Finite element analysis in implant dentistry: State of the art and future directions

doi:10.1016/j.dental.2023.04.002

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第39卷，第6期，2023年6月，第539-556页

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摘要

目的

讨论有限元（FE）建模在种植牙科中的研究现状，突出其主要特点和当前局限性，并提出建议以指导未来的研究方向。

方法

通过 PubMed、Web of Science、Scopus、Science Direct 和 Google Scholar 等数据库，使用特定关键词进行文献检索。在对标题、摘要和全文进行评估后，根据纳入和排除标准筛选出相关文献。本综述共纳入了 147 项研究。

结果

迄今为止，针对骨-牙种植体系统的有限元分析主要是通过分析几种类型的种植体进行的；建模时仅考虑部分骨组织，并假设其为各向同性材料，尽管骨组织实际具有各向异性行为；在多数情况下假设种植体与骨组织完全骨结合；仅考虑作用于种植体的轴向或斜向压力载荷；忽略了肌肉力以及骨重建过程。最后，目前在牙科领域尚无有限元建模的标准化方法。

意义

FE 建模是一种有效的计算工具，可用于研究种植体的长期稳定性。最终目标是将此类技术应用于临床实践，以帮助牙医进行诊断和治疗。为此，未来的研究应深入探讨载荷在微观尺度上对骨-种植体复合体的影响，这是一个尚未充分研究的关键因素。因此，采用一个多尺度模型可能具有重要意义，该模型可通过多个长度尺度整合此类信息。这将有助于获取有关种植体设计、骨应力分布以及骨生长之间关系的信息。最后，为了使 FE 建模能够高度预测种植体的长期稳定性，采用标准化方法将是必要的。

图形摘要

关键词

牙科种植体

有限元建模与分析

骨-种植体界面

骨整合

骨重建

1. 引言

1.1 背景

有限元（FE）方法是一种用于分析和模拟物理现象的数值技术。它通常用于产品设计中，以减少对物理原型和实验的需求，并优化组件以更高效、更具成本效益地创造更优质的产品。FE 方法依赖数学模型来理解和量化各种物理现象，包括结构与流体行为、机械应力、热传导、波传播以及生物细胞的生长 [1]。由于技术的发展，有限元方法已成为一种日益有用的工具，未来也具有巨大的发展潜力：据估计，2022 年全球仿真与分析软件市场规模为 182.5 亿美元，并预计将以 14.92%的复合年增长率增长，到 2027 年达到 366.8 亿美元。

在医学领域，有限元（FE）方法对于研究难以通过体内或体外实验进行分析的复杂生物力学系统特别有用。该方法可用于预测组织在不同刺激条件下（包括健康和病理状态）的机械响应，并评估结构变化 [2]，[3]，[4]。此外，有限元方法也已广泛应用于医疗器械的设计过程中，用于分析新型器械的功能及其可能引发的并发症 [5]。

多年来，人们已对有限元（FE）方法在牙科领域的应用开展了大量研究 [6]。这些研究大多集中于分析种植体在功能负载下的力学行为，旨在评估种植体组件和骨骼在宏观尺度上的应力和应变场 [7]，[8]。然而，机械载荷会在微观尺度上引发适应性变化，影响骨骼的稳态与重建过程。这些变化显著影响骨-种植体界面处的应力和应变场分布 [9]，[10]，[11]。通过数值建模捕捉这些效应可能引发新一轮的多学科方法，旨在理解并控制生物整合，并改善牙科种植体的长期存活率。

1.2 研究目的

本文综述旨在全面概述有限元（FE）方法在种植牙领域的应用。文中讨论了所有主要方面，并提出了克服当前局限性的建议。最后，对未来的展望进行了探讨。

1.3 文献检索

通过 PubMed、Web of Science、Scopus、ScienceDirect 和 Google Scholar 进行了电子检索。目的是研究骨-牙种植体系统的力学响应。使用的关键词包括：“牙种植体”、“有限元分析”、“有限元建模”、“骨-牙种植体界面”、“骨整合”和“骨改建”。纳入标准为：过去二十年内以英文发表的全文论文（包括计算机模拟、体外和体内研究以及各种类型的综述）和会议论文，且研究重点为有限元分析和骨特性。排除标准为：不涉及牙科应用或以非英文发表的文献。根据文章标题和摘要进行初步筛选后，进一步评估全文，以确定符合纳入和排除标准的研究。最终共有 147 项研究被纳入本综述。

1.4. 概述

综述的结构安排如下。第 2 节提供了有限元（FE）建模与分析的批判性和详细概述，用于研究牙科种植体的力学行为。具体而言，对 FE 建模与分析中涉及的所有方面进行了探讨，包括：几何结构（第 2.1 节）、计算离散化（离散化方法）（第 2.2 节）、本构关系（第 2.3 节）、骨重建（骨改建）（第 2.3 节）、骨-种植体界面（第 2.5 节）以及载荷与边界条件（第 2.6 节）。此外，第 2.7 节还概述了主要的有限元分析结果。在第 3 节中，提供了当前关于载荷对牙科种植体稳定性及骨组织响应影响的研究进展。探讨了如何进一步解释这一关键因素以将有限元分析提升到“下一步”。最后，在第 4 节中给出了克服当前有限元建模局限性的建议。

2. 牙种植体-骨系统的有限元建模与分析

以下将详细探讨种植体-骨系统有限元建模与分析的主要步骤，旨在提供对主要方面和主要结果的全面概述，并讨论目前尚未解决的问题。

为便于理解，下文中使用的解剖学和几何术语如图 1 和图 2 所示。具体而言，图 1(a)展示了一部分右侧下颌骨，其中图 1(b)所示的骨段被提取出来。近中、远中、前庭和舌侧方向在图 1 中均有标注。图 2 展示了牙种植体及其组成部分。通常，用于单颗牙修复的牙种植体包括种植体本体（即植入骨内的基台）、上部结构、基台与上部结构之间的连接部分以及义齿冠（见图 2(a)）。连接部分的解剖结构各不相同。其抗旋转装置的几何形状可以是六边形、八边形、三叶形或其他形式。连接方式可以是螺丝固位、锥形配合或粘结固位。根据所选用的制作材料，牙冠可以是整体式的，或者由基底结构和外部上部结构组成（例如金属-陶瓷冠）。牙冠可以通过粘结剂固定在基台上，或者通过穿通螺丝直接与种植体连接。在后者的情况下，牙冠表面设有孔洞，以便接近穿通螺丝。图 2(b) 显示了种植体插入骨段的情况。

2.1 几何结构

生成有限元模型的第一步是几何表示。接下来将详细分析骨组织和种植体的几何结构。

2.1.1 骨几何结构

重建的骨几何结构可分为简化几何结构和基于图像的几何结构。

在简化几何结构的背景下，骨组织已被建模为具有不同形状的二维（2D）或三维（3D）对象，如表 1 [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29]，[30]，[31]，[32]，[33]，[34]，[35]，[36]，[37]，[38]，[39]。

表 1．用于骨组织的简化几何模型的主要特征。NR：未报告；COR：皮质骨；TRAB：松质骨；h：高度；l：近远中方向长度；w：颊舌向宽度；D₁、D₂、D₃ 和 D₄ 表示 Lekholm 与 Zarb[40] 提出的分类，该分类根据皮质骨与松质骨之间的关系定义了四种骨类型。

参考文献	骨几何结构	特征
Chun 等人 [13]	二维矩形	仅限 COR
Empty Cell	20（h） × 16（）mm²
黄等人 [14]	三维块体	仅适用于 D₁ 的 COR；
Empty Cell	10 × 10 × 15 毫米 ³	对于 D₂，1.2–1.6 mm 厚的冠根比（COR）
Empty Cell		内部致密的骨小梁；
Empty Cell		对于 D₃，厚度为 0.6–0.8 mm 的 COR
Empty Cell		内部致密的骨小梁；
Empty Cell		厚度为 0.6–0.8 mm 的 D₄
Empty Cell		内侧松质骨
Mellal 等人 [15]	三维块体	1 毫米厚的冠部修复体（COR）
Empty Cell	20（h） × 10（w） × 30（） mm³	内部区域的 TRAB
林等人 [17]	三维块体	1.5毫米厚的修复体
Empty Cell	24（高）×12（宽）×15（？）毫米³	内侧的 TRAB
北川等人 [18]	三维块体	COR 和 TRAB 建模
Empty Cell		皮质骨厚度：1.0 mm 下颌上缘，
Empty Cell	24.5（h） × 12.5（w） × 10（） mm³	颊侧边缘2.5毫米，舌侧边缘2.0毫米，
Empty Cell		下颌下缘3.0毫米
塞维迈等人 [19]	三维块体	COR 和 TRAB 模型
Empty Cell	24.2（h）毫米，16.3（w）毫米	模拟的 D₁、D₂、D₃和 D₄型骨组织
Heckmann 等人 [20]	三维块体	COR 和 TRAB 建模，
Empty Cell		COR NR 的厚度值
Akca 等人 [21]	三维圆柱体	1 毫米厚的冠部修复体（COR）
Empty Cell	（13毫米高度）	12 毫米高度的 TRAB
Kurniawan 等人 [22]	三维块体	2、1 和 1 毫米的 COR
Empty Cell		对于 D₂、D₃和 D₄型骨组织，分别进行分析；
Empty Cell		内部区域的骨小梁（TRAB）及其密度
Empty Cell		对于 D₂和 D₃型骨较高，而对于 D₄型骨较低
田等人 [23]	三维块体	1.5毫米厚的修复体
Empty Cell	20（h） × 9（w） × 20（） mm³	内侧的 TRAB
Chang 等人 [24]	三维长方体	对于 D₂型骨，采用 2 毫米厚的冠部修复体（COR）
Empty Cell	20（高）×10（宽）×10（长）mm³	内部致密的骨小梁；
Empty Cell		对于 D₄型骨，1 mm 厚的修复体边缘圆角
Empty Cell		内侧松动的 TRAB
吴等人 [25]	三维块体	3 毫米厚的冠部修复体（COR）
Empty Cell	43.5（高）× 30（宽）× 41（长）毫米³	在三维块体的上部
Empty Cell		COR 层下方的 TRAB
上田等人 [26]	三维块体	六个不同厚度值的应力中心
Empty Cell		（范围 0.5–2 mm）
Empty Cell		内侧的 TRAB
Poovarodom 等人 [27]	三维块体	2 毫米厚的冠部修复体（COR）
Empty Cell	12.5（h） × 7.6（w） × 10（）mm³	内侧的 TRAB
Öztürk 等人 [28]	三维块体	1.5毫米厚的修复体
Empty Cell	20（高）× 20（宽）× 20（长）mm³	内侧的 TRAB
达特拉克等人 [29]	三维块体	厚度为 2 毫米的冠部修复体（COR）
Empty Cell	30（h） × 10（w） × 20（）mm³	内侧的 TRAB
沙什等人 [30]	三维块体	厚度为 2 毫米的冠部修复体（COR）
Empty Cell	20（高）×15（宽）×15（长）毫米³	内侧的 TRAB
奥利维拉等人 [31]	三维块体	三种不同厚度值的应力中心
Empty Cell		（0.5、1 和 2 毫米）
Empty Cell		内侧的 TRAB
Empty Cell		密度分为两个水平（高和低）
巴达利亚等人 [32]	三维形状	1.5毫米厚的修复体
Empty Cell		内侧的 TRAB
吴等人 ^[33]	三维块体	2 毫米厚的冠部修复体（COR）
Empty Cell	25（高）× 12（宽）× 10（长）mm³	内侧的 TRAB
帕拉奇尼等人 [34]	三维块体	厚度为 2 毫米的冠部修复体（COR）
Empty Cell	17 mm（h）和 12 mm（w）	内侧的 TRAB
Nokar 等人 [35]	三维块体	厚度为 2 毫米的冠部修复体（COR）
Empty Cell	25（h） × 23（）	内侧的 TRAB
达特拉克等人 [36]	三维块体	COR 和 TRAB 建模
Empty Cell		COR NR 的厚度值
Barbosa 等人 [37]	三维块体	1 毫米厚的冠部修复体（COR）
Empty Cell		内侧的 TRAB
Alemayehu 等人 [38]	三维块体	COR 和 TRAB 建模
Empty Cell		皮质骨非旋转区的厚度值
米洛内等人 [39]	三维块体	COR 和 TRAB 建模
Empty Cell		COR NR 的厚度值

