## Introduction

Acetabular dysplasia is one of the most influential factors to progress to osteoarthritis (OA) of the hip joint due to insufficient bony coverage of the femoral head1,2,3,4. As a characteristic of acetabular dysplasia, it has been reported that the iliac bone has an inward morphological abnormality5. The posterior part of the iliac bone constitutes the sacroiliac joint (SIJ). In the pelvis, the SIJ has a small range of motion to work as a shock absorber between the spine and lower extremities6,7. In humans, the SIJ plays a crucial role in the ability to walk upright on two legs (bipedal walking)8, as it dissipates loads effectively. Although movement in the SIJ is limited, one of the representative movements of the SIJ is nutation and counter-nutation. This is a forward and backward rotation of the sacrum on the sagittal plane, respectively. According to the principles of manual medicine for the treatment SIJ function, the SIJ would be in the most-stable position when the sacrum is in the counter-nutation position relative to the ilium, indicating a closed-packed position of the SIJ9. The SIJ consists of an anterior cartilaginous region and a posterior ligamentous (syndesmotic) region10. Particularly, the posterior ligaments play an important role in load-bearing and transmission in the pelvis8. Repetitive and/or unexpected movements may cause minor subluxation of the SIJ, which has been hypothesized to lead to joint dysfunction11. In general, a joint dysfunction can be defined as a functional disorder of the joint without known specific cause12, i.e., modern medical standards including imaging equipment and surgical technology are unable to detect the morphological changes9. When SIJ dysfunction occurs, nerve endings, mainly in the posterior ligaments of the joint13, may be involved in the generation of pain as an alarming sign of SIJ dysfunction. In approximately 15%-30% of patients with lower back or gluteal pain, the dysfunctional SIJ can be identified as a cause14,15. The diagnosis of the pain originating from the SIJ is often delayed owing to a lack of specific imaging findings, and many patients experience chronic pain without appropriate treatment. Several clinical reports mentioned that hip disorders could affect the SIJ condition and pain16,17,18. The mechanical effects on both the cartilaginous and the ligaments of the SIJ in patients with acetabular dysplasia may differ from those in the healthy pelvis.

Peri-acetabular osteotomy is a mainstay in the surgical treatment of acetabular dysplasia to prevent the progression of OA of the hip. This surgery transects the iliac and ischial bones that are in contact with the hip joint; here the osteotomized bone is rotated outward and anteriorly to improve the bony and cartilaginous coverage of the femoral head. Acetabular osteotomy, such as rotational acetabular osteotomy and periacetabular osteotomy, has been performed and good, long-term results have been reported clinically in improving hip pain and suppressing OA progression19,20,21,22. It is believed that acetabular osteotomy alters stress distribution at the hip joint23, but is unclear to date whether mechanical changes to the SIJ will occur.

In this given study, stress distribution of the whole pelvis was analyzed using preoperative and postoperative models of the osteoligamentous pelvis of four patients who underwent unilateral spherical periacetabular osteotomy24.

The study aimed to investigate the stress environment of the SIJ in acetabular dysplasia. It was hypothesized that the inwardly-rotated innominate bone in acetabular dysplasia causes increased the stress at the SIJ, and the changes induced by peri-acetabular osteotomy will aid decrease this stress.

## Methods

### Model creation and mesh generation

Finite element models of acetabular dysplasia pelves (Fig. 1) were created based on computed tomography (1-mm slice thickness) of preoperative (‘pre model’) and postoperative (‘post model’) pelves of four female patients (18–41 years old). All methods were carried out in accordance with relevant guidelines and regulations, and all patients provided the signed informed consent for use of the data. This study was approved by Institutional Review Board committee of Kanazawa Medical University. The bone and cartilage components were segmented in MECHANICAL FINDER ver. 10 (Research Center of Computational Mechanics, Inc., Tokyo, Japan) including the lumbar vertebra, the sacrum, both hip bones and proximal ends of both femora, as well as both SIJ cartilages, the pubic symphysis, both hip joint cartilages and the intervertebral disks. These geometries were modified and the SIJ cartilage was adapted to the bone shape in SpaceClaim 2021R1 (Cybernet Systems Co., Ltd., Tokyo, Japan). All models were then imported into ANSYS 2021R1 (Cybernet Systems Co., Ltd.). The twelve types of ligaments and two types of muscles surrounding the pelvis were modelled by a total of 210 spring elements8 and 20 beam elements, respectively (Fig. 2). The ligaments were defined in a way where they could respond only when they are subjected to tensile loads. Tensile forces of 720 N and 100 N were applied on the gluteus medius muscle and iliacus muscle on both sides, respectively25. Meshing was performed using tetrahedral elements consisting of ten nodes each (Supplementary Table S1). All figures of finite element models were displayed by ANSYS 2021R1 and modified by Microsoft PowerPoint on Microsoft 365.

