Table 1 shows the results of six simulations, each of which measures the strategic resistance of a voting rule as the share of trials in which (beginning with sincere voting) it is impossible for any coalition of voters to secure a preferred outcome by voting insincerely. All simulations assume a spatial model with three candidates, and employ 100,000 trials; but they differ in the number of issue dimensions (ranging from one to four) and voters (ranging from nine to 999). The top panel reports these parameters as well as the share of trials in which a Condorcet winner exists, while the table’s main body reports strategic resistance scores for 15 voting rules (counting Dodgson-Hare only once). Rules are arranged in descending order of their average strategic resistance score.
In all six simulations, Dodgson-Hare performs strictly better than all other voting rules considered here, provided the assumption that the cooperative victim strategy (denoted as CVS) is not feasible. On the other hand, if it is feasible, Dodgson-Hare receives low-to-middling marks for strategic resistance: higher than Borda, but lower than plurality. Hence, Dodgson-Hare may be said to display extraordinarily frequent strategic resistance in this model if and only if one agrees that the cooperative victim strategy is unrealistic.The FEA of the Nuss procedure proceeds in the following order: selecting various displacement controls of the metal bar, fabricating 3D models of the chest wall and metal bar, applying material properties to the detailed tissues of the chest wall and metal bar, and setting boundary conditions. To describe the movement of the actual chest wall and metal bar as closely as possible, emphasis was placed on the conditions for the anterior translational, rotational, and equilibrium displacement of the metal bar. This study was conducted in accordance with the guidelines and regulations. The patients’ parents provided written consent for their children to participate in the study and allowed access to the patients’ medical images. This study was approved by the Institutional Review Board (IRB) of Pusan National University Hospital (IRB No. 1910-017-084). The patients’ parents provided written informed consent for their children to participate in the study and allowed access to the patients’ medical images.
Finite element model (FEM)
To fabricate a 3D model of the chest wall, a medical image of the patient must be obtained. In patients with PEX, a typical type of symmetry was selected to reduce the error according to the unusual asymmetric type. A 15-year-old male patient visited the Pusan National University Hospital and underwent the Nuss procedure. The tissues of the chest wall were separated and extracted from the preoperative CT images. Preoperative CT images taken at 2-mm intervals were selected for the convenience of 3D model fabrication. Among the tissues that comprise the chest wall, the sternum, ribs, costal cartilage, and intercostal muscle (ICM) have a mechanical effect on displacement control. In the CT images, tissues other than the ICM that could not be discerned were selected and extracted. The CT image was imported into Mimics 23.0 (Materialise, Leuven, Belgium), and each tissue was separated, masked, and three-dimensionalized. Early 3D models had rough surfaces and irregular holes; in this study, the surface was smoothed and filled because a rough surface lowers the convergence and accuracy of FEA.
The reason for considering the ICM was to simulate the interaction between the ribs more accurately. However, as the ICM could not be extracted from the CT images, it was assumed that the ICM existed based on the central location of the cross-section of the upper and lower ribs and costal cartilage. In the CT images of several ribs and costal cartilage, lines connecting the center of each section were drawn. Two central curves that matched the curvature of the ribs, and a square-shaped curved surface on both sides were constructed. This curved surface was defined as the ICM by giving it a certain thickness. The thickness of the ICM is approximately 1.97–4.85 mm according to the position difference of the anterior-posterior (AP) and superior-inferior (SI)24. Clinically, in this study, it was determined that the difference in muscle movement was small, and the average value was set equally to 3.39 mm.
A 3D model similar to the actual metal bar was produced for comparison with the results of the Nuss procedure. The 3D model of the metal bar was designed based on the height, width, thickness, and angle of the metal bar applied in the postoperative medical image (12.7 × 279.4 × 2.8 mm)25. The 3D model was designed using Inventor 2021 (Autodesk, Mill Valley, USA), a software suitable for the 3D design of mechanical models. The designed metal bar was inserted into the chest wall model containing the ICM. The insertion point of the metal bar model was designated as the height at which the chest depression started and at the 4th intercostal section, which is the insertion position of the metal bar applied to the actual patient. The insertion shape of the metal bar will differ depending on the displacement control method of the metal bar. In the anterior translation, the concave metal bar was inserted into the chest wall because the concave metal bar was moved linear motion without rotation. The center of the metal bar was placed in contact with the sternum in the thoracic cage. In the rotation, the concave metal bar was rotated within the chest wall and displaced in a convex shape, and the metal bar was placed in the chest wall in a concave shape at the position of the metal bar insertion described above. As the ICM of the chest wall was inserted to control rib movement, we decided to exclude the relationship between the ICM and the metal bar. Accordingly, the ICM at the point where the metal bar was inserted into the chest wall was removed by the radius of contact when the metal bar rotated.
