## 2 Model-Based Analysis of MSK Function  
### Model Structure  
Musculoskeletal models are, at their core, a mathematical representation of the MSK system (Figure 2). Here, we will describe the general structure of commonly used MSK models. On the simplest level, models contain simplified representations of bones, connected by a series of joints which are then actuated by muscle tendon units (MTU) typically represented as line actuators that have force-producing properties. Each bone of the skeletal system is represented as a rigid body with a geometry, a mass, and inertial properties. Joints connect two adjacent bodies and define their articulation (i.e., location and orientation of joint axis around which the body moves) and permitted motions (i.e., degrees of freedom (DOF)). Several different joint definitions are used within MSK models which differ in the number of DOFs (i.e., movement directions). These range in complexity from allowing no movement (weld joint e.g., fibrous joint between the tibia and fibula), permitting a single rotation (hinge joint, e.g., dorsi-/plantar-flexion of the ankle), allowing three rotations (complex ball and socket joint, e.g., hip), all the way up to joints permitting three rotations and three translations (6 DOF joint, e.g., tibiofemoral). Joint kinematics describe how bony and soft tissues limit joint movement (e.g., defining hip rotation range of motion due to the hip capsule). As a result, the permitted range of motion of each DOF is defined, representing these bony and soft tissues structure function, without explicitly modelling  
Figure 2: Left: Example of generic full-body MSK model [1] detailing each model component. Displayed are each of the bones, an example of an experimental marker set (pink dots), and the biceps femoris long head, vastus medialis and medial gastrocnemius muscle tendon units. Right: Example of a more complex model [2,3] with representations of ligaments and cartilage surfaces of the tibiofemoral and patellofemoral joint. them. Some models introduce more complex joint definitions, (e.g., through kinematic coupling) which relates an undetermined (i.e., difficult or impossible to measure) DOF to a measured DOF. A specific example of this kinematic coupling is the patellofemoral joint [4]. Given the difficulties in reliably tracking its movement with markers, patella movement is defined as a function of the knee angle rather than based on measured marker positions. Muscle tendon units span joints and actuate them by developing force along their line of action (i.e., path). In MSK models, muscles and tendons are modelled as a combined unit, hence the term muscle tendon unit. The path of each MTU is defined by an origin and insertion point, but also the intermediate pathway can be specified to represent wrapping over soft tissue or bony structures. This intermediate pathway can be defined using a combination of static or moving points, or analytical surfaces [5]. The pathway determines the elongation of the muscle as a function of the joint movement and by extension the moment arm [6]. The force-producing capacity of the MTU is modelled based on a Hill-type model of muscle contraction [7]. This model describes generic curves describing force generation of active and passive MTU components as a function of length and velocity, thereby capturing the effects of length and contraction velocity on force generation. These generic force-generating curves are made actuator specific by assigning MTU-specific parameters (e.g., maximal isometric force, optimal fibre length, tendon slack length, pennation angle and maximal fibre contraction velocity) to appropriately scale and tune the force-generating capacity. These MSK models can then be used in combination with 3D experimental gait data (i.e., marker positions, and ground reaction forces) to estimate biomechanical parameters during dynamic function--for example, joint angles--but also muscle activations and forces ultimately allowing calculation of joint contact forces.