### Material properties

The material properties used in this study were cited from previous studies8,26,27 (Table 1). All tissues were defined as being uniform and isotropic materials for simplification. The hyper-elastic material properties were defined using Mooney-Rivlin model, which is the strain energy density function W given by the following formula, as a complete non-compressional body.

$$W = C_{10} \left( {I_{1} - 3} \right) + C_{01} \left( {I_{2} - 3} \right) + C_{11} \left( {I_{1} - 3} \right)\left( {I_{2} - 3} \right).$$

Mimicking double-leg stance, 300 N and 600 N of loads were applied via the lowers ends of both femora, which were shortened to two thirds of the total length, and the base of the sacrum, respectively. The superior aspect of the second sacral spine and both femora were fixed in space. For contact type, all surfaces in contact were defined as “bonded”, which means the surfaces are fixed to each other.

### Measured parameters

Acetabular head index (AHI, A/B) (Fig. 3) and sharp angle (Fig. 4) are indices for an assessment of acetabular dysplasia and are the ratio of acetabular coverage to the femoral head28 and the angle between the horizontal line and a line from the tip of the pelvic tear drop to the lateral edge of the acetabular roof29, respectively. Upper coverage (upper) and posterior coverage (posterior) of the femoral head diameter were measured. The resultant displacement of the pelves, equivalent (Von Mises) stress of the SIJ cartilages, the normal stress on lateral axes, the angles of rotation on the SIJs, the maximum elastic force of spring probes and acetabular head indices were investigated. Equivalent stress is a scalar value that is calculated from normal stresses and shear stresses without any distinction between tension and compression. Normal stress on lateral axes indicates the stress on the normal direction of the contact surfaces. Maximum elastic force of the spring elements was measured and summed for each of the ligaments as loads on ligaments.

## Results

### Acetabular head indices improved mainly in upper coverage of acetabulum

The AHIs on the side which underwent surgery improved − 1% to 22% (average 12.5%) in the upper area coverage and − 4% to 12% (average 5%), and in the posterior area coverage (Table 2). Coverage improved on both sites only in patient 1. Patients 2 and 4 improved in upper coverage, and patient 3 improved in the posterior coverage. The sharp angle on the side which underwent surgery improved from 51.8° to 42.5°. The average improvement was 9.3°.

### Acetabular dysplasia pelves tended to rotate inward, called the ‘inflare’

The displacement vector diagrams (Fig. 5a) of the pelves showed that the post models of all patients showed the innominate bones were deformed posteriorly. In the pre models derived from the patient datasets 1, 3 and 4, however, the innominate bones were deformed laterally to anteriorly. On the other hand, patient 2 in the pre models showed that the innominate bone was deformed posteriorly more than the post models. In the pre models of patients 1, 3 and 4 on acetabular dysplasia, the pelves rotated inward in the state of inflare as indicated by the red arrows in Fig. 5b. In the post models, the iliac crest was more rotated outward than in the pre models as indicated by the blue arrows in Fig. 5b. The maximum displacement on the iliac crest of surgical sides decreased by approximately 19% from pre models to post models in patients 1 and 3 which had a high improvement in posterior AHI more than 10%.

### Sacral nutation decreased when increased acetabular coverage

In the surgical sides of post models, the rotation of SIJs decreased in patients 1 and 3, and increased in patients 2 and 4 compared to each pre models (Fig. 6). In addition, sacral nutation movement is decreased with posterior acetabular coverage and the overall improvement of the coverage led to counter-nutation (upper: r =  − 0.39, posterior: r =  − 0.68) (Supplementary Figure S1).

### The equivalent stress and compressive stress on sacroiliac joints cartilage decreased when increasing acetabular coverage

The equivalent stress of SIJ cartilage decreased on the posterior regions in patient 1, and increased in the superior regions in all four cases (Fig. 7a). Maximum equivalent stress of SIJ cartilage ipsilateral to the surgical sides of pre models decreased by 80% in patient 1, while it increased by 67-2834% in the other cases compared to the post models (Fig. 7c). Normal stresses yielded that the SIJs were compressed on the inferior and extended on the superior regions (Fig. 7b). Minimum normal stress on the surgical sides decreased by 90% in patients comparing the pre models to the post models, and increased by 36–11,204% in the other cases (Fig. 7d). In patients 1 and 3, the posterior coverages were highly increased (10% and 12% increase), and the stresses decreased or slightly increased. Patients 2 remained the posterior coverage (2% increase) and patient 4 were decreased (4% decrease). These models showed slightly or dramatically increment. Thus, posterior acetabular coverage improvement reduced the maximum equivalent stress and minimum normal stress (equivalent stress: r = 0.26 (upper), 0.71 (posterior); normal stress: r = 0.13 (upper), 0.83 (posterior)) (Supplementary Figures S2 and S3)﻿.