即使采用了简化的形状，几何模型中也考虑了皮质骨和松质骨部分。一般来说，松质骨核心位于内部，周围包裹着厚度在 0.5 mm 至 3 mm 之间的皮质骨壳层。在一些研究中，依据 Lekholm & Zarb 的分类方法，对皮质骨和松质骨的贡献进行了探讨，并据此考察了不同的骨质质量，该分类方法在种植牙领域被广泛采用。这是一种定性分类方法，根据皮质骨与松质骨之间的关系定义了四种骨类型（D₁、D₂、D₃和 D₄）。D₁型骨的特征是均质的致密骨；而 D₂型骨则由致密的皮质骨厚层包围着致密的松质骨核心组成。 D₃ 指围绕致密小梁骨的一薄层皮质骨，而 D₄ 则对应于围绕低密度小梁骨核心的一薄层皮质骨。根据所研究的骨类型，在几何模型中使用了不同的皮质厚度值，如表 1 所列。

由于数字成像技术的发展，在一些工作中，几何形状不是重建简化的几何形状，而是从诊断图像数据中得出，例如计算机 <a id=5>断层扫描（CT），允许更真实的重建 [41]，[42]，[43]， [44]， [45]， [46]， [47]， [48]， [49]， [50]， [51]， [52]， [53]， [54]， [55]， [56]，[57]，[58]，[59]，[60]，[61]，[62]，[63]，[64]。

在简化几何模型和通过 CT 图像重建的几何模型中，骨骼均被模拟为连续介质材料，忽略了骨小梁网络的结构。然而，骨的失效和适应过程是在单个骨小梁水平上发生的。因此，已有研究证实，考虑详细的骨小梁结构是评估骨-种植体界面应力和应变的关键因素，从而可以评价种植体的稳定性和成功率[65]，[66]。骨小梁网络的结构可以通过微 CT 图像获取，从而生成能够反映骨小梁精细结构的有限元模型。Marcian 等人[67]，[68]利用基于微 CT 构建的有限元模型评估了在不同骨整合阶段，不同种植体几何形态下承重所引起的微应变和位移。他们证明了应变分布高度依赖于小梁骨的结构，并指出部分骨整合可能是种植体长期稳定性的一个潜在风险。在 [69]、[70]、[71] 中也展示了在有限元模型中考虑复杂小梁结构的重要性，表明忽略这一因素会显著影响对种植体-骨界面性能仿真的准确性。尽管显微 CT 图像提供了构建复杂小梁骨计算模型的可能性，但考虑到高剂量辐射暴露的问题，无法对活体患者进行显微 CT 扫描，从而限制了有限元模型在临床常规中的应用。因此，将临床环境下获取的诊断图像与去模糊算法 [72]，[73] 相结合，以减少临床图像的部分容积伪影并恢复强度数据，可能有助于检测有限元模型中需包含的骨小梁结构，从而成为解决上述问题的一种方案。这在口腔医学领域需要深入研究，并可能为迄今被低估的复杂骨应变分布模式提供新的认识。

2.1.2 种植体几何结构

牙种植体的几何形状通常来源于由制造商提供的三维计算机辅助设计数据。表 2 综述了文献中已分析的牙种植体情况，包括制造商、种植体名称、组成部分及其几何特征。如表 2 所示，已有研究对来自不同制造商、具有不同几何特征的多种类型种植体进行了分析，体现出显著的多样性。此外，就构成各种植体的组件而言，种植体也不具有同质性。种植体的几何形状是评估骨-种植体界面应力和应变的重要因素。事实上，当周围骨发生改建时，其几何特征可能影响种植体的性能，从而加剧骨吸收 [69]。从表 2 可以看出，基台与种植体本体之间的连接类型是研究力学响应时需要考虑的另一个重要因素。这种连接方式会影响种植体本体-基台以及骨-种植体界面的力传递机制 [74]。已有研究报道，锥度连接螺钉固位的连接方式由于承受较大的过载，会导致大量的骨吸收 [74]。

表 2 不同研究中分析的牙种植体概述。NR：未报告；A：基台；BI：种植体主体；S：螺丝；C：牙冠（CF：冠框架，PS：义齿上部结构）；D：直径；L：长度；TL：螺纹长度；TP：螺纹螺距；TD：螺纹深度；P：节距；D₁：下部直径；D₂：更大直径；DBT：螺纹间距；L_tot：种植体总长度；d：种植体最大直径；l_BI：骨-种植体界面长度。

参考文献	种植体制造商	种植体名称	种植体部件	几何特征（毫米）	评论
塞维迈等人 [19]	士卓曼	ITI	A，BI，C（CF+PS）	D＝4.1；L＝10	–
Akca 等人 [21]	士卓曼	士卓曼牙科种植体	A，BI	BI：D＝4.1 & L＝10	–
Kurniawan 等人 [22]	NR	NR	A，BI	BI D = 3.8 & L = 10	–
田等人 [23]	NR	NR	A，BI	D = 4.1；L = 10	四个牙种植体模型
Empty Cell					直线型 BI 连接至直线型 A
Empty Cell					直线型 BI 连接至角度型 A
Empty Cell					倾斜的种植体与直的种植体相连
Empty Cell					倾斜的 BI 连接到一个倾斜的 A
Chang 等人 [24]	诺贝尔生物护理	诺贝尔生物护理 MK III 组织水平	A，BI，S，C	BI L = 11.5；A L = 7.5 且 D = 4.1	–
Empty Cell	士卓曼	诺贝尔生物护理骨水平		BI L = 10；A L = 6.5 & D = 3.4
Empty Cell		士卓曼 SLA 骨水平		BI L = 10；A L = 6 且 D = 3.5
Empty Cell		士卓曼 SLA 软组织水平		BI L ＝ 10；A L ＝ 6 & D ＝ 4.5
Öztürk 等人 [28]	百奥 Horizon 的	种植体内部	A、BI、S、C（CF+PS）	直径（D）= 4；长度（L）= 12	A 连接设计：内六角
Empty Cell	Camlog	螺纹线 K 系列		D = 4.3；L = 13	管中管
Empty Cell	士卓曼	骨水平		D = 4.1；L = 12	交叉健身
Empty Cell	捷迈邦美	锥形螺纹杆		D = 4.1；L = 13	摩擦配合
达特拉克等人 [29]	NR	诺贝尔	A，BI，S，C	BI D = 3.5 & L = 15	–
Empty Cell		安柯来司
Empty Cell		生物视野
Empty Cell		西韦
奥利维拉等人 [31]	SOADCO	必要圆锥	A，BI，C	直径 D = 4；长度 L = 10	–
Empty Cell		维加
Empty Cell		有限元分析在种植牙中的应用：现状与未来方向 - ScienceDirect
Nokar 等人 [35]	士卓曼	ITI	A，BI，C（CF+PS）	BI L = 12 & DBT = 1.25 & TD = 0.35；	–
Empty Cell				A L = 5.5
达特拉克等人 [36]	NR	NR	A，BI，S，C	D＝3.5；L＝14	分析了五个骨整合种植体（BI）
Empty Cell				D = 3.4；L = 15
Empty Cell				D = 3.5；L = 14
Empty Cell				D = 3.5；L = 15
Empty Cell				D = 3.5；L = 15
Alemayehu 等人 [38]	士卓曼	标准型	A，BI，C	BI D = 4.1 & L = 14 & P = 0.8	五种螺纹设计：
Empty Cell					方形、支撑、倒置支撑
Empty Cell					梯形，三角形
巴吉等人 [43]	士卓曼	ITI 标准 (1)	A，BI	L_tot ＝ 16，d ＝ 4.10，l_BI ＝ 7.5，TP ＝ 1.15，TD ＝ 0.24
Empty Cell	诺贝尔生物护理	ITI 标准 (2)	A，BI	L_tot＝17，d＝3.30，l_BI＝9.0，TP＝0.98，TD＝0.20	–
Empty Cell	登士柏费亚 dental	诺贝尔直接种植体	A，BI	L_tot＝16，d＝4.50，l_BI＝9.0，TP＝0.73，TD＝0.21
Empty Cell		布拉内马克系统	A，BI，S	L_tot＝14，d＝3.75，l_BI＝12，TP＝0.60，TD＝0.27
Empty Cell		安科卢斯	A，BI	L_tot ＝ 11，d ＝ 4.50，l_BI ＝ 11，TP ＝ 1.06，TD ＝ 0.20
奥斯瓦尔等 [52]	NR	NR	A，BI	D＝4；L＝12；TL＝0.5	三种螺纹设计：
Empty Cell				TP = 1.2；TD = 0.42	V 形、支柱形和反向支柱形
Aslam 等人 [59]	NR	NR	A，BI，S	D = 4.5；L = 11	–
Wolff 等人 [69]	士卓曼	士卓曼 BL	BI	直径（D）= 4.1，长度（L）= 10	–
Empty Cell		士卓曼 S	BI	直径 D = 4.1，长度 L = 12
Empty Cell		士卓曼 SP	BI	D = 4.1 且 L = 10
Empty Cell	Camlog	Camlog Plus	BI	D = 3.8，4.3，5 & L = 9，11
Empty Cell	奥齿泰科技	阿斯特拉速固种植体	BI	直径（D）= 4，长度（L）= 11
Kayabasi 等人 [76]	士卓曼	ITI	A，BI，C（CF+PS）	BI D = 4.1 & L = 12；A L = 4	–
de Cos Juez 等人 [77]	NR	NR	BI	D = 6.8；P = 1,4,7；	–
皮尔莫达里安等人 [84]	NR	NR	A，B I，C	D＝4.1；L＝8.5–13	螺纹螺距的变化为 0.6–1 mm
Eazhil 等人 [102]	诺贝尔生物护理	替换选择锥形	A，BI	D = 3.5–5.0；L = 13–16	不同 D 和 L 的分析
布里苏埃拉-贝拉斯科等人 [103]	士卓曼	士卓曼标准	A，BI，C（CF+PS）	BI D = 4.8 & L = 10；A D = 4.8 & L = 5.5	–
Bachiri 等人 [105]	士卓曼	ITI	A，BI，C（CF+PS）	BI D = 4.1 & L = 14；	–
Empty Cell				A D₁ = 2.6 & D₂ = 3.6 & L = 7.2
塔尔马佐夫等人 [106]	士卓曼	标准加组织水平 RN	A，BI	BI：D＝4.1 & L＝10	BI：0.2 mm TD 和 1.2 mm P
Empty Cell				A：L = 5.5
巴赫拉米等人 [108]	NR	NR	A，BI，S，C	BI L = 11.5 & TP = 0.6	–
奇奇乌等人 [110]	美格真	任一外部	A，BI，S	NR	–
Empty Cell		AnyOne 内部	A，BI，S
Empty Cell		任何一人一站式	A，BI，S
科卡等人 [114]	士卓曼	ITI	A，BI，C（CF+PS）	D＝4.1；L＝10	–
丁等人 [117]	士卓曼	ITI	A，BI，S，C	BI D = 4.8 且 L = 10	–
de Moraes 等人 [119]	康纳克斯系统假体有限公司（Conexão Sistemas de Prótese Ltda）提供种植体	NR	A，BI，S，C	D = 3.75；L = 8.5	外六角种植体
Empty Cell	Odontofix 牙科材料工业贸易有限公司负责牙冠制作			D = 3.75；L = 8.5	外六角种植体
Empty Cell				D = 3.75；L = 8.5	莫尔斯锥度种植体
Empty Cell				D = 5；L = 8.5	外六角种植体
Empty Cell				D = 5；L = 8.5	外六角种植体
Empty Cell				D=5；L=8.5	莫尔斯锥度种植体
刘等人 [127]	登士柏种植体	安柯来司	A，BI	BI D＝4 & L＝13；AD＝3.75 & L＝6	–
Empty Cell	安多吉尔有限责任公司	安多吉尔	A，BI，S	BI D = 4 & L = 13；A D = 4 & L = 6
普雷姆纳特等人 [128]	鸿誉	标准螺纹种植体	A，BI，C	螺纹种植体的骨整合程度 D = 4 和 L = 10	–
Empty Cell		标准圆柱形种植体		BI od 圆柱形种植体 D=4.1，L=9
Covani 等人 [132]	瑞典 Martina	优质种植体-种植体 1	A，BI，S，PS	D = 3.8；L = 13.8	种植体 1：A-BI 连接采用带有小基台的内六角结构
Empty Cell		优质种植体-种植体 2			种植体 2：无领内六角连接 A-BI
Empty Cell		高端种植体-种植体 3			种植体 3：无六角结构的内部连接 A-BI
Empty Cell		高端种植体-种植体4			种植体 4：外部六角连接 A-BI

