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Mathematical modeling for the formation of teleost vertebrae based on comparative morphological observation

坂下, 美咲 大阪大学 DOI:10.18910/82342

2021.03.24

概要

Elucidating the mechanism by which the shape of bones is formed is essential for understanding the development of vertebrates because the body shape of vertebrates is determined by the skeleton that is composed of differently shaped bones. To support the body of vertebrates, bones must withstand various external loads, such as those imposed by gravity and muscle tension. Previous cytological studies have reported that the cellular activities involved in bone formation vary in response to external loads, supporting the paradigm that bones adapt these shapes to external loads (Wolff’s law). In fact, by simulating these cellular activities through mathematical models, the internal trabecular structure of bones can be successfully reproduced, thereby demonstrating that the internal structure of bones adapts to external loads. In contrast, there have been few attempts to simulate the external structure of bones, which determines vertebrate morphology. However, Wolff’s law may be applied to the external shape of bones because cells of the same type form both the internal and external structures of bones. In this study, I aimed to elucidate the formation of the external shape of bones based on Wolff’s law. To simulate the external shape of bones, understanding the morphological characteristics of bones is needed. Therefore, I performed the comparative morphological observation and the computer simulation of teleost vertebrae.

In teleosts, the internal structure of the vertebrae is invariable, exhibiting an hourglass shape,whereas the lateral structure supporting the internal structure differs among species. By examining the vertebrae of 32 species from 10 orders of Teleostei using high-resolution micro-CT scans, I found considerable variation in the lateral structure of teleost vertebrae. Furthermore, I discovered two structural characteristics that are shared among most of the examined species. One was the sheet- like trabeculae that extend radially from the center of the vertebral body with a constant thickness. The other was the presence of hollow spaces on the internal parts of the lateral structures. The combination of different arrangements of sheet-like trabeculae and internal hollow spaces formed different shapes of the lateral structure of the vertebral body. This finding shows the possibility that the shape variation of teleost vertebrae can be explained using a single mathematical model. Next, I constructed a three-dimensional topology optimization model of teleost vertebrae.

Topology optimization is a mathematical method that generates the optimal shape according to external loads, that algorithm is similar to the cellular activities in bone formation. To reproduce the teleost vertebrae, I applied different external loads to the internal hourglass-shaped part. The simulations produced a variety of three-dimensional structures, some of which exhibited several structural features similar to the teleost vertebrae. Also, by adjusting the geometric parameters,such as the width of the hourglass shape, I reproduced the variation in the teleost vertebrae shapes. These results indicate that the simulation using topology optimization can successfully reproduce the external shape of teleost vertebrae, suggesting that teleost vertebrae adapt these shapes to external loads.

The results of this study show the possibility that the external shape of bones is determined by external loads. According to this finding, both internal and external structures can be explained based on Wolff’s law. Moreover, as the external shape of bones varies among the vertebrate species, the mathematical model of this study can be utilized to understand how these different shapes adapt to external loads. Therefore, this study provides a new perspective for an in-depth understanding of bone morphology.

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