Kenzen Takeuchi

A SMOOTHING METHOD FOR SHAPE OPTIMIZATION: TRACTION METHOD USING THE ROBIN CONDITION

This paper presents an improved version of the traction method that was proposed as a solution to shape optimization problems of domain boundaries in which boundary value problems of partial differential equations are defined. The principle of the traction method is presented based on the theory of the gradient method in Hilbert space. Based on this principle, a new method is proposed by selecting another bounded coercive bilinear form from the previous method. The proposed method obtains domain variation with a solution to a boundary value problem with the Robin condition by using the shape gradient.

Buckling and bone modeling as factors in the development of idiopathic scoliosis.

STUDY DESIGN Computational analysis using the finite-element method was used to examine a possible etiology of idiopathic scoliosis. OBJECTIVES To compare changes in the coronal and the transverse planes of idiopathic thoracic scoliosis with changes produced in a finite-element buckling model, and to investigate the influence of bone modeling on the buckling spine. SUMMARY OF BACKGROUND DATA Although it is now widely accepted that growth is related strongly to the onset and progression of scoliosis, the pathomechanism or etiology of idiopathic scoliosis still is not clear. A previous study showed that a buckling phenomenon caused by anterior spinal overgrowth can produce scoliosis, and that the fourth buckling mode matched the clinical characteristics associated with the thoracic type of idiopathic scoliosis. The fourth buckling mode occurs when the first, second, and third buckling modes are prevented. METHODS The spinal finite-element model used in this study consisted of 68,582 elements and 84,603 nodes. The transverse changes seen in the computed tomography images of 41 patients with idiopathic thoracic scoliosis (apex, T8; average Cobb angle, 52.5 degrees) were compared with those produced in the fourth buckling mode. Bone modeling (bone formation and resorption) was simulated as heat deformation caused by changes in temperature. The bone formation and resorption were simulated, respectively, by positive and negative volume changes in proportion to the stress that occurred in the buckling spine. RESULTS Computed tomography images of scoliosis show that as the scoliosis becomes more severe, the thoracic cage decreases on the convex side of the curve and increases on the concave side. The opposite thoracic cage deformation was obtained in the fourth buckling mode. In patients with scoliosis, the sternum essentially remains in its original position with respect to the vertebrae, but in the linear buckling model, it shifted in the direction of vertebral body rotation. In contrast to clinical data, the incremental deformation resulting from bone formation corrected the original curve, and the thoracic cage distorted. On the other hand, incremental deformation resulting from bone resorption worsened the original curve, and the thoracic cage distorted in a manner similar to that described by the clinical data. CONCLUSIONS This computational investigation suggests that scoliotic changes in the spinal column triggered by the buckling phenomenon are counteracted by bone formation, but worsened by bone resorption. The authors hypothesized that scoliosis progressed with resorption of loaded bone. However, it is unclear whether this hypothesis applies to a living body in practice because of the effects from additional factors.