Bone defects are one of the major challenges in orthopedics, with approximately 2.2 million bone graft procedures performed worldwide each year and the annual cost of these procedures approaching $2.5 billion [1,2]. Tissue transplants have been used in humans for tissue repair since at least 1660 [3]. Allogeneic bone grafting is considered an effective option for bone repair, but the problem of allograft immune rejection seriously affects its use in clinical practice. However, the widely used artificial bone substitutes sometimes produce foreign body reactions due to the mismatch in biocompatibility and mechanical properties and the inability to participate in normal metabolic activities [4]. For this reason, new methods and techniques are explored to solve the challenges of bone defect treatment.

Tissue engineering is a new discipline that has emerged in recent years with the development of modern biomaterials technology and cellular biotechnology and other emerging technologies to develop biological substitutes for repairing and maintaining or promoting functional and morphological recovery of tissues or organs after injury. Tissue engineering research consists of four main elements: signaling molecules, target cells that respond to the regulation of the signaling molecules, scaffold materials, and recipient beds with good blood supplies [5]. Bone tissue scaffolds not only provide a three-dimensional environment for cell growth and metabolism but also play a supporting role. Thus, how to construct a scaffold that meets the requirements is one of the priorities of bone tissue scaffold research. The ideal bone scaffold should have properties such as bioactivity, biodegradability, biocompatibility, mechanical support, good porosity, and the ability to deliver material, and the scaffold gradually degrades as new bone tissue proliferates and grows until it completely replaces the scaffold and grows into new bone [6,7,8,9,10]. However, the traditional methods for producing bone tissue scaffolds, such as particle elution, electrostatic spinning, phase separation/freeze-drying, and gas forming, cannot precisely control the size of the scaffold pores, and the connectivity of the pores cannot be guaranteed, which cannot simulate the anisotropy of natural bone [11]. The structural shape also cannot match the anatomical morphology of the bone defect site, and the preparation of personalized bone tissue scaffolds cannot be realized [12,13].

Bone tissue engineering has benefited from the development of additive manufacturing technologies in the past few years. Scaffold geometric parameters such as porosity, pore size, shape, and interconnectivity can be precisely controlled by additive manufacturing techniques. Tissue scaffold structures can be fabricated to meet functional requirements. The additive manufacturing techniques that can produce tissue scaffolds include selective laser sintering (SLS), selective laser melting (SLM), electron beam melting (EBM), and binder jetting (BJ) [14].

With the increasing maturity of additive manufacturing technology and computer-aided design (CAD) technology, it has become possible to achieve the design and fabrication of complex controllable porous scaffolds. To better mimic the function of natural bone structures, various methods have been proposed for the design of bone tissue scaffolds. 

Bone is a three-dimensional non-homogeneous structure with complex characteristics ranging from macroscopic to the nanoscale. A sound design approach is needed that combines structural stiffness with fluid permeability so that the scaffold is both permeable enough to transport nutrients and rigid enough to resist physical loading. Completely solid metals are not compatible because they are impermeable, and the Young’s modulus of solid metals is much higher than the bone modulus of the human body. Because of the mismatch in stiffness, a stress shielding phenomenon will occur, resulting in scaffold failure. Recently, porous structures have been introduced into orthopedic surgery to replace damaged bone tissues. If the structure of porous metals can be digitally designed and fabricated with advanced manufacturing techniques, they can be designed to replicate the properties of bones [17].

It is almost impossible to analyze quantitatively the properties of conventional porous scaffolds because they are composed of a large number of randomly shaped pores. In order to obtain a simplified model, researchers assume that the scaffold is composed of periodically repeating unit cells. CAD, image-based design, and implied surfaces are the conventional design strategies for typical cyclical scaffolds. The CAD-based approach is the most commonly used method in scaffold design, mainly using various CAD tools such as Unigraphics NX (UG),Computer Aided Tri-Dimensional Interface Application (CATIA), and Pro/ENGINEER (Pro/E) [18,19]. To simplify the design process based on CAD, several specialized design software packages have been developed containing libraries of widely used construction units. Based on the scaffolding libraries, the Computer Aided System for Organizing Scaffolds (CASTS) was developed. The aim was to achieve effective automation of the entire design process for the desired topology [20,21]. Shape functions are used to construct porous supports with implicit functional surfaces or irregular polygon models in mathematical modeling, breaking through the geometric limitations of conventional porous units. Although scaffolds with ideal stiffnesses and permeability can be obtained using these methods, these methods require extensive experiments to achieve the desired performance. Moreover, the final results obtained may not be optimal [22].

