Department of Mechanical Engineering, Indian Institute of Technology Ropar, Satish Dhawan Block, Room No. 308, Rupnagar, Punjab, 140001, India.
Biomech Model Mechanobiol. 2019 Oct;18(5):1335-1349. doi: 10.1007/s10237-019-01147-z. Epub 2019 Apr 5.
While physiological loading on lower long bones changes during bone development, the bone cross section either remains circular or slowly changes from nearly circular to other shapes such as oval and roughly triangular. Bone is said to be an optimal structure, where strength is maximized using the optimal distribution of bone mass (also called Wolff's law). One of the most appropriate mathematical validations of this law would be a structural optimization-based formulation where total strain energy is minimized against a mass and a space constraint. Assuming that the change in cross section during bone development and homeostasis after adulthood is direct result of the change in physiological loading, this work investigates what optimization problem formulation (collectively, design variables, objective function, constraints, loading conditions, etc.) results in mathematically optimal solutions that resemble bones under actual physiological loading. For this purpose, an advanced structural optimization-based computational model for cortical bone development and defect repair is presented. In the optimization problem, overall bone stiffness is maximized first against a mass constraint, and then also against a polar first moment of area constraint that simultaneously constrains both mass and space. The investigation is completed in two stages. The first stage is developmental stage when physiological loading on lower long bones (tibia) is a random combination of axial, bending and torsion. The topology optimization applied to this case with the area moment constraint results into circular and elliptical cross sections similar to that found in growing mouse or human. The second investigation stage is bone homeostasis reached in adulthood when the physiological loading has a fixed pattern. A drill hole defect is applied to the adult mouse bone, which would disrupt the homeostasis. The optimization applied after the defect interestingly brings the damaged section back to the original intact geometry. The results, however, show that cortical bone geometry is optimal for the physiological loading only when there is also a constraint on polar moment of area. Further numerical experiments show that application of torsion along with the gait-analysis-based physiological loading improves the results, which seems to indicate that the cortical bone geometry is optimal for some amount of torsion in addition to the gait-based physiological loading. This work has a potential to be extended to bone growth/development models and fracture healing models, where topology optimization and polar moment of area constraint have not been introduced earlier.
在骨骼发育过程中，下肢长骨的生理负荷会发生变化，但骨横截面要么保持圆形，要么从近乎圆形缓慢变为椭圆形和大致三角形等其他形状。骨骼被认为是一种最优结构，它通过最佳分布骨量（也称为沃尔夫定律）来实现强度最大化。该定律最恰当的数学验证之一是基于结构优化的公式，其中总应变能最小化，同时受到质量和空间的约束。假设骨骼发育过程中的横截面变化和成年后的骨稳态变化是生理负荷变化的直接结果，那么这项工作就会研究哪种优化问题的公式（统称为设计变量、目标函数、约束条件、加载条件等）可以得出与实际生理负荷下的骨骼相似的数学最优解。为此，提出了一种用于皮质骨发育和缺损修复的高级基于结构优化的计算模型。在优化问题中，首先针对质量约束最大化整体骨刚度，然后针对同时约束质量和空间的极惯性矩约束最大化整体骨刚度。研究分两个阶段完成。第一阶段是骨骼发育阶段，此时下肢长骨（胫骨）的生理负荷是轴向、弯曲和扭转的随机组合。对这种情况应用拓扑优化并施加面积矩约束，得到的横截面为圆形和椭圆形，与生长中的老鼠或人类的骨骼相似。第二个研究阶段是成年后的骨稳态阶段，此时生理负荷具有固定模式。在成年老鼠的骨骼上施加一个钻孔缺陷，这将破坏骨稳态。有趣的是，缺陷后的优化将受损部分恢复到原始完整的几何形状。然而，结果表明，只有在极惯性矩约束的情况下，皮质骨几何形状才是生理负荷的最优选择。进一步的数值实验表明，在步态分析生理负荷的基础上施加扭转，可以改善结果，这似乎表明除了基于步态的生理负荷外，皮质骨几何形状还对一定量的扭转是最优的。这项工作有可能扩展到骨骼生长/发育模型和骨折愈合模型，因为之前没有引入拓扑优化和极惯性矩约束。
