# Walking in the Absence of Anterior Cruciate Ligament: The Role of the Quadriceps and Hamstrings
## 2 Materials and Methods  
A three-dimensional dynamic musculoskeletal model [10] was used to simulate the behavior of the knee joint during walking in different testing conditions. The model consisted of an articulated system of rigid bodies linked together by a series of rotation actuators (motors) that were animated by kinematic data corresponding to an unperturbed, normal walking cycle. The data that was used was extracted from our repository and referred to the average of 14 strides from 5 normal subjects who were males aged between 24 and 36 years and walking barefoot at their natural velocity. The SAFLo (Servizio di Analisi dellaFunzionalita Locomotoria) marker set protocol [17] was used and it provided the kinematic data corresponding to the 6 degrees of freedom (d.o.f.) of the pelvis (anterior/posterior, medial/lateral, vertical displacements, anteversion/retroversion, frontal plane tilt and horizontal rotation), the 3 d.o.f. of the hip joint (flexion/extension, adduction/abduction and internal/external rotation), the 2 d.o.f. of the knee joint (flexion/extension and internal/external rotation) and the 2 d.o.f. of the ankle joint (plantar/dorsiflexion and pronation/supination). These data were normalized in time and referred to an ideal cycle duration of 1 s. Masses of body segments, corresponding to an ideal subject that was 1.72 m tall and had 70 kg of body mass, were obtained from an anthropometric table [18]. The inverse dynamics problem solution provided us with the knee joint moments corresponding to this ideal subject. Digital models of the femur and the tibia, which were obtained through MRI images of a Caucasian male (age: 42 years, body height: 1.72 m, body mass: 70 kg) were attached to the thigh and the shank segments, respectively, in such a way that the only knee flexion-extension was imposed, while the remaining five d.o.f. were left unconstrained, where they were controlled by the interaction of the femoral condyles and tibial plateau surfaces and the knee joint ligaments (reproduced by several springs with non-linear characteristics) and resulting from the forward dynamics computation. The result was that, during the dynamic simulation of walking, the relative motion of the tibia and femur depended on the morphology of the femoral condyles in contact with the tibia plateaus, on the tension of the ligaments, on the muscle forces, and on the external forces and moments (ground reaction force, weight, and inertia of the shank and foot). The 3D model of the knee included a detailed ligament structure (Figure 1), which was composed of the anterior cruciate ligament (ACL, specifically removed for the purpose of this study), the posterior cruciate ligament (PCL), the lateral collateral ligament (LCL), the medial collateral ligament (subdivided in superficial and deep components (MCL and MCL-deep, respectively)) and the fibrous capsule, represented by its antero-lateral (Cap-Ant-L), posterior-lateral (Cap-Post-L) and posterior-medial (Cap-Post-M) bundles. An exhaustive description of this model can be found in the cited publication [10]. In the present study, the springs representing the two components of the ACL (Figure 1a) were set at zero stiffness, which corresponded to removing the ACL from the model ('NoACL' condition), without any change in the external forces and moments applied. As in the previous work, the muscle forces were estimated using a static optimization algorithm, which applied the minimization of the maximum force criterion in relation to the muscle physiological cross-sectional area [19]. The forces were transmitted directly to the bones as if the tendons were inextensible. Then, the force actuators correspon  
Figure 1: Smoothed tibial (dark orange) and femur (dark green) articular surfaces that were rigidly constrained to their respective bones. Straight springs representative of ligaments connected the femur and tibia. Posterior view (**a**): ACL-a, ACL-p, PCL-a and PCL-p are the anterior and posterior bundles of the ACL and PCL, respectively.