Currently, there are numerous types of topology optimization methods. In the case of the SIMP method, it is a material interpolation model that allows for the existence of intermediate relative densities (between 0 and 1), penalizes the material density, and filters low-density cells to obtain accurate topological results. This method, which is of great power, can design complex structures with multiscale features [23].

The SIMP method was used to design the porous structure using the customized morphology and mechanical properties of trabecular bone to design a three-dimensional structure with gradient porosity similar to the pore structure [24]. In order to obtain the desired porosity and elastic properties, a homogenization-based algorithm was used to design a three-dimensional bone scaffold [25]. The authors also demonstrated that the method can produce a porous structure that matches the anisotropic stiffness of human trabecular bone using recognized biomaterials. Porous structures with maximum permeability have been designed by this method [26]. Researchers also optimally designed multifunctional porous material microstructures for two competing properties, namely, stiffness and fluid permeability [27]. The topological optimization technique was used to minimize the difference between the effective elastic tensor of the optimized scaffold and the elastic tensor of natural bone. By comparison, it was concluded that bone remodeling was optimal when the elastic tensor of the bone scaffold was slightly higher than that of natural bone [28]. Despite the advantages of the SIMP method, the resulting optimized structures generally suffer from numerical instabilities such as tessellation, grid dependence, and grayscale cells. Moreover, there is no way for pore connectivity to be ensured by the material model in the microstructure design, and additional non-physical constraints are needed during the optimizations [29].

Voronoi tessellation method (VTM) is a way of modeling irregular open-hole structures that can be used to delineate spatial regions [30].

In previous studies, a method that allows for the design of interconnected porous lattices was proposed, which can mimic specific tissue characteristics to achieve bone regeneration scaffolds [31]. The method combined the anatomical shape of the defect by controlling the porosity and pore size of the scaffold. The advantage of this method is to provide geometrical heterogeneity, thus resulting in a very biomimetic shape.

A parametric design method for lattice porous structures was proposed based on the design characteristics of Voronoi structures [32]. Deviations in model porosity and surface area are ensured because the uniform distribution of seed points has a high stability in the lattice cells. By this method, not only lattice structures with uniformly fractionated or graded distribution of porosity can be generated, but also lattice structures with customized porosity according to each cell can be generated. A VTM-based structural design method was proposed in order that the dominant elastic modulus of the scaffold could be controlled, and the stress shielding between the scaffold and the bone could be reduced [33]. A gradient scaffold suitable for the natural bone modulus can be obtained by this method, which also improves the stress shielding. It was found that the stochastic structure can be defined independently by nodal connectivity Z, strut density d, and strut thickness t during the design phase. The relative density, stiffness, and ultimate strength of the structure can also be predicted based on the parameters [34]. The advantage of the stochastic structure is that the single integrated model can define the structure to achieve a broad range of design requirements, even as a gradient of properties within the same component.

The relationship between porosity or apparent elastic modulus and compressive strength of irregular porous structures cannot be simply generalized, as an increase in one parameter leads to a decrease in the other parameter. A more complex relationship may exist based on the complex irregular porous structure of VTM and needs to be further investigated [35].

Machine learning (ML) has proven to be a valuable method for research in various fields. It is used to discern patterns from complex data sets and is a branch of artificial intelligence. ML algorithms are used in areas such as image and speech recognition, spam detection, and drug discovery [36]. The properties of materials or structures can already be predicted by machine learning models, and new materials with the desired properties can also be designed.