The constructed 3D model of the chest wall and metal bar must be converted to FEM for analysis. The ICM was a surface type, and the remainder of the constitution was formed as a body type in the 3D model. Accordingly, the ICM was converted into an FEM consisting of two-dimensional shell elements, and the remainder was converted into a 3D tetrahedral or hexahedral element. Because the metal bar is a mechanical model based on exact dimensions, a hexahedral element was applied. As the sternum, ribs, and costal cartilage had a complex and irregular shape, a tetrahedral element was applied. To understand the impact of ICM, the results were derived by the model with and without ICM, respectively. The numbers of elements and nodes in the model without ICM were 162,362 and 307,781, respectively, and those in the model with ICM were 176,730 and 371,185, respectively (Table 1). In addition, the mesh sensitivity for this chest wall model has been established previously and is considered an appropriate form of FEM25.
Table 1
Element type and number of elements and nodes for each tissue in the finite element (FE) model of the chest wall
Tissues
|
Element type
|
Number of elements
|
Number of nodes
|
Sternum
|
Tetrahedron
|
4,907
|
8,333
|
Ribs
|
48,137
|
95,595
|
Costal cartilage
|
94,594
|
283,097
|
Intercostal muscle (ICM)
|
Square
|
28,760
|
92,398
|
Metal bar (Titanium)
|
Hexahedron
|
332
|
2,762
|
Total
|
|
176,730
|
371,185
|
Material properties
Material properties should be accurately applied to understand the exact behavior of the constructed FEM. The model in this study consists of a chest wall and a metal bar, and the chest wall is made up of several tissues, such as the sternum, ribs, costal cartilage, and ICM. Additionally, the internal components of the sternum and ribs are divided into cortical and cancellous bones, with somewhat different material properties. The chest wall is deformed due to the Nuss procedure but is not fractured or excessively damaged. In addition, the metal bar was not deformed owing to its high elastic modulus. Accordingly, only the linear section was considered for determining the material properties of each tissue. The elastic modulus and Poisson’s ratio of each tissue were applied as described previously (Table 2)26-29. It was also assumed that all the tissues were homogeneous and isotropic.
Table 2
Elastic material properties of each tissue in the finite element (FE) model
Materials
|
Young’s modulus (MPa)
|
Poisson’s ratio
|
Sternum
|
(Cortical bone)
|
11,500
|
0.3
|
(Cancellous bone)
|
40
|
0.45
|
Ribs
|
(Cortical bone)
|
5,000
|
0.3
|
(Cancellous bone)
|
40
|
0.45
|
Costal cartilage
|
37.5
|
0.3
|
Intercostal muscle (ICM)
|
10.3
|
0.3
|
Metal bar (Titanium)
|
200,000
|
0.29
|
As the sternum and ribs are divided into cortical and cancellous bones, this boundary should be clearly applied to the FEM. Because the cortical bone had a constant thickness surrounding the cancellous bone, it was formed on the sternum and ribs. The thickness of the sternal cortical bone was measured on the CT image of the patient, but the boundary of the costal cortical bone was unclear. Accordingly, the ribs were referred to in a previous study. The average thickness of the sternal cortical bone was 2.1 mm, and there was a slight difference in the thickness of costal cortical bone depending on the location. Therefore, a value between 0.685 and 0.725 mm was applied depending on the location of the ribs30.
