Model construction
A 3D finite element model of the SIJ was developed. Three-dimensional models of the sacrum, ilia and femurs were reconstructed from the computed tomography (CT) images of a healthy male volunteer (34 years old, 170 cm in height, and 65 kg in weight) using Mimics 20.0 (Materialise Company, Leuven, Belgium), and the cortical and cancellous regions of the bones were distinguished. Axial slices 0.5-mm thick spanning the entire pelvis were selected for model construction. All surface models were meshed using Geomagic 2013 (Raindrop Company, Marble Hill, USA). The SIJ is composed of cartilage and the end-plate of the sacrum and the ilia, with their surrounding ligaments. The cartilage was reconstructed with a uniform thickness; the regions of the articular surfaces were based on CT images, and the thicknesses of the cartilage were acquired from the literature. The sacral and iliac cartilages had thicknesses of 2 mm and 1 mm, respectively. The bone end-plate thicknesses of the sacral and iliac parts of the cartilage were assumed to be 0.23 mm and 0.36 mm, respectively. The gap between the two cartilages was set at 0.3 mm [12]. The material properties chosen from previous studies [12, 28] are summarized in Table 1.
The anterior sacroiliac ligament (ASL), long posterior sacroiliac ligament (LPSL), short posterior sacroiliac ligament (SPSL), interosseous sacroiliac ligament (ISL), sacrospinous ligament (SS), and sacrotuberous ligament (ST) complexes were modelled as 3D tension-only truss elements. The attachment regions were chosen according to the literature [12]. Two fresh cadaver dissections were used to observe the ligaments’ positions and orientations. The ASL was made up of numerous thin bands that spanned the ventral surface of the SIJ, connecting the lateral aspect of the sacrum to the margin of the auricular surface of the ilium. The LPSL extended from the posterior superior iliac spine to the third and fourth transverse tubercles of the back of the sacrum. The SPSL lay deep to the LPSL and consisted of large fibres attaching the lateral aspect of the dorsal sacral surface to the tuberosity of the ilium. The ISL lay in the intra-articular space and was composed of a series of short, strong fibres connecting the tuberosities of the sacrum and ilium. The SS was a thin triangular ligament that connected the ischial spine to the lateral border of the sacrum. The ST was behind the sacrospinous ligament, which attached the ischial tuberosity to the lateral border of the sacrum. The material properties of each ligament were obtained from the literature [28]. In total, the pelvic-femur model contained 727,474 elements and 275,399 nodes. Figure1 shows the intact model with ligamentous attachments.
Three common manipulations were selected based on their popularity and validity. The point and orientation of the applied forces were determined by previous studies [29, 30]. In addition, the magnitudes of the forces were determined by determining the manipulative power of five therapists using a biomechanical testing machine. The detailed loading and boundary conditions, as well as the x-, y-, and z-axes, are described in Figure 2. The compressive stresses and displacements of the SIJ and the ligament strains for the three manipulations were then investigated using Abaqus 2018 (Dassault Systemes S. A Company, Massachusetts, USA).
Manipulation of hip and knee flexion
The patient lay supine while the therapist flexed the patient's hip and knee as much as possible with pronation. Then, the therapist pushed down the knee, at which point the left hip joint was assumed to be fully constrained. The most posterior regions of the sacrum and the posterior superior iliac spine were fixed. The left hip was flexed to 155° and was intorted to 35°. A compressive (downward) force of 600 N along the ventral-dorsal direction was simultaneously applied at the end of the left femur.
Manipulation of oblique pulling
The patient was in the side-lying position, and the therapist stood at the patient's ventral side. The therapist placed one hand on the dorsal side of the sacrum to fix the patient’s position and placed the other hand on the anterior superior spine, pushing the ilium towards the back. Thus, most regions of the sacrum and the right iliac crest were fixed. Then, a push force of 600 N along the ventral-dorsal direction and parallel to the left SIJ surface was applied to the left anterior superior spine.
Manipulation of lower limb hyperextension
The patient lay in a prone position, and the leg being treated was hyperextended at the hip so that the anterior superior spine could just lift off the bed. Then, the therapist applied a downward force to the iliac crest being treated. In this manner, the right lateral region of the ilium and the right pubic tubercle were fixed. Then, a push force of 600 N along the dorsal-ventral direction and parallel to the left SIJ surface was applied on the left iliac crest. The point of the applied force was the midpoint between the highest point of the iliac crest and the posterior superior spine.
Mesh convergence study
To evaluate the degree of accuracy of our FE model, a detailed mesh convergence study was conducted. Four FE models were developed. The number of elements and nodes for each mesh resolution is shown in Table 2. The meshes shown in Figure 3 were named as mesh 1, mesh 2, mesh 3 and mesh 4, respectively. Following boundary conditions and material properties, loads, and constraints described in detail in the above sections, MHKF, MOP and MLLH were applied to these meshes. The results of the maximum stress and maximum displacement were numerically estimated for each of the meshes.
Model validation
To validate the developed models, two tests were performed. For the pelvic model, the distribution of the principal strain of the pelvis was compared with that indicated in the study of Zhang [31]. Zhang et al analyzed the distribution of principal strain on the cortical bone of the pelvis for the single-legged stance. In this model, the distribution of the principal strain of the pelvis was investigated under the same loading and boundary conditions.
For the sacrum model, the relationship between load and displacement was compared to that reported in cadaveric [32] and computational studies [12, 33]. In the cadaveric experiment, the bilateral ilia were fixed. Five translational forces (anterior, posterior, superior, inferior, and mediolateral) of 294 N and three moments (flexion, extension, and axial rotation) of 42 Nm were applied separately to the centre of the sacrum. The displacements of a node lying in the mid-sagittal plane between the inferior S1 and superior S2 vertebral endplates were calculated. In this model, the displacement was estimated under the same loading.