种植体几何形态在影响种植体周围骨反应方面起着重要作用。此外，设计患者特异性和部位特异性的牙种植体至关重要。目标应是使种植体与患者骨组织的条件相匹配，从而在骨-种植体界面创建一个有利于骨整合的生物力学环境。因此，有必要进一步开展相关研究，开发能够考虑患者特异性骨条件的数值模型，以辅助个性化和部位特异性牙种植体的设计。在此背景下，三维增材制造技术可被认为在制造患者特异性和部位特异性种植体方面具有重要作用，能够以相对较低的成本较为容易地制造复杂的种植体。

2.2. 计算离散化

接下来生成 FE 模型的步骤是对骨骼和牙科种植体进行计算离散化，即网格划分。对于骨骼和牙科种植体，已采用六面体单元 [14]、[20]、[26] 以及线性或二次四面体单元四面体单元 [17]、[23]、[24]、[25]、[27]、[28]、[35]、[38]、[42]、[50]、[54]、[56]、[64]、[75]、[76]、[77]、[78]、[79]、[80]、[81]、[82]、[83]、[84]、[85] 进行分析。在大多数情况下，使用的是四面体单元。这与其能够更好地适应复杂几何形状的能力有关 [86]。通常，单元类型（线性或二次）以及相应的形函数的选择取决于具体问题。二次单元允许每个单元内部存在抛物线型位移场，从而使应力和应变在每个单元内线性变化，因此能够更准确地捕捉单元内部的应力和应变梯度。相反，线性单元内的应力和应变则为常量。此外，二次单元能够更精确地描述几何曲率，从而更好地表达几何形状。然而，高阶单元由于节点数量及相应自由度的增加，会导致模型的计算需求上升。当研究重点是获得精确的应力/应变分布时，即使大量线性单元可能以较低的计算成本实现高应力梯度的解析并获得准确结果，二次单元仍可能是更优的选择。在线性单元建模接触、裂纹萌生与扩展时，可避免使用二次形函数时出现的问题（例如收敛问题、体积锁死）。

有限元分析结果的准确性取决于单元和节点的数量。众所周知，单元数量越多，结果越精确，但计算成本也越高。因此，为了确定网格单元的尺寸以及确保有限元分析结果准确且不受网格尺寸变化影响所需的单元数量，需要进行收敛性研究。在这类研究中，通过改变网格单元的尺寸，直到某个输出参数收敛到某一特定值，即继续增加单元数量对输出参数影响较小，但会显著增加计算时间。因此，收敛性研究对于在结果准确性和计算成本之间找到良好平衡至关重要。

2.3. 本构行为

材料的力学行为通过本构模型来描述。这些模型根据模型的复杂程度依赖于不同数量的参数。众所周知，最简单的本构模型是各向同性线弹性模型。在这种情况下，本构关系由以下方程表示(1)

σ = C : ϵ,

其中 σ 为二阶柯西应力张量，

C

为四阶弹性张量，具有次对称性和主对称性且为正定张量，ϵ 为二阶微小应变张量。

C

的对称性将独立材料常数的数量减少至 21 个。对于各向同性材料，即其物理特性在所有方向均相同的情况下，

C

的独立材料常数进一步减少至两个：杨氏模量 E 和泊松系数 ν。虽然在假设种植体具有各向同性本构行为的情况下是合理的，但对于骨组织来说可能并不合适。事实上，骨组织具有一定的各向异性程度，因此采用更简化的本构模型可能会显著影响有限元分析结果，尤其是在骨-种植体界面应力分布方面的结果。

接下来将详细讨论骨和种植体的本构模型及其模型参数的取值。

2.3.1 骨组织

由于骨的各向异性特性和具有个体及部位特异性，其力学行为较为复杂。如表 3 所示，大多数有限元研究（FE studies）采用了简化方法，将皮质骨和松质骨建模为线弹性各向同性均质材料。在这种情况下，只需使用两个材料常数即可完全定义材料行为，即弹性模量 E 和泊松比 ν。分析表 3 可知，当考虑各向同性均质行为时，皮质骨的杨氏模量范围为 9.6 GPa 至 22.8 GPa，松质骨则为 0.210 GPa 至 1.37 GPa。此外，皮质骨的泊松比范围为 0.18 至 0.457，松质骨则为 0.05 至 0.322。

表 3 骨组织的本构行为及材料参数（E 以 GPa 为单位）。IH：各向同性均质；OH：正交各向同性均质；TIH：横观各向同性均质；D₁、D₂、D₃ 和 D₄ 表示 Lekholm & Zarb [40] 分类法，该分类基于皮质骨与松质骨之间的关系定义了四种骨类型；NP：不存在；x：径向方向（皮质骨厚度方向）；y：切向（周向）；z：纵向方向。对于正交各向同性模型，若未特别说明，则假定 ν_xy = ν_yx，ν_xz = ν_zx，ν_yz = ν_zy。