Boundary conditions
In the Nuss procedure, the main boundary conditions affecting the chest wall are set by mechanical judgment by referring to the actual deformity of the chest wall. When the sternum is anteriorly translated by a metal bar, the manubrium of the sternum shows a relatively small amount of movement. The manubrium is restricted by the sternoclavicular joint where the manubrium and clavicle are in contact. Therefore, to derive the correct amount of anterior translation, the mechanical behavior of the sternoclavicular joint was analyzed, and controlled conditions were applied to the manubrium in contact with the clavicle25. When the metal bar is displaced, the spine and the most posterior part of the chest wall should be fixed to allow the sternum to translate. Accordingly, a fixed condition without movement was set at the costovertebral joint, where the vertebra and ribs were in contact. Separation between the tissues does not occur when the tissues of the chest wall are deformed. Therefore, the contact conditions between all the tissues were set as the bonded conditions. The metal bar was in contact with the sternum, causing displacement. In the anterior translation, the sternum and metal bar were in contact with each other and anteriorly translated simultaneously. Accordingly, the relationship between the two bodies was set as the bonded condition. In contrast, rotation is a movement in which the anterior translates while sweeping the sternum. At this time, a friction condition occurs between the two bodies and the friction coefficient should be determined. Several soft tissues exist in the anterior mediastinum and serve to significantly lower the coefficient of friction. In this study, because the coefficient of friction was very low, it was determined that this friction force had no significant effect on the mechanical behavior of the chest wall. Accordingly, a frictionless condition was set between the metal bar and sternum in the rotation.
By applying the established FEM, material properties, and boundary conditions, the analysis was performed according to the anterior translation, rotation-equilibrium, and rotation-equilibrium with ICM scenarios. The analysis was determined by structural analysis of large deformations by referring to the boundary conditions and degree of deformation of the chest wall. In this study, because the two displacements were sequentially performed in the equilibrium scenario, a multistep analysis was applied. The rotational displacement was described as 1st step, the equilibrium displacement as the 2nd step, and the confirmation of convergence of the equilibrium displacement as the 3rd step in scenario selection. These steps were performed sequentially, and the computational time was set to 1s per step for a total of 3s. ANSYS 2019 R1 (Ansys Inc., Canonsburg, USA) was used for the FEA, and the computer power was set to be suitable for various analysis conditions. The computer’s processor is an Intel(R) Xeon(R) CPU 2.40 GHz applied with 28 cores, and the RAM capacity is 64 GB.
Physical quantities for validation of the virtual Nuss procedure
As a result of the FEA, the accuracy of each scenario was analyzed by comparing it with the results of patients who underwent the Nuss procedure. For comparison, each result was derived by selecting the anterior sternal translation and Haller index (HI) after the Nuss procedure. In addition, the equivalent stresses of the sternum and metal bars, which are the subjects of the behavior of the chest wall, were derived for the mechanical comparison analysis of each scenario. Finally, for quantitative comparison of each derived value, the relative difference between them was numerically expressed. The computational times for each scenario were as follows: Anterior translation: 0.05 h, Rotation-equilibrium: 10.90 h, and Rotation-equilibrium with ICM: 5.08 h.
After the Nuss procedure, the sternum was anteriorly translated, and the chest wall was restored close to the normal state. At this time, given that the sternum should be translated in an appropriate amount, the amount of anterior translation at the point where the metal bar and sternum are in contact is important. Therefore, the result of the actual Nuss procedure and the result of the scenarios in this study were compared to determine the amount of anterior sternal translation of the sternum. The distance between the same points preoperatively and postoperatively was measured based on the position at which the metal bar and sternum were in contact. The accuracy of each scenario was quantified by numerically expressing the relative difference between the measured value for each scenario and the measured value after the Nuss procedure.
In the case of chest wall deformities, such as PEX or pectus carinatum, the HI was used to numerically determine normality. The left/right lengths and anteroposterior length of the inner chest wall were measured in the section with the most severe deformation of the chest wall. The value obtained by dividing the left/right length by the anteroposterior length is the HI. The closer the HI after the Nuss procedure to the HI of the normal chest wall, the more accurate the result of the surgery31. The lengths of the chest walls were measured from the cross-sectional image of the actual patient after the Nuss procedure and the results of each scenario. The HIs were calculated using these values, and the accuracy was quantified by numerically expressing the relative difference between the calculated value of the actual Nuss procedure and the scenarios.