Finite-element model development
Magnetic resonance images were obtained of a left knee (number of excitations = 1, echo train length = 3, slice thickness = 2 mm, slice spacing = 2 mm, matrix = 240 × 320, field of view = 140 mm with axial T1 fat saturation (fat-sat) (repetition time (TR) = 712 ms, echo time (TE) = 12 ms), coronal T1 fat-sat (TR = 730 ms, TE = 10 ms), and sagittal T1 fat-sat (TR = 796 ms, TE = 10 ms); Siemens TrioTim, Washington, DC, USA) of a single male subject (42 years; 75 kg). Thirty-five millilitres of contrast-enhancing fluid was then injected into the knee (12 ml of Isovue 300, 0.15 ml of gadolinium, 18 ml of normal saline and 4.85 ml of 0.5% ropivacaine). Following 2–3 min of unloaded knee flexion and extension, weight-bearing low-dose cone beam computed tomography (SCT) images (0.30 mm isotropic voxel size; 20 × 35 × 35 cm field of view; CurveBeam, Warrington, PA, USA) were taken to provide kinematic data for approximately 0°, 20° and 30° of knee-joint flexion. The participant was positioned with the tips of the great toes, patellae, and the anterior superior iliac spines coplanar to each other and the feet were 10° externally rotated. The work was carried out in accordance with the World Medical Association Declaration of Helsinki; however, as it was not a systematic investigation, it was not classified as research on human subjects and, therefore, did not require Institutional Research Board approval.
Images were imported into MIMICS (v. 17.0, Materialise, Leuven, Belgium) for segmentation. The femur, tibia, femoral cartilage, medial and lateral tibial cartilages, and medial (MM) and lateral (LM) menisci were segmented in the sagittal plane to capture the anteroposterior curvature of the articulating surfaces (Fig. 1a). Smoothing and Boolean subtractions between intersecting geometries were applied in 3-Matic (v. 9.0, Materialise, Leuven, Belgium); the inferior surfaces of the menisci were assumed to be congruent with the tibial articular surface. The menisci were imported into Solidworks (v. 2015, Dassault Systems, Waltham, MA, USA), where the anterior and posterior segments were terminated with flat surfaces to facilitate attachment of the insertional ligaments. The medial and lateral epicondyles and the adductor tubercle were identified on the segmented femur and correspondingly on the SCT images. Custom-written code (Matrix Laboratory, MATLAB, The Mathworks Inc., Natick, MA, USA) was used to calculate the transformations required to position the bones correctly so that they matched the kinematics of the weight-bearing SCT scans.
Geometries were meshed with linear tetrahedral elements in Mentat (v. 2013, MSC Software Corporation, Newport Beach, CA, USA). Contact between cartilage surfaces and between cartilage and meniscal surfaces allowed tangential slip with a friction coefficient of 0.02 . No tangential motions were permitted between cartilage and bone interfaces. Meniscal insertional ligaments were modelled as linear non-compressive springs with insertion sites determined from the MR images. Stiffnesses for the lateral anterior, lateral posterior, medial anterior, and medial posterior ligaments were 216, 130, 169, and 207 N/mm, respectively . Articular cartilage was assumed to be linearly elastic and isotropic with a Young’s modulus of 13 MPa  and a Poisson’s ratio of 0.42 . The menisci were modelled as linearly elastic and transversely isotropic with Young’s moduli, Poisson’s ratios and shear moduli of 20 MPa, 0.2 and 8.3 MPa, respectively, in-plane, and 150 MPa, 0.3 and 57.7 MPa circumferentially . Bones were modelled as rigid . The tibia was constrained in all six degrees of freedom. The femur was constrained in flexion–extension, internal–external rotation, and anteroposterior translation and unconstrained in medial–lateral translation and varus–valgus angulation.
Validation of finite-element analysis and modelling intact menisci
Simulations were run in MSC.Marc (v. 2013, MSC Software Corporation, CA, USA). Compressive loads of 375 N (representing approximately one-half body weight) and 750 N (representing approximately full body weight) were applied along the tibial long axis at 0°, 20° and 30° of knee-joint flexion. The results of meniscal movement from the simulations with one-half body weight were compared against those observed in the weight-bearing SCT images.
Modelling meniscal tears
Meniscal tears (Fig. 1b) were created using 3-Matic on the finite-element mesh of the intact tissue. The menisci were divided into anterior, middle and posterior thirds circumferentially (Fig. 1c), and inner, middle and outer thirds radially (white avascular, red–white, and red vascular zones, respectively). Longitudinal tears were placed only at the center of the red zone, where they are most commonly observed . Stable and unstable full-thickness longitudinal tears were modelled as 7 and 14 mm in length, respectively [4, 17]. Radial tears were located in the white zone, with a stable length of one-sixth of the rim width and an unstable length of one-half of the rim width . All meniscal tear simulations were conducted with a compressive load of one times body weight. Maximum principal values of stress (max PCS) sampled from the inner to the outer rim in the anterior, middle, and posterior thirds of both MM and LM were used as an indicator of hoop stress. As the dominant load transmission in the articular cartilage is through compression, peak minimum principal stress is reported.