参考文献	本构行为	皮质骨材料参数	小梁骨材料参数
Mellal 等人 [15]	IH	弹性模量 E = 15，泊松比 ν = 0.3	E = 1，ν = 0.3
北川等人 [18]	IH	弹性模量 E = 13.7，泊松比 ν = 0.3	弹性模量 E = 1.37，泊松比 ν = 0.3
塞维迈等人 [19]	IH	弹性模量 E = 13.7，泊松比 ν = 0.3	致密 E = 1.37，ν = 0.3
Empty Cell			低密度 E = 1.10，ν = 0.3
田等人 [23]	IH	弹性模量 E = 13.7，泊松比 ν = 0.3	弹性模量 E = 1.37，泊松比 ν = 0.3
吴等人 [25]	IH	弹性模量 E = 16.7，泊松比 ν = 0.3	弹性模量 E = 0.759，泊松比 ν = 0.3
帕拉奇尼等人 [34]	IH	弹性模量 E = 14.7，泊松比 ν = 0.3	弹性模量 E = 1.47，泊松比 ν = 0.3
Nokar 等人 [35]	IH	弹性模量 E = 14.8，泊松比 ν = 0.3	弹性模量 E = 1.85，泊松比 ν = 0.3
巴吉等人 [43]	IH	弹性模量 E = 13.7，泊松比 ν = 0.3	上颌骨 E = 0.5，ν = 0.3
Empty Cell			下颌骨 E = 1, ν = 0.3
丁等人 [45]	IH	弹性模量 E = 13.7，泊松比 ν = 0.3	弹性模量 E = 1.37，泊松比 ν = 0.3
Vairo 等人 [46]	IH	D₂E = 13.7，ν = 0.3	D₂E = 0.5，ν = 0.3
Gacnik 等人 [48]	IH	弹性模量 E = 13.7，泊松比 ν = 0.3	弹性模量 E = 1.37，泊松比 ν = 0.3
Empty Cell	OH	E_x＝10.630，E_y＝12.510，E_z＝19.750	E_x = 0.210，E_y = 1.148，E_z = 1.148
Empty Cell		ν_xy＝0.313，ν_yz＝0.226，ν_xz＝0.246	ν_xy = 0.055，ν_yz = 0.322，ν_xz = 0.055
Empty Cell		ν_yx＝0.368，ν_zy＝0.357，ν_zx＝0.457
奥斯瓦尔等 [52]	IH	弹性模量 E = 13.7，泊松比 ν = 0.3	弹性模量 E = 1.37，泊松比 ν = 0.3
Verri 等人 [53]	IH	弹性模量 E = 13.7，泊松比 ν = 0.3	弹性模量 E = 1.37，泊松比 ν = 0.3
金等人 [57]	IH	弹性模量 E = 13，泊松比 ν = 0.3	弹性模量 E = 0.69，泊松比 ν = 0.3
金等人 ^[58]	IH	弹性模量 E = 13，泊松比 ν = 0.3	弹性模量 E = 0.69，泊松比 ν = 0.3
Aslam 等人 [59]	颌骨皮质骨	E_x＝12.6，E_y＝12.6，E_z＝19.4	E_x＝1.15，E_y＝0.2106，E_z＝1.15
Empty Cell	TIH 小梁骨	ν_xy＝0.3，ν_yz＝0.253，ν_xz＝0.253	ν_xy = 0.055，ν_yz = 0.01，ν_xz = 0.322
Empty Cell		ν_yx = 0.3，ν_zy = 0.39，ν_zx = 0.39	ν_yx＝0.01，ν_zy＝0.055，ν_zx＝0.322
李等人 [61]	IH	弹性模量 E = 13，泊松比 ν = 0.3	弹性模量 E = 1.3，泊松比 ν = 0.3
贾法里安德霍迪等 [62]	IH	弹性模量 E = 13.7，泊松比 ν = 0.3	弹性模量 E = 0.24，泊松比 ν = 0.3
朴等人 [64]	IH	弹性模量 E = 13.7，泊松比 ν = 0.3	D₃E = 1.3，ν = 0.3
Empty Cell			D₄E = 1.10，ν = 0.3
林内茨基等人 [80]	IH	D₁E = 13.7，ν = 0.3	NP（仅皮质骨）
Empty Cell		D₂E = 13.7，ν = 0.3	D₂E = 1.0，ν = 0.3
Empty Cell		D₃E = 13.7，ν = 0.3	D₃E = 1.0，ν = 0.3
Empty Cell		D₄E = 1.0，ν = 0.3	D₄E = 0.2，ν = 0.3
Cinel 等人 [82]	IH	弹性模量 E = 13.7，泊松比 ν = 0.3	弹性模量 E = 1.37，泊松比 ν = 0.3
皮尔莫里迪安等人 [84]	IH	弹性模量 E = 13.7，泊松比 ν = 0.3	弹性模量 E = 1.37，泊松比 ν = 0.3
Losada 等人 [85]	IH	弹性模量 E = 13.4，泊松比 ν = 0.31	弹性模量 E = 1.37，泊松比 ν = 0.31
王等人 [96]	OH	E_x＝10.8，E_y＝13.3，E_z＝19.4	E_x = 3，E_y = 1，E_z = 0.2
Empty Cell		ν_xy = 0.309，ν_yz = 0.224，ν_xz = 0.249	ν_xy = 0.3，ν_yz = 0.3，ν_xz = 0.3
Empty Cell		ν_yx＝0.381，ν_zy＝0.328，ν_zx＝0.445
Ciucciu 等人 [101]	OH	E_x＝9.6，E_y＝9.6，E_z＝17.8	E_x ＝ 0.144，E_y ＝ 0.099，E_z ＝ 0.344
Empty Cell		ν_xy = 0.55，ν_yz = 0.3，ν_xz = 0.3	ν_xy＝0.23，ν_yz＝0.11，ν_xz＝0.13
布里苏埃拉-贝拉斯科等人 [103]	IH	D₂E = 15，ν = 0.3	D₂E = 1，ν = 0.25
巴赫拉米等人 [108]	IH	弹性模量 E = 14.5，泊松比 ν = 0.323	弹性模量 E = 1.37，泊松比 ν = 0.3
奇乔等 [110]	OH	E_x = 9.6，E_y = 9.6，E_z = 17.8	E_x ＝ 0.144，E_y ＝ 0.099，E_z ＝ 0.344
Empty Cell		ν_xy = 0.55，ν_yz = 0.3，ν_xz = 0.3	ν_xy = 0.23，ν_yz = 0.11，ν_xz = 0.13
科卡等人 [114]	IH	弹性模量 E = 13.4，泊松比 ν = 0.3	弹性模量 E = 1.37，泊松比 ν = 0.3
丁等人 [117]	OH	E_x＝12.7，E_y＝17.9，E_z＝22.8	E_x＝0.511，E_y＝0.114，E_z＝0.907
Empty Cell		ν_xy = 0.18，ν_yz = 0.28，ν_xz = 0.31	ν_xy = 0.22，ν_yz = 0.30，ν_xz = 0.31
Ausiello 等人 [123]	IH	弹性模量 E = 13.7，泊松比 ν = 0.3	E = 0.5，ν = 0.3
刘等人 [127]	IH	弹性模量 E = 13.4，泊松比 ν = 0.3	弹性模量 E = 1.37，泊松比 ν = 0.3
普雷姆纳特等人 [128]	IH	弹性模量 E = 14.8，泊松比 ν = 0.3	D₁E = 9.5，ν = 0.3
Empty Cell			D₂E = 5.5，ν = 0.3
Empty Cell			D₃E = 1.6，ν = 0.3
Empty Cell			D₄E = 0.69，ν = 0.3
丁等人 [129]	OH	E_x＝12.7，E_y＝17.9，E_z＝22.8	E_x＝0.511，E_y＝0.114，E_z＝0.907
Empty Cell		ν_xy = 0.18，ν_yz = 0.28，ν_xz = 0.31	ν_xy＝0.22，ν_yz＝0.30，ν_xz＝0.31
Khened 等人 [130]	IH	弹性模量 E = 13.7，泊松比 ν = 0.26	弹性模量 E = 1.37，泊松比 ν = 0.31

为了更好地体现骨的力学行为，一些作者采用了正交各向异性材料模型（表 3），该模型的特点是在三个相互垂直的方向上具有不同的材料属性。Dhatrak 等人 [29] 进行了一项数值研究，比较了在垂直、侧向和斜向三种不同载荷条件下，四种不同种植体周围骨-种植体界面在骨的各向同性和正交各向异性材料模型下的应力分布情况。他们发现，相较于各向同性材料模型，正交各向异性材料模型中的 Von Mises 应力有所增加，并由此得出结论：正交各向异性模型比各向同性材料模型更适合预测骨-种植体界面的应力分布。然而，这一结论尚需通过结合数值模拟与实验研究的方法加以验证。Taheri 等人也研究了采用正交各向异性材料属性的影响 [87]。在他们的研究中，对皮质骨和松质骨使用并比较了两种材料模型（即各向同性和正交各向异性模型），得出结论：采用各向同性材料模型会忽略骨应力和位移的重要部分。这些被忽略的部分可能在未来的种植体设计中起关键作用。

在材料特性方面另一个需要强调的问题与骨内材料分布均匀性的假设有关，这一假设可能并不合适。众所周知，骨具有非均匀的密度场，因此其杨氏模量分布也具有非均匀性 [86]，并且这种非均匀性可以根据图像（CT 或显微 CT）进行建模。在此背景下，Gacnik 等人 [48] 建立了与空间和骨密度相关的各向同性和正交各向异性材料模型，用于描述人体下颌骨的特性，其中骨密度的变化直接来源于 CT 图像，并将这些模型与相应的均质模型进行了性能比较。他们发现，在骨密度依赖的正交各向异性模型中，冯·米塞斯等效应力分布值相较于其他模型更高，并得出结论认为，考虑到骨密度的空间分布和骨密度依赖性的正交各向异性模型可能更为合适，因为这种模型通过骨密度反映了骨的各向异性。Marcian 等人 [71] 使用微 CT 图像并假设骨组织为各向同性且采用线弹性模型，比较了不同骨组织力学行为（骨组织中均匀与非均匀杨氏模量分布）的模拟场景。对于均匀模型，他们考虑了两种不同情况，分别采用了两个 E 值，即 E = 15 GPa 和 E = 5 GPa。对于非均匀情况，他们采用了四种不同的模型：

•
谢弗尔宾模型(2) $E = \{\begin{matrix} 0.050 & P G I < 17000 \\ 13.636 (P G I \cdot 2.21 \times 1 0^{- 4} - 3.30) & P G I \geq 17000 \end{matrix}$
•
凯勒模型(3) $E = \{\begin{matrix} 0.050 & P G I < 17000 \\ 10.5 {(P G I \cdot 2.21 \times 1 0^{- 4} - 3.30)}^{2.29} & P G I \geq 17000 \end{matrix}$
•
斯奈德模型(4) $E = \{\begin{matrix} 0.050 & P G I < 17000 \\ 3.891 {(P G I \cdot 2.21 \times 1 0^{- 4} - 3.30)}^{2.39} & P G I \geq 17000 \end{matrix}$
•
井川模型(5) $E = \{\begin{matrix} 0.050 & P G I < 17000 \\ 17.486 {(P G I \cdot 2.21 \times 1 0^{- 4} - 3.30)}^{1.596} & P G I \geq 17000 \end{matrix}$

其中，E 以 GPa 为单位表示，PGI 是从图像中获得的像素灰度强度。他们证明了骨材料表示类型对螺钉中的应力具有显著影响。种植体内部的应力差异可能会影响对种植体寿命的预测。Hussein 等人 [63] 研究了不同材料赋值的影响。具体而言，作者比较了具有非均匀 E 分布的有限元模型与均匀有限元模型。他们展示了不同材料赋值之间的不同结果，突出了结果对种植体位置的依赖性。

2.3.2 种植体

对于牙科种植体，其本构行为通常被假定为均质、各向同性和线弹性。表 4 显示了用于牙科种植体各个组件的不同材料，以及它们的杨氏模量 E 和泊松比ν的数值。大多数研究考察了基台螺丝固位的种植体系统。基台、种植体主体和螺丝通常被认为由钛及其合金制成，其杨氏模量范围介于 102 GPa 至 136 GPa 之间。钛具有良好的力学性能和生物相容性。然而，钛的主要问题是其灰色外观可能引起美学方面的困扰。因此，人们提出了其他用于种植体主体的材料，例如氧化锆。基台通常由钛或氧化锆制成，尽管在某些情况下也使用了其他材料，例如 Alemayehu 等人进行的研究中所使用的金制基台[38]。

表 4 假设牙科种植体具有各向同性和线性弹性本构行为时的材料类型及材料参数。E：杨氏模量（GPa）；ν：泊松系数；NP：不存在；NR：未报告；CF：冠部结构；PS：修复上部结构。

