. With regard to hip and knee total joint arthroplasties, current implant designs preserve these degrees of freedom to restore joint functionality.

The rotational motion of the hip joint has been quantified during numerous functional activities including gait, sit-to-stand, and stair climbing. Hip joint rotations during walking are shown in Figure 6 based on findings from a 2019 study.⁵ The stance phase (leg in contact with the ground) encompasses approximately 60% of the gait cycle and starts with heel strike and finishes with toe-off. The swing phase (leg in no contact with the ground) makes up approximately 40% of the gait cycle and starts with toe-off and ends with another heel strike. Figure 6 illustrates that the hip joint extends in the sagittal plane by almost 35° during the stance phase, before flexing back during the swing phase. Abduction/adduction angles as well as internal/external rotation angles both cover a range of 10° during the gait cycle. The first half of the stance phase is characterized by hip abduction and internal rotation, whereas the second half of the stance phase is characterized by a hip adduction and external rotation in order to return to its equilibrium position. Because the hip joint is a ball-and-socket joint, the hip rotations require a relative sliding motion of the femoral head with regard to the acetabulum. Relative sliding motion occurs during functional daily activities, with the largest sliding motion required for flexion/extension rotations.

The complex motions of the knee joint during functionally strenuous activities have been studied in the literature. Representative data for flexion/extension, internal/external rotation, and anterior/posterior translation during gait are presented in a study⁶ and are shown in Figure 7. These data are derived from average patient data and used to program knee simulators for the assessment of wear patterns of knee arthroplasties. With regard to the gait cycle as shown in Figure 7, the knee is almost fully extended at heel strike, with the tibia being slightly rotated with regard to the femur. During the first 50% of the stance phase, knee flexion angles of up to 20° can be observed, whereas the tibia demonstrates internal rotation and anterior translation relative to the femur. During the last 50% of the stance phase, the knee demonstrates almost full extension, with the tibia rotating externally and displacing posteriorly relative to the femur. With regard to the swing phase, a large amount of knee flexion occurs (almost 60°) before the knee returns to full extension in time for the next gait cycle. The largest loads and contact stresses occur during the stance phase, in which the leg is in contact with the ground.

Because of the complex interplay of knee rotations and knee translations during functional daily activities, the knee joint demonstrates a combination of rolling and sliding motions. As a result, the location of contact between tibial and femoral components of the joint replacement moves during functional daily activity. The precise motion of the contact location is highly dependent on the design of the articulating surfaces. One example of a potential motion pattern for a total knee arthroplasty design is shown in Figure 8. The medial condylar contact pathway and the lateral condylar contact pathway both typically follow a narrow “8” shape. There are differences with regard to the pathway between the medial and lateral condyle as
shown in Figure 8, which facilitates the internal/external rotation of the femoral component relative to the tibial component.

The numerical outputs of musculoskeletal models require substantial validation and verification to provide clinical utility. For that purpose, experimental methods including instrumented implants and the telemetric transmission of load, strain, or pressure data have been successfully developed. With regard to the hip joint, most experimental studies used strain gauges attached to the neck of the femoral component of the hip arthroplasty⁶,⁷ (Figure 9). The output of the strain gauge is transmitted to a receiver where data analysis for forces acting on the femoral head can be performed. In terms of the knee joint, a limited number of prior experimental studies tested instrumented knee implants (distal portion of the femur) with a telemetric transmission of measured data.⁸ A different approach has been used, whereby the tibial knee prosthesis was instrumented for the measurement of knee loads during functional tasks.⁹ The most complete database for multiple joint replacements (hip, knee, shoulder, spine) and numerous functional tasks is provided by OrthoLoad (https://orthoload.com/).

With regard to in vivo loads of the musculoskeletal system during functional daily activities, recent studies showed in vivo shoulder loads of up to one to two times the body weight.⁹,¹⁰ In vivo joint forces in the lower extremities are higher and often reach multiple times the body weight during routine daily activities. In vivo joint loads of up to two to eight times the body weight have been recorded for the hip and knee joint during strenuous activities, including running and jumping,¹⁰,¹¹,¹²,¹³,¹⁴,¹⁵ as shown in Table 1.