## 2 Materials and Methods  
### The Model Implementation  
Our model was implemented on the SimWise-4D platform (Design Simulation Technologies, DST, Canton, MI, USA). This software is designed for the analysis of complex dynamic systems, in which rigid bodies are connected by different kinds of joint constraints, and the interaction between object surfaces is solved in the modality called \"collision\". By this software, we have created a knee joint model which includes bones, ligaments and muscles. As already described in a previous paper [29], models of femur, tibia and patella were obtained through segmentation of MRI using commercial software (Amira 5.3.3, Visage Imaging, Inc., San Diego, CA, USA). MRI images referred to a Caucasian male (age: 42 years, body height: 1.72 m, body mass: 70 kg), with no signs of pathology. Contact surfaces were smoothed by using Meshmixer (version 3.5.47, Autodesk, San Rafael, CA, USA). To reduce the simulation time, portions of the distal femur and proximal tibia were split from the rest of the bones and then attached to them by rigid constraints (Figure 1). In this way, the algorithms for surfaces interaction had to work on these small bone portions only. The friction coefficient between tibia and femoral surfaces was set at 0.01 [33]. Ligaments were represented by non-linear springs attached to the bones at specific points that were identified with the support of a text of joint physiology [34] and other publications [6; 22; 26; 29; 35]. Several springs were used for each ligament, to represent the different fascicles whichmay be tensioned in different working conditions. The whole ligaments structure included (see Figure 1): (a) the anterior cruciate ligament (ACL) subdivided into the two compartments, anterior (ACL-a) and posterior (ACL-p); (b) the posterior cruciate ligament (PCL) subdivided in anterior (PCL-a) and posterior (PCL-p) bundles; (c) the lateral collateral ligament (LCL) subdivided into three components, LCL-1, LCL-2 and LCL-3 (respectively anterior, intermediate and posterior); (d) the medial collateral ligament (MCL), composed of the superficial bundles MCL-a, MCL-i and MCL-p (anterior, intermediate and posterior) and by the deep components MCL-da (anterior) and MCL-dp (posterior); (e) the anterolateral fibrous capsule (Cap-Ant-L); and (f) the two portions of the posterior capsule: Cap-Post-L and Cap-Post-M (lateral and medial, respectively). The elastic characteristics of these springs were expressed in the form adopted by many authors [6; 28; 36] that included a quadratic rise of the force up to a predefined strain limit, and a linear behavior for larger deformations:  
\[f=\left\{ \begin{array}{cc}0.25K\varepsilon^{2}/\varepsilon_{l},&0\leq \varepsilon\leq 2\varepsilon_{l}\\ K(\varepsilon-\varepsilon_{l}),&\varepsilon>2\varepsilon_{l}\\ 0,&\varepsilon<0\end{array} \right. \tag{1}\]  
Figure 1: The two bone portions considered for the contact algorithm: the distal femur (red, that was rigidly attached to the femur) and the proximal tibia (blue, rigidly attached to the tibia). The forces produced by the interaction of the surfaces are transmitted to the constraints that can measure the resultant force and the moment. The ligaments connecting femur and tibia are represented by straight springs having non-linear characteristics (see text). \(K\) is the stiffness (N/m) in the linear section of the curve; \(\varepsilon\) is the strain, (L \(-\) L\({}_{0}\))/L\({}_{0}\); \(\varepsilon_{I}\) is the strain limit that corresponds to the intercept of the linear tract with the abscissa. According to [6] the strain limit was assumed \(\varepsilon_{I}=0.03\). The separation of the quadratic behavior from the linear behavior occurs at \(\varepsilon=2\varepsilon_{I}\). One fundamental parameter that strongly conditions the ligament force is the rest length L\({}_{0}\). According to [26], we assumed that in a reference position corresponding to knee full extension the strain of the ligament \(\varepsilon_{ref}\) was known. These values are reported in the cited publication.