参考文献	基台材料参数	种植体材料参数	螺钉材料参数	冠部材料参数
黄等人 [14]	钛	钛	NP	NP
Empty Cell	E = 103，ν NR	弹性模量 E = 103，泊松比 ν NR
林等人 [17]	钛	钛	钛	NP
Empty Cell	弹性模量 E = 110，泊松比 ν = 0.35	弹性模量 E = 110，泊松比 ν = 0.35	弹性模量 E = 110，泊松比 ν = 0.35
北川等人 [18]	钛	钛	金合金	NP
Empty Cell	弹性模量 E = 107，泊松比 ν = 0.33	弹性模量 E = 107，泊松比 ν = 0.33	弹性模量 E = 91，泊松比 ν = 0.33
塞维迈等人 [19]	钛	钛	NP	钴铬（CF）
Empty Cell	弹性模量 E = 110，泊松比 ν = 0.35	弹性模量 E = 110，泊松比 ν = 0.35		弹性模量 E = 218，泊松比 ν = 0.33
Empty Cell				瓷 (PS)
Empty Cell				弹性模量 E = 82.8，泊松比 ν = 0.35
Akca 等人 [21]	材料 NR	材料非放射性	NP	NP
Empty Cell	弹性模量 E = 110，泊松比 ν = 0.35	弹性模量 E = 110，泊松比 ν = 0.35
Kurniawan 等人 [22]	钛	钛	NP	NP
Empty Cell	弹性模量 E = 110，泊松比 ν = 0.35	弹性模量 E = 110，泊松比 ν = 0.35
田等人 [23]	钛	钛	NP	NP
Empty Cell	弹性模量 E = 110，泊松比 ν = 0.3	弹性模量 E = 110，泊松比 ν = 0.3
Chang 等人 [24]	钛	钛	钛	金合金
Empty Cell	弹性模量 E = 102，泊松比 ν = 0.3	弹性模量 E = 102，泊松比 ν = 0.3	E = 102，ν = 0.3	弹性模量 E = 95，泊松比 ν = 0.3
吴等人 [25]	钛	钛	钛	NP
Empty Cell	弹性模量 E = 104，泊松比 ν = 0.3	弹性模量 E = 104，泊松比 ν = 0.3	弹性模量 E = 104，泊松比 ν = 0.3
上田等人 [26]	钛	钛	NP	金合金
Empty Cell	弹性模量 E = 110，泊松比 ν = 0.3	弹性模量 E = 110，泊松比 ν = 0.3		弹性模量 E = 100，泊松比 ν = 0.3
达特拉克等人 [29]	钛	钛	钛	材料非线性
Empty Cell	弹性模量 E = 110，泊松比 ν = 0.35	弹性模量 E = 110，泊松比 ν = 0.35	弹性模量 E = 110，泊松比 ν = 0.35	弹性模量 E = 70，泊松比 ν = 0.19
巴达利亚等人 [32]	材料非保留（NR）	材料非保留（NR）	NP	NP
Empty Cell	E = 106，ν = 0.35	弹性模量 E = 106，泊松比 ν = 0.35
Nokar 等人 [35]	Ti-6Al-4v	Ti-6Al-4v	NP	钴铬合金（CF）
Empty Cell	弹性模量 E = 105，泊松比 ν = 0.33	弹性模量 E = 105，泊松比 ν = 0.33		弹性模量 E = 220，泊松比 ν = 0.30
Empty Cell				长石质（PS）
Empty Cell				弹性模量 E = 61.2，泊松比 ν = 0.19
Alemayehu 等人 [38]	金	钛	钛	瓷
Empty Cell	弹性模量 E = 136，泊松比 ν = 0.37	弹性模量 E = 110，泊松比 ν = 0.34	弹性模量 E = 110，泊松比 ν = 0.34	弹性模量 E = 68.9，泊松比 ν = 0.28
巴吉等人	钛	钛	钛	NP
Empty Cell	弹性模量 E = 114，泊松比 ν = 0.34	弹性模量 E = 114，泊松比 ν = 0.34	弹性模量 E = 114，泊松比 ν = 0.34
傅等人 [47]	NP	钛或氧化锆	NP	瓷
Empty Cell		弹性模量 E = 110，泊松比 ν = 0.35（钛）		弹性模量 E = 70，泊松比 ν = 0.19
Empty Cell		E = 210，ν = 0.31 适用于氧化锆
Gacnik 等人 [48]	钛合金（Ti-6Al-4V）	钛合金（Ti-6Al-4V）	NP	NP
Empty Cell	弹性模量 E = 110，泊松比 ν = 0.33	弹性模量 E = 110，泊松比 ν = 0.33
Sarfaraz 等人 [50]	钛合金	钛合金	钛	瓷
Empty Cell	弹性模量 E = 110，泊松比 ν = 0.33	弹性模量 E = 110，泊松比 ν = 0.33	弹性模量 E = 110，泊松比 ν = 0.33	弹性模量 E = 70，泊松比 ν = 0.19
奥斯瓦尔等 [52]	钛	钛	NP	NP
Empty Cell	弹性模量 E = 110，泊松比 ν = 0.3	弹性模量 E = 110，泊松比 ν = 0.3
金等 [57]	钛合金5级	钛4级	钛-5级	氧化锆
Empty Cell	弹性模量 E = 114，泊松比 ν = 0.33	弹性模量 E = 105，泊松比 ν = 0.34	弹性模量 E = 114，泊松比 ν = 0.33	弹性模量 E = 205，泊松比 ν = 0.19
李等人 [61]	钛	钛	钛	氧化锆
Empty Cell	弹性模量 E = 103，泊松比 ν = 0.33	弹性模量 E = 103，泊松比 ν = 0.33	弹性模量 E = 103，泊松比 ν = 0.33	弹性模量 E = 200，泊松比 ν = 0.31
Park 等人 [64]	钛	钛	钛	NR
Empty Cell	弹性模量 E = 110，泊松比 ν = 0.34	弹性模量 E = 110，泊松比 ν = 0.34	弹性模量 E = 110，泊松比 ν = 0.34	弹性模量 E = 140，泊松比 ν = 0.28
马尔西亚等人 [68]	NP	钛	NP	NP
Empty Cell		弹性模量 E = 110，泊松比 ν = 0.3
de Cos Juez 等人 [77]	NP	钛	NP	NP
Empty Cell		弹性模量 E = 110，泊松比 ν = 0.33
林涅茨基等人 [80]	钛	钛	NP	NP
Empty Cell	弹性模量 E = 114，泊松比 ν = 0.34	弹性模量 E = 114，泊松比 ν = 0.34
皮尔莫达里安等人 [84]	钛	钛	NP	瓷
Empty Cell	弹性模量 E = 110，泊松比 ν = 0.35	弹性模量 E = 110，泊松比 ν = 0.35		弹性模量 E = 68.9，泊松比 ν = 0.28
Losada 等人 [85]	钛	钛	钛	瓷
Empty Cell	弹性模量 E = 110，泊松比 ν = 0.32	弹性模量 E = 110，泊松比 ν = 0.32	弹性模量 E = 110，泊松比 ν = 0.32	弹性模量 E = 68.9，泊松比 ν = 0.19
巴希尔等人 [105]	钛	钛	NP	长石瓷 FF
Empty Cell	弹性模量 E = 110，泊松比 ν = 0.3	弹性模量 E = 110，泊松比 ν = 0.3		弹性模量 E = 82.8，泊松比 ν = 0.35
巴赫拉米等人 [108]	钛	钛	钛	瓷
Empty Cell	弹性模量 E = 110，泊松比 ν = 0.31	弹性模量 E = 110，泊松比 ν = 0.31	弹性模量 E = 110，泊松比 ν = 0.31	弹性模量 E = 82.8，泊松比 ν = 0.33

2.4 骨改建

骨组织可以被视为一种能够适应载荷条件的自优化结构。事实上，骨组织在持续经历重塑过程（即吸收和形成过程）中不断改变其几何和材料特性以适应所受载荷 [88]。因此，载荷对骨重塑具有直接的控制作用 [89]。特别是，载荷是胶原纤维取向的一个关键因素。Traini 等人 [91]、[92]、[154] 通过组织学证据深入研究了这种载荷效应，证明了受载的牙种植体会改变骨的微观结构，从而导致胶原纤维呈现主要横向取向的排列。尽管负载如何影响参与骨重建过程的骨细胞活动的机制尚不明确，但应力和应变（或应变能密度）通常可被视为骨重建的机械刺激。Frost [93] 指出，重建、建模和修复决定了骨结构对不同需求的适应性。这种情况发生在四个递增机械应力水平（区域）的一般背景下。相邻区域之间的转换由微应变的阈值来界定。每种适应过程的激活都需要达到某一阈值的微应变，定义为最小有效应变（MES）。图 3 展示了这些不同的区域以及每个区域中发生的相应现象。具体而言，当应变处于废用区或吸收区时，骨发生吸收，导致骨丢失；而在生理载荷区或超载区时，则发生骨形成，从而维持甚至增加骨量。

因此，种植体周围骨的骨改建反应是进行种植体生物力学分析时需要考虑的一个重要因素，这有助于获得关于骨-种植体系统力学行为的更详细信息，并提高牙种植体的长期疗效。然而，目前很少有研究在有限元模型中纳入骨改建过程。一般来说，骨改建过程可以用数学方法表示为(6)

\frac{d ρ}{d t} = B (M_{s} - M_{s_{r e f}}),

其中，ρ 表示骨密度，t 是时间，B 是一个常数，M_s 是机械刺激，而

M_{s_{r e f}}

是机械刺激的一个平衡点。如果 M_s 的值小于

M_{s_{r e f}}

，则

\frac{d ρ}{d t}

为负值，表明骨密度随时间减少，即发生骨吸收；如果 M_s 的值大于

M_{s_{r e f}}

，则

\frac{d ρ}{d t}

为正值，骨密度随时间增加，从而诱导骨形成。

骨吸收是义齿种植体中的一个主要问题，因为它会导致骨-种植体界面处的松动，影响牙种植体的完整性和稳定性。虽然骨吸收通常被认为是加载不足所致，但界面处的过度加载也被认为是促成因素之一。众所周知，高载荷会在骨组织中引起连续损伤 [94]。然而，骨组织在一定程度上可以自行修复损伤。然而，当载荷过高时，自我修复机制无法跟上损伤增长的速度，从而发生超载吸收。因此，为了评估牙种植体的长期有效性，数值模型需要考虑的另一个方面是骨-种植体界面的重建反应。在此背景下，Li 等人 [94] 提出了一种新的骨重建数学模型，该模型考虑了由于超载而可能发生的吸收，其表达式如下：(7)

\frac{d ρ}{d t} = B (\frac{U}{ρ} - k) - D {(\frac{U}{ρ} - k)}^{2}, 0 \leq ρ \leq ρ_{c b}

其中，

\frac{d ρ}{d t}

表示密度的变化，B 和 D 为常数，

\frac{U}{ρ}

表示单位骨质量的应变能，是驱动骨改建的刺激因子，k 为阈值，ρ_cb 为骨的最大密度。当刺激较小时，第一项线性项占主导地位，导致 ρ 的变化率为负，即低载荷吸收；当刺激较大时，第二项二次项成为主导，同样导致 ρ 的变化率为负，即高载荷吸收。Li 等人 [94] 通过使用该改建模型，并因此考虑了过载导致的骨吸收，证明了此类模型在预测牙科种植体机械行为方面的优越能力。Chou 等人 [95] 实现了一种改建算法，以探讨种植体设计对骨改建的影响，证明改建能够预测非均匀密度分布，从而预测弹性模量分布，其中种植体轮廓对此分布具有一定影响。Wang 等人 [96] 通过实施一种三维正交各向异性骨改建算法，评估了种植前后牙槽骨中改建的影响，在该算法中，应变能量密度 U 被视为驱动改建过程的刺激因素，从而导致骨密度的局部变化（ρ）。改建过程由以下方程描述：(8)

