Fig 1.
(A) Structure of the fish vertebrae. Abbreviations: auc: autocentrum; ns: neural spine; na: neural arch; vb: vertebral body; ha: hemal arch; hs: hemal spine. (B–G) Volume-rendered micro-computed tomography (micro-CT) images of left lateral views of the caudal vertebral bodies of (B) Pagrus major, (C) Acropoma hanedai, (D) Zenopsis nebulosa, (E) Muraenesox cinereus, (F) Scarus forsteni, and (G) Macroramphosus sagifue. (B’–G’) Schematic illustrations of the morphological features of (B–G) that displays (B) a single thick longitudinal plate-like ridge (thick trabecula), (C) a single thin longitudinal plate-like ridge, (D) two longitudinal plate-like ridges, (E) a transverse plate-like ridge, (F) hump-like structures rising on the edge of the vertebral body, (G) tarp-like triangle ridges extending from the center to the edge of the vertebral body. (B” and D”) Transverse sections at the midpoint of the vertebral bodies of (B”) P. major and (D”) Z. nebulosa. These images indicate that the lateral ridges extend from the vertebral body center. Scale bars: (B–F) 1 mm, and (G) 500 μm. In (B” and D”), the interval between scale markers is 1 mm.
Fig 2.
(A) Transparent view of entire structure. The light gray domain corresponds to the autocentrum. (B) Quarter cross-section of analysis domain on zx plane (dark gray plane in (A)). (C) Entire structure. The yellow-colored area is the design domain. (D) Transverse section at midpoint of analysis domain. The light gray wedge-shaped regions represent the vertebral arches. The geometric parameters are described in Table 1.
Table 1.
Parameter settings to define geometry of analysis domain in Figs 2–6.
Fig 3.
Optimization for compressive loads to autocentrum.
The uppermost images are diagrams of the loads. The arrows indicate the load directions. The blue-colored area indicates where the load was applied. The images in the second row from the top show diagrams of the boundary condition; the boundary condition is the same in the two different compressive load cases. We imposed the symmetric boundary condition u ⋅ n = 0 to the pink-colored area on the xy, yz, and zx planes and used an eighth model (left). The middle and right images in this row indicate the areas to which the symmetric boundary condition was imposed on the yz and zx plane, respectively. The area of the symmetric boundary condition on the xy plane is similar to that on the zx plane. The images in the third, fourth, fifth rows are the entire structures, transverse sections, and top views of the optimization results. The numbers in the leftmost column show the volume fraction.
Fig 4.
Optimization for bending loads to autocentrum.
The arrangement and color scheme of the images are the same as those described in Fig 3. The boundary condition is the same in the three different bending load cases. We imposed the symmetric boundary condition u ⋅ n = 0 on the xy and yz planes (left image in the second row from the top) and used a quarter model. The middle and right images in the second row from the top are the areas to which the symmetric boundary condition was imposed on the yz and xy plane, respectively. The magnified image of the center on the yz plane is shown. We imposed the displacement constraint uy = 0 to the point (0, CR, 0) (red dot) when the bending loads were applied to half of the concave surface of the hourglass-shaped domain in y ≤ 0 (the blue-colored area). When we applied the bending loads to half of the area in y ≥ 0, we imposed the same displacement constraint to the point (0, −CR, 0). In optimization for the diagonal bending loads, the results are displayed for w = 0.8.
Fig 5.
Optimization for shear and torsional loads to autocentrum.
The arrangement and color scheme of the images are the same as those described in Fig 3. For the shear loads, we imposed the symmetric boundary condition u ⋅ n = 0 on the xy plane (left image in the second row from the top) and used a half model. The right diagram is the area to which the symmetric boundary condition is imposed on the xy plane. Magnified images of the center of the analysis domain are shown. We imposed two displacement constraints: ux = 0 to the points (0, −CR, 0) and (0, CR, 0) (the red dots), and uy = 0 to the edge y2 + z2 = CR2 (z ≥ 0, the red line). For the torsional loads, we imposed the displacement constraints u = 0 to the edge y2 + z2 = CR2 (the red line).
Fig 6.
Optimization for tensile loads to vertebral arches.
The arrangement and color scheme of the images are the same as those described in Fig 3. For the tensile load in left–right axis direction, we imposed the symmetric boundary condition u ⋅ n = 0 to the pink-colored areas on the xy, yz, and zx planes and used an eighth model. For the tensile load in dorsal–ventral axis direction, we imposed the symmetric boundary condition to the pink-colored areas on the xy and zx planes and used a quarter model. We also imposed the displacement constraint ux = 0 to the edge y2 + z2 = CR2 (y ≥ 0 and z ≥ 0, the red line). The bottom right of the second row shows the magnified image of the center of the analysis domain.
Fig 7.
Dependence of lateral structure of vertebral bodies on analysis domain geometry.
(A and B) Adjusting the gap width of vertebral arches η. The load case was (A) the bending loads to the autocentrum surface in the horizontal direction and (B) the tensile load to the vertebral arches in the left–right axis direction. The optimization results are displayed every five degrees. In (A), the transverse sections and top views are also displayed. (C) Adjusting the vertebral body lengths in the cranial–caudal direction. The value of θ is displayed. The volume fraction Vf is 0.33 in (A), 0.3 in (B), and 0.4 in (C).