Machine learning techniques (MLTs) were combined with parametric finite element analysis (FEA) to further optimize the geometry of the short-stemmed hip scaffold to reduce proximal femoral stress shielding [37]. The minimization algorithm was used to obtain the optimal geometry, which allows unseen values of selected parameters of the hip brace geometry to be explored. The combination of FEA, MLT, and search pattern optimization algorithms can significantly reduce the computational cost [37]. An efficient method for the design optimization of scaffolds in biological tissues was proposed, which reduced computational time [38]. The optimization problem for the design of the geometry of titanium scaffolds was formulated by introducing a probabilistic model of mesoscale cortical bone. With this advanced algorithm, this very difficult constrained nonconvex optimization problem can be solved in the presence of uncertainty in biomechanics. A new method for designing layered materials using machine learning was proposed [39]. A database of hundreds of thousands of structures from FEA was used for training, along with a self-learning algorithm for discovering high-performance materials in which inferior designs are eliminated to obtain superior candidates. It further demonstrated that coarse graining can be replaced by machine learning, which means that materials can be analyzed and designed without the use of complete microstructural data. New material designs can be discovered and fabricated by this new paradigm for intelligent additive manufacturing with several orders of magnitude improvement in computational efficiency over conventional methods. A Generative Adversarial Network (GAN) model was proposed to learn the Inverse Homogenization (IH) mapping from attributes to cell shapes that can be used to optimize functionally graded cell structures [40]. Machine learning algorithms were used to predict the most suitable polymer/blend for cartilage replacement, using as input a range of tensile modulus, elongation at break, and tensile strength of natural cartilage [41].

However, as the number of parameters increases, the time required for MLT increases exponentially. As a data-driven model, IH-GAN cannot generate the shape and properties of cells outside of a given training data distribution. This may limit the performance of the optimized cell structure [40].

遗传算法因其高效率而广泛用于许多结构优化设计[42]。因此，在当前的许多研究工作中，遗传算法被用来获得最佳的支架结构。例如，开发了一种新的计算机方法，该方法结合了FEA和GA来设计支架，方法是选择支架纤维直径和纤维间距，以显示每个降解阶段所需的刚度[43]。采用克里金法（KRG）方法，利用非支配排序遗传算法II（NSGA-II）进行多目标优化，得到具有最佳抗压性的分层三维多孔（H3DP）结构的优化设计[44]。基于椭圆体建模的多约束背包问题的结构，采用混合遗传算法求解了多孔结构的逆模型。生成的仿生骨支架具有良好的生物活性、更好的机械性能和均匀的降解速率[45]。提出了一种超材料逆向工程的数值方法，该方法将渐近均质化方案与遗传算法相结合，该算法可以使用目标顺应性张量中包含的完整参数集确定最佳内部材料模式[46]。通过集成有限元分析和多目标GA，为无水泥股骨支架开发了一种新颖的多目标定制形状优化方案[47]。提出了一个优化框架，以全自动方式生成现成的术前计划解决方案[48]。它基于一种遗传算法，能够解决具有非线性约束的多目标优化问题。基于GA的搜索通过最小化反向传播神经网络（BPNN）预测的微动来优化支架。与初始设计相比，基于GA搜索获得的最佳MMG提供了更好的初级稳定性[49]。

虽然遗传算法是一个非常强大的工具，但在遗传算法优化脚手架的过程中，需要评估每一代的约束和适应度函数是耗时的，特别是在计算结构复杂的支架时。

上述四种方法的优缺点比较如表1所示。除此之外，还有其他方法可以优化骨支架。水平集算法以跟踪相边界为中心，从而有效地描述平滑边界以控制拓扑变化[50]。对基于水平集的拓扑优化进行了更系统、更全面的研究[51]。为了获得具有最大磁导率的材料，针对由纳维-斯托克斯和麦克斯韦方程组控制的周期性材料设计问题，开发了一种可变水平集技术[52]。双向进化结构优化（BESO）方法允许在优化过程中同时添加和去除材料[53]。本文仅综述了部分骨支架优化方法，但并未介绍所有方法。也没有提到骨和骨植入物方程的力学分析和优化。