\{\begin{matrix} \frac{d ρ}{d t} = B (\frac{U}{ρ} - (1 - δ) K_{r e f}) & if \frac{U}{ρ} < (1 - δ) K_{r e f} \\ \frac{d ρ}{d t} = 0 & if (1 - δ) K_{r e f} < \frac{U}{ρ} < (1 + δ) K_{r e f} \\ \frac{d ρ}{d t} = B (\frac{U}{ρ} - (1 + δ) K_{r e f}) & if (1 + δ) K_{r e f} < \frac{U}{ρ} < K_{O L} \\ \frac{d ρ}{d t} = B (K_{O L} - \frac{U}{ρ}) & if \frac{U}{ρ} > K_{O L} \end{matrix}

其中

\frac{U}{ρ}

为单位骨质量的应变能密度，B 为改建速率常数，K_ref 和 K_OL 分别为改建参考值，δ 为惰性区的带宽。Wang 等人从表 3 中报道的正交各向同性杨氏模量和泊松比初始值出发 [96]，通过改建方程（式（8））计算了每个时间步的骨密度变化。随后，根据新的密度值，按照以下方程更新正交各向同性弹性模量（E₁、E₂ 和 E₃，单位为 MPa）(9)

\{\begin{matrix} E_{1} = 6382 + 255 (- 23930 + 24000 ρ) \\ E_{2} = - 13050 + 13000 ρ \\ E_{3} = - 23930 + 24000 ρ \end{matrix}

(10)

\{\begin{matrix} E_{1} = 2349 ρ^{2.15} \\ E_{2} = 1274 ρ^{2.12} \\ E_{3} = 194 ρ \end{matrix}

其中，公式（9）适用于皮质骨（1.2 < ρ < 2.0，其中 ρ 以 g/cm³ 表示），而公式（10）适用于松质骨（0.2 < ρ < 1.2，其中 ρ 以 g/cm³ 表示）。他们发现完整模型与植入模型之间的应力、应变和密度分布存在显著差异，因此得出结论：必须考虑骨重建过程，才能更深入地了解骨-牙种植体系统的力学响应。Santonocito 等人 [97] 开发了一种骨重建算法，能够预测种植体周围骨组织的外部和内部骨重建过程。外部骨重建通过允许皮质骨外表面的节点在模拟过程中改变其位置，以模拟骨沉积或吸收现象。通过改变小梁骨和皮质骨的密度来模拟内部重构，并通过以下定律表示(11)

\frac{d ρ}{d t} = k_{r} \frac{d r}{d t} S_{v} (ρ) ρ_{t}

其中，

\frac{d r}{d t}

是骨沉积/吸收速率，它是组织应力的函数，而组织应力是驱动骨改建的刺激因素；ρ_t 表示完全矿化组织的密度；k_r 表示活性骨表面常数，假设其值为 0.2；S_v(ρ) 表示骨表面积密度，定义为给定骨样本中的骨表面积与总体积（包括实心和空隙部分）之比。在每个时间步长中，模型会根据外部改建导致的新骨几何结构进行更新，同时材料属性也会因内部改建而更新。Santonocito 等人 [97] 展示了其骨改建算法预测种植体随时间推移性能的能力，强调了此类算法在帮助临床医生预防种植体失败并为每位患者制定适当种植修复治疗方案方面的重要性。

2.5 骨-种植体界面

大多数有限元研究均假设骨-种植体界面处于100%骨整合状态。这意味着皮质骨和松质骨与种植体完全结合，在种植体-骨界面既不能滑动也不能分离 [19], [21], [24], [26], [30], [34], [37], [43], [46], [48], [52], [55], [59], [63], [64], [67], [81], [82], [98], [99], [100], [101], [102], [103], [104], [105], [106], [107]。然而，这在临床情况下可能不会发生。因此，在模型中需要考虑更复杂的接触情况及其对种植体向骨组织传递载荷的影响。

为了模拟骨-种植体界面不同的结合条件，有限元模型中使用了不同的摩擦接触算法。Tang 等人 [78] 定义了四个接触区域：种植体-骨、种植体-基台、种植体-螺钉和基台-螺钉。他们在所有钛-钛界面上设定了摩擦系数（μ）为 0.3，皮质骨-种植体界面为 0.65，松质骨-种植体界面为 0.77。Bahrami 等人 [108] 研究了表面粗糙处理对下颌骨种植体负载时骨-种植体界面应力分布的影响。他们假设种植体组件（种植体本体、基台、螺钉和牙冠）之间为完全结合状态，而种植体的冠部（与皮质骨接触）被视为抛光表面（μ = 0.4），更根尖部分（与小梁骨接触）则被视为等离子喷涂或多孔珠状表面（μ = 1）。在他们的研究中，Wu 等人 [25] 考虑了基台与种植体主体之间的界面，其摩擦系数 μ 为 0.3；种植体主体与皮质骨和小梁骨之间的界面，其摩擦系数 μ 分别为 0.4 和 0.8。Ramos 等人 [109] 在研究中采用的骨与种植体主体之间的摩擦系数 μ 为 0.3。Kim 等人 [57]、[58] 认为皮质骨和小梁骨之间、种植体主体与骨之间以及基台与牙冠之间的界面实现了完全骨整合。在种植体主体与基台之间，以及基台与螺丝之间采用了 0.5 的摩擦系数。Cicciú等人 [110] 假设骨-种植体和骨-骨界面实现了完全骨整合，并将牙科种植体金属表面之间的接触建模为摩擦系数 μ 为 0.3。Wu 等人所开展的研究中在 Alemayehu 等人[33]的研究中，种植体-基台主体、种植体主体-螺丝以及基台-螺丝界面的摩擦系数设定为 0.3；而种植体主体与骨之间的界面，皮质骨和松质骨的摩擦系数分别采用 0.65 和 0.77。Alemayehu 等人[38]在骨-种植体界面实施了摩擦接触，其中种植体与皮质骨之间的摩擦系数为 0.65，种植体与松质骨之间的摩擦系数为 0.77。Losada 等人[85]假设螺丝与金属支架之间、皮质骨与松质骨之间以及咬合材料与支架之间为绑定接触条件，而在种植体主体与松质骨、种植体主体与皮质骨以及螺丝与种植体主体之间设定了 0.3 的摩擦系数。Dhatrak 等人的研究中定义了一个摩擦接触条件（μ=0.3），用于模拟骨-种植体界面状况。 [36] cortical bone 和 trabecular bone 被建模为绑定接触，而骨-种植体界面则采用摩擦系数 μ = 0.2 进行建模，见于 [97]。Fiorillo 等人 [111] 对骨-种植体界面采用摩擦系数为 0.15 的摩擦接触进行建模，而 cortical bone 与 cancellous bone 之间的接触则设定为绑定接触。

2.6 载荷与约束

文献中报道的研究所采用的载荷和约束条件如表 5 所示。就约束条件而言，通常将骨段的近中面和远中面在所有自由度上进行约束。就载荷而言，大多数有限元研究考虑了静态载荷，不仅模拟了垂直载荷和水平力，还考虑了组合载荷（例如，斜向咬合力），以模拟更加真实的受力条件，从而获得更真实的力学响应。然而，牙种植体不仅受到静态载荷的作用，还会受到动态载荷的影响 [112]，这些动态载荷也应被考虑以分析种植体断裂或疲劳失效的潜在原因。目前，仅有少数有限元研究探讨了动态载荷的影响。Kayabasi 等人 [76] 的研究表明，对于具有倒锥形螺纹形状的牙种植体，动态载荷可使种植体内部的应力增加 10–20%。Alemayehu 等人 [38] 比较了不同螺纹设计和咬合载荷方向下种植体及种植体周围骨的静态、准静态和动态响应。他们发现，在动态载荷条件下，种植体及其周围骨组织中的应力显著高于准静态和静态载荷条件，因此得出结论：需要通过动态分析来正确理解某种种植体设计在临床应用中的性能影响。Geramizadeh 等人 [113] 研究表明，与静态载荷相比，动态载荷会在种植体周围骨组织中产生更高的应力（约高 5%-10%），这表明有必要对牙种植体进行瞬态分析。

表 5 文献中报道的有限元研究中所使用的静态载荷与约束概况。FC：完全约束；MS：近中面；DS：远中面；VL：垂直载荷；HL：水平载荷；LL：侧向载荷；OL：斜向载荷。

参考文献	载入	载荷位置	约束条件	约束的位置
塞维迈等人 [19]	总载荷 300 N 的垂直向载荷（VL）	冠部颊尖处150牛顿力 +	FC	每种设计的 x 轴上的最终单元
Empty Cell		150 牛顿远中窝洞向力
Kurniawan 等人 [22]	压力（100 MPa）	基台	FC	骨的主应力和位移
Chang 等人 [24]	1. 轴向力（100 N）	牙冠	NR	NR
Empty Cell	2. 倾斜载荷（100 N，30^∘）
吴等人 [25]	LL（190 N，相对于种植体轴线 30^∘）	基台	FC	骨的主应力和位移
上田等人 [26]	1. VL	牙冠	FC	骨的主应力和位移
Empty Cell	2. OL（向种植体轴线舌侧倾斜 15°，60 N）
巴达利亚等人 [32]	1. 轴向载荷（100 N）	基台	NR	NR
Empty Cell	2. 咬合力（30 牛）
达特拉克等人 [36]	三种载荷条件：	牙冠	FC	位于骨块底部三分之一处的节点
Empty Cell	1. 垂直向（100 N）（咀嚼力），从冠方向根尖方向
Empty Cell	2. 在颊舌方向施加 40 N 的 LL 载荷
Empty Cell	3. 植体轴向 45^∘ 倾斜时的 OL（100 N）
Barbosa 等人 [37]	1. 轴向载荷（150 N）	牙冠	FC	骨的主应力和位移
Empty Cell	2. 倾斜载荷（150 N，30^∘）
Alemayehu 等人 [38]	近远中向联合载荷，	位于远处的虚拟参考点	FC	种植体与下颌骨之间的界面
Empty Cell	颊舌向和根尖方向	从牙合面起3毫米处
Natali 等人 [41]	OL（200 牛垂直载荷 + 20 牛水平载荷）	种植体基部	FC	骨的主应力和主应变
巴吉等人 [43]	OL（22^∘）：沿 LL 方向的	基台	FC	骨的主应力和位移
Empty Cell	颊舌向轴（100 N）＋压入向力（250 N）
丁等人 [45]	垂直向和颊舌向倾斜	基台	FC	位于髁突关节面的节点
Empty Cell	（45度角，150牛）			以及咀嚼肌的附着区域
Vairo 等人 [46]	OL（68°倾斜载荷；VL 250 N + LL 100 N）	基台	FC	骨的主应力和主应变
傅等人 [47]	OL（与种植体轴成 45^∘ 角）	牙冠	FC	骨的主应力和位移
Sarfaraz 等人 [50]	VL(100 N)	牙冠	NR	NR
Empty Cell	OL（与种植体纵轴成 45^∘，100 N）
杉浦等人 [51]	颊舌向 OL（30^∘，200 N）	基台	FC	骨的主应力和位移
奥斯瓦尔等 [52]	垂直载荷（100 牛）	基台	FC	骨的主应力和位移
Verri 等人 [53]	1. 轴向载荷（200 N）	1. 在四个牙尖顶点处	FC	骨块的前表面和后表面
Empty Cell	2. 咬合力（100 牛顿）	2. 两个牙尖顶点处
Shirazi 等人 [55]	垂直载荷（100 牛顿）	基台	FC	基底支撑骨
Aslam 等人 [59]	轴向载荷（200–800 N）	基台	FC	NR
Empty Cell	咬合力（50–150 牛顿，与长轴呈 45^∘）
Didier 等人 [60]	垂直载荷（120 牛）	基台	NR	NR
李等人 [61]	1. 咬合垂直向力（100 N）	牙冠	NR	NR
Empty Cell	2. 咬合载荷（100 N）（与种植体轴向成 30^∘ 角）
贾法里安德霍迪等 [62]	压力（1.117 MPa）	牙冠	FC	小梁骨的主应⼒和主应变
Limbert 等人 [65]	轴向载荷（100 N）	种植体	FC	μCT 骨块的前后表面
马尔西亚等人 [67]	轴向载荷（200 N）	种植体	NR	NR
de Cos Juez 等人 [77]	有限元分析在种植牙中的应用：研究现状与未来方向 - ScienceDirect	种植体基部	NR	NR
Empty Cell	HL（15N）
林内茨基等人 [80]	OL（118.2 N）75^∘ 斜向载荷	基台	FC	下颌骨模型圆柱面上的节点
皮尔莫达里安等人 [84]	OL（45°，180 N）	冠部中心点	FC	皮质骨的主应力和位移
Empty Cell	轴向载荷（180 N）
李等人 [94]	咬合力（400 牛）	种植体	FC	下颌骨节段的两个切割端
Santonocito 等人 [97]	咀嚼力 VL（100 N）	基台	FC	方形骨系统底面
孔等人 [99]	轴向（200 N）和颊舌向（100 N）载荷	瓷修复体	FC	骨的主应力和位移
奇乔等人 [101]	1. 400 N 的纯牵引力	与牙齿接触的种植体表面	FC	模型的外侧表面
Empty Cell	2. 400 N 的纯压力
Empty Cell	3. 400 N 的弯曲力
Empty Cell	4. 400 N 的混合拉伸-弯曲
Empty Cell	5. 400 N 的复合压缩弯曲
Eazhil 等人 [102]	OL（118.2 N）与牙合平面呈 75^∘ 角	基台	NR	NR
布里苏埃拉等人 [103]	在前庭-舌侧方向加载（150 N），	牙冠中央窝	NR	NR
Empty Cell	相对于种植体轴线的6°
巴赫拉米等人 [108]	垂直载荷（200 N）+ 侧向载荷（20 N）	牙冠的咬合面	FC	皮质骨的主应力和位移
奇乔等人 [110]	轴向载荷（800）N	基台	FC	骨块的外侧和底面
Fiorillo 等人 [111]	垂直载荷（400 牛顿）	种植体周围骨组织的应力分布	FC	骨的主应力和位移
Chou 等人 [116]	1. 垂直载荷（500 N）	基台	FC	骨的主应力和位移
Empty Cell	2. HL（250 牛顿）
施维塔拉等人 [118]	1. 沿种植体长轴方向施加的垂直载荷（100 N）	牙冠	NR	NR
Empty Cell	2. 咬合力（100 N）（与种植体长轴成 30°角）
de Moraes 等人 [119]	1. 轴向载荷（200 N）	1. 牙尖内斜面上的四个点	FC	骨的主应力和主应变
Empty Cell	咬合力（100 牛顿）	2. 舌侧牙尖内部斜面的两个加载点
Ausiello 等人 [123]	OL（350 N）12^∘ 斜向载荷	基台	FC	松质骨的下表面
阿尼图亚等人 [124]	1. 轴向载荷（200 N）	牙冠	FC	骨块的外表面
Empty Cell	2. OL（200 N，30^∘ 斜向载荷）
Chang 等人 [125]	OL（100 牛顿的垂直载荷 + 20 牛顿的水平载荷）	牙冠	FC	骨的主应力和位移
刘等人 [127]	1. 垂直载荷（50 牛或 100 牛或 150 牛）	基台	FC	NR
Empty Cell	2. HL（50 牛或 100 牛）
Khened 等人 [130]	OL（100 N）30^∘ 斜向载荷	牙冠	FC	FE 骨模型底部的节点
贾法里安等人 [131]	200 牛顿的纵向种植体轴向力 (VL)	基台	FC	骨的主应力和位移
Empty Cell	+ 40 N 的 HL（颊舌向），呈 15^∘ 角度
Poovarodom 等人 [133]	OL（200 N）30^∘ 斜向载荷	咬合面水平偏移2毫米	FC	骨的主应力和位移
Empty Cell		从中心向颊侧

2.7 有限元结果

大多数有限元研究集中于评估种植体及其周围骨内的 Von Mises 应力分布 [20]，[23]，[43]，[44]，[48]，[49]，[52]，[83]，[84]，[100]，[101]，[114]，[115]，[116]，[117]，[118]，[119]，[120]。这些研究显示，应力场受种植体几何形状、连接设计、种植体材料、加载类型以及骨-种植体界面性质等因素的显著影响。就种植体的几何特征而言，种植体直径、骨-种植体界面长度以及螺纹间距、形状和深度等因素在负荷传导机制及其引起的应力分布中起着重要作用。已有研究报告指出，增加种植体直径可减少牙槽嵴骨的应力 [45], [99], [119], [121]。在改善应力分布模式方面，种植体直径比其长度更为重要 [45], [102]。Baggi 等人在其有限元研究中发现，研究发现，增加种植体直径可降低皮质骨中的应力值和应力集中区域，而增加种植体长度则可使松质骨中的应力分布更为有效。有限元研究表明，螺纹的存在确保了骨与种植体之间更大的接触面积，从而改善了种植体周围区域的应力分布。Karaman 等人通过有限元分析研究了改变种植体微沟槽轮廓对周围骨组织应力传递的影响，结果显示方形轮廓的螺纹比 V 形轮廓的螺纹更稳定。在 Ausiello 等人进行的有限元研究中，螺纹宽度和厚度被证明是减少骨组织内诱发应力和损伤的主要因素。有限元研究还强调了骨组织质量和数量对应力分布及种植体失效的影响。 D4 型骨质的特点是皮质层较薄且小梁骨密度较低，对种植体提供的稳定性有限，且骨-种植体界面较小，增加了失败风险。 Anitua 等人 [124] 通过有限元分析研究了两个变量（即种植体直径和骨质质量）对骨应力的影响。他们发现，无论种植体直径如何，骨质质量的改善均会导致骨应力降低。然而，增加种植体直径对降低骨应力的作用更为显著。

有限元分析还被应用于拓扑优化领域以设计新型牙科种植体。拓扑优化旨在通过有限元分析，为一组载荷和约束条件生成优化的材料分布。该过程通过迭代进行，以评估新设计的性能，在保持种植体结构稳定性和功能特性的前提下，去除冗余材料。在此领域，Chang 等人 [125] 通过应用拓扑优化发现，与传统种植体相比，牙科种植体的体积可减少 17.9%，而生物力学性能相似，例如种植体的应力、种植体-骨复合体的应力、更低的位移以及更高的刚度。这种体积的减小可能因为空间增加而有利于骨组织生长。Gupta 等人 [126] 研究表明，牙科种植体体积减少 32%至 45%仍可保持其全部功能。

3. 载荷对种植体稳定性的影响：微尺度层面作为有限元建模与分析演进的关键因素

现代牙科无法脱离对如何控制机械条件以优化口腔种植体周围骨整合速度和质量的理解。迄今为止，我们已知骨具有根据其暴露的载荷大小、频率和持续时间来调整其质量和结构的能力 [134], [135]。如先前在第 2.4 节和图 3 中所述，机械静力学理论表明，当载荷施加于骨上时，其建模和重塑的刺激取决于应变的大小，因此也取决于载荷的大小。载荷的频率同样可以发挥重要的作用。骨质量可以通过数量相对较少但具有高应变的加载循环 [136]，或通过大量具有低应变的加载循环来维持 [137]。载荷的实体决定了激活细胞的数量，而应变的水平则决定了成骨细胞活性的强度，即应变的成骨作用效果会随着应变频率的增加而增强[138]，[139]。Turner 提出了以下数学方程来解释这一情况[140]：(12)

Ψ = K_{1} \sum_{i = 1}^{n} ϵ_{i} f_{i}

其中，应变刺激Ψ与一个比例常数 K₁ 有关，取决于应变的大小（ϵ）和加载频率 f。当载荷频率为零（f=0）时，应变刺激消失（Ψ=0）[140]。因此，静态载荷不会对骨适应机制的激活产生任何影响。Halldin 等人对兔子的组织学研究以及有限元分析结果 [141]，[142] 表明，骨密度的增加并不会对原始骨量产生负面影响，即使应变超过了皮质骨的极限应变和屈服应变，这也表明静态骨应变不会影响骨吸收。他们还指出，种植体稳定性（通过移除扭矩测量）随时间下降的原因可能是骨应力松弛、骨吸收和骨重建共同作用的结果。在种植体植入后，这种下降的主要原因很可能是黏弹性骨应力松弛所致。随后发生吸收和重塑，因为致密骨再血管化需要一定时间。种植体植入后立即开始骨改建和重塑以修复手术创伤。预应力骨在重塑过程中逐渐被消除，稳定性降低是由于原始骨重塑所致，而新骨形成增加骨整合后种植体稳定性随之提高。

骨结合种植体的成功在很大程度上取决于在宏观尺度和微观尺度水平上形成机械嵌合。有限元（FE 仿真及其与实验结果的验证依赖于宏观尺度水平的评估。只有少数研究 [9]、[11]、[141]、[142]、[143]、[144] 中从微观尺度角度进行了生物力学仿真。当牙种植体处于功能状态时，载荷会传递至骨骼。根据骨支持情况、骨质量、种植体表面粗糙度、弹性模量以及种植体材料的性能、种植体直径和种植体壁厚的不同，载荷会在种植体与骨界面处产生压应力、张应力和剪切力 [11]。这些力的作用需要在微观尺度上，即种植体螺纹间区域进行研究，因为它们可能会改变影响种植体骨整合因素的系统化分析范式。事实上，鉴于有限元（FE）软件技术的发展及其计算能力的提升，以及可用于编程和验证未来 FE 模型的新临床证据的出现，当前必须深化微观尺度层面的研究。

剪切强度必须进行研究，主要由于微动（fretting）现象的存在。微动是一种发生在微观尺度上的摩擦腐蚀过程。具体而言，它是在两个材料之间的接触区域在载荷作用下，由于振动或其他外力产生微小相对运动时发生的一种磨损过程 [145]。根据 Perona 等人的研究 [146]，定义微动时，种植体微动幅度必须小于 100μm。总体而言，微动对种植体-骨界面的影响是组织层面损伤与生物反应的综合结果。微动损伤可在一定程度上被活体组织修复（即适当的微动），从而引发细胞增殖、细胞外基质的新生以及骨多细胞形成单位的生成。然而，当微动损伤超过活体组织的修复能力时，就会发生骨吸收或种植体松动或其他负面生物反应（即不当微动）：成骨细胞生长受抑制、成纤维细胞生长增加、免疫系统激活以及骨吸收 [145], [147]。适当或不当微动的临界微动阈值被设定在 50 至 150 μm 之间。可能 100 μm 是避免纤维整合和种植体不稳定性的临界水平 [148]。剪切强度越大，种植体产生的微动越小。Hansson 等人 [11] 建立了一个数学模型，用以估算在剪切状态下允许种植体与骨之间产生间隙时的骨-种植体界面剪切强度，以试图模拟临床情况。具有优化微几何结构的粗糙表面将显著提高界面剪切强度。未来，将这种模型整合到有限元微观模型中 [143]，并在体内条件下对该模型进行测试，以确定微动对接合过程的实际影响，并能够预测种植体与支撑骨之间的机械嵌合。

关于 Hansson 等人进行的二维有限元研究[9]，[144]清楚地表明，就抗压强度和抗拉强度而言，种植体螺纹几何形状是影响微观尺度下负载分布以及支撑骨所受应变的关键因素。骨组织中的最大抗拉应力位于螺纹顶部外侧，而最大抗压应力则出现在螺纹下侧斜面过渡到圆顶部位的外侧。将这些发现与 Traini 等人所进行的组织学研究[90]，[91]，[149]，[150]，[151]，[152]，[153]的结果进行比较是非常有趣的。他们证明了螺纹形状是影响横向和纵向骨胶原纤维数量的一个因素。[155] 较多的横向纤维位于螺纹下缘下方，而纵向纤维则分布在螺纹之间的空间。纤维的方向在螺纹顶部外侧发生改变。总体而言，在骨基质中，横向排列的胶原纤维更能抵抗压应力，而纵向排列的胶原纤维则更能抵抗剪切应力。此外，一项体内研究指出，在即刻负载的种植体周围以横向排列的胶原纤维为主，而在未负载的种植体周围则以纵向排列的胶原纤维为主。[91] 这明确表明负载会影响骨胶原纤维的排列方向，从而有助于骨的力学性能的实现。与体内评估相结合，采用更复杂的有限元分析来评估载荷、骨应变和骨胶原取向之间的相关性，将是非常有趣的。对于骨整合种植体而言情况如此，而对于即刻负重或早期负重的种植体来说，这种相关性甚至更加显著。然而，正如 Puleo 和 Nanci 先前所述 [156]，为了更全面地理解骨组织的生物力学反应，仍需对骨-种植体界面处发生的事件进行更深入的研究。正如这些作者所报道的，形态学研究揭示了骨-种植体界面的异质性。无论种植体材料如何，经常观察到的一个特征是存在一个无纤维的界面区，类似于天然骨界面中的类骨质线（cement lines）和界膜层（laminae limitantes）。这一现象应得到更深入的研究，并最终将其纳入有限元分析中，因为它可能改变传统的生物力学范式，即认为种植体与骨组织是“融合”在一起的，中间没有任何介于两者之间的结构。总之，在微尺度水平上模拟和预测骨反应的机会可以辅助临床医生在诊断和随访阶段的工作：在其他所有影响因素相同的情况下，通过改变施加的载荷，可能实现对骨反应的调控。

4. 未来展望与结论

有限元建模为研究牙种植体、骨组织及其相互作用的力学行为提供了更为完善的理论基础，有助于实现种植体的长期稳定与成功。许多研究已朝此方向展开。然而，对骨-种植体相互作用、种植体相关问题的深入理解以及最优种植体设计的开发仍需进一步研究。为此，可以考虑以下建议：首先，从几何角度来看，与其仅考虑种植体周围的颌骨段，更合理的方式可能是建立整个颌骨模型，以体现其对种植体所受力的贡献。这种模型可以通过 CT 或磁共振等诊断图像获得，从而不仅能获取患者特异性的几何结构，还能获得材料属性的真实分布。实际上，与其假设骨密度和弹性模量 E 呈均匀分布，更合理的方法是将骨组织按照密度及其对应的 E 值，建模为一种非均质材料。此外，这可能有助于捕捉与患者的年龄和健康状况相关的骨组织特定条件，这些因素可能会影响骨-种植体之间的相互作用。骨密度显著降低的病理性骨与健康骨相比，其与种植体的相互作用方式不同，从而影响种植体的稳定性。另一项建议涉及有限元分析中应采用的边界条件。对整个颌骨进行建模可以纳入肌肉力量，以考虑咀嚼运动过程中肌肉对骨-种植体系统力学响应的影响。对肌肉力量的建模可能需要对肌肉内纤维走行路径进行详细描述。该信息可通过磁共振扩散张量成像 [157] 获得，该技术为获取肌肉体积内纤维排列情况提供了一种有前景的方法。另一个建议是引入骨改建算法，该算法可能允许研究骨几何形态和密度随时间的变化，从而分析骨-种植体系统力学行为的长期演变。特别是，骨改建算法应能够模拟骨-种植体界面的微观力学行为，并反映其在宏观尺度上对骨-种植体系统力学性能的影响，进而影响其长期稳定性。为此，可借助多尺度模型，通过多个长度尺度考虑组织成分的组织学排列。将这种骨组织特性纳入模型有助于获得有关种植体设计、骨应力分布以及骨生长之间关系的信息。硬件技术的不断进步将有助于克服当前在开发此类复杂模型时的计算限制。另一个需要强调的方面是牙种植体的设计。有限元分析结合拓扑优化技术有助于设计出在减少材料用量的同时仍能保持种植体功能不变的种植体。材料的减少可以因空间的可用性而促进骨生长，从而提高种植体的稳定性。最后，作者认为，为了全面理解种植体的失效机制及其周围力学环境，应研究应变，并采用基于应变的准则进行断裂分析。对骨样本进行的实验测试已明确表明，骨失效主要是一种受应变控制的事件 [158]，[159]。

需要强调的一个重要方面是，目前针对牙种植体有限元建模的验证研究还很少 [8]。然而，确保数值模型能够准确预测其设计所要复制的物理现象，如应变/应力场等物理量，是一个至关重要的方面，特别是在将数值模型用于临床目的时更是如此 [160]。此外，在验证的背景下，由于实验测量结果可能会受到较大不确定性的影响，因此应开展敏感性分析。因此，有必要开展敏感性研究，以了解不确定性对模型结果的影响。根据作者的观点，未来的研究应明确地解决牙种植体数值模型的验证问题。

有限元建模与分析的最终目标是开发一种有效的计算工具，供牙科医生在临床实践中用来分析骨-种植体系统的力学响应，从而评估其长期稳定性，以决定更适合患者的种植体。在此背景下，应采用基于机器学习的方法，快速实现有限元模型的生成与分析，以开发可在临床实践中应用的技术。对于临床应用而言，该技术必须具备快速性、高度自动化，并能提供用于个性化决策的准确指标。此外，与建模涉及的所有方面相关的有限元建模标准化对于基于有限元模型的临床适用性至关重要。

资助

由欧洲联盟 - 欧洲社会基金 - 联合研究中心（PON）研究与创新计划 2014–2020（行动 IV.6）- FSE-REACT EU 提供资助。

CRediT 作者贡献声明

克里斯蒂娜·法尔辛内利（Cristina Falcinelli）：构思，方法论，撰写－原始草稿，撰写－审阅与编辑。 弗朗切斯科·瓦伦特（Francesco Valente）：构思，方法论，撰写－原始草稿，撰写－审阅与编辑。 马尔科·瓦斯塔（Marcello Vasta）：撰写－审阅与编辑，资金获取，监督。 托尼诺·特雷尼（Tonino Traini）：构思，撰写－审阅与编辑，监督。所有作者最终批准了该稿件，并同意对工作的所有方面负责。

致谢

Cristina Falcinelli 和 Marcello Vasta 感谢意大利数学物理国家小组（GNFM-INdAM）的支持。Cristina Falcinelli 感谢欧洲联盟 - 欧洲社会基金 - 2014–2020 年研究与创新操作计划（PON）（行动 IV.6）- FSE-REACT EU 提供的资金支持。

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有限元模型：通往个性化牙科时代计算机建模之路
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This article reviews the applications of Finite Element Models (FEMs) in personalized dentistry, focusing on treatment planning, material selection, and CAD-CAM processes. It also discusses the challenges and future directions of using finite element analysis (FEA) in dental care.
This study synthesizes current literature and case studies on FEMs in personalized dentistry, analyzing research articles, clinical reports, and technical papers on the application of FEA in dental biomechanics.
Sources for this review include peer-reviewed journals, academic publications, clinical case studies, and technical papers on dental biomechanics and finite element analysis. Key databases such as PubMed, Scopus, Embase, and ArXiv were used to identify relevant studies.
Studies were selected based on their relevance to the application of FEMs in personalized dentistry. Inclusion criteria were studies that discussed the use of FEA in treatment planning, material selection, and CAD-CAM processes in dentistry. Exclusion criteria included studies that did not focus on personalized dental treatments or did not utilize FEMs as a primary tool.
FEMs are essential for personalized dentistry, offering a versatile platform for in-silico dental biomechanics modeling. They can help predict biomechanical behavior, optimize treatment outcomes, and minimize clinical complications. Despite needing further advancements, FEMs could help significantly enhance treatment precision and efficacy in personalized dental care.
FEMs in personalized dentistry hold the potential to significantly improve treatment precision and efficacy, optimizing outcomes and reducing complications. Their integration underscores the need for interdisciplinary collaboration and advancements in computational techniques to enhance personalized dental care.
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¹: 以上作者对本研究的贡献相同。

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种植牙中的有限元分析：研究现状与未来方向

摘要

目的

方法

结果

意义