Laboratory for Structural NMR Imaging, Department of Radiology, University of Pennsylvania Medical Center, 3400 Spruce Street, Philadelphia, Pennsylvania 19104, USA.
Med Phys. 2013 May;40(5):052303. doi: 10.1118/1.4802085.
Bone strength is the key factor impacting fracture risk. Assessment of bone strength from high-resolution (HR) images have largely relied on linear micro-finite element analysis (μFEA) even though failure always occurs beyond the yield point, which is outside the linear regime. Nonlinear μFEA may therefore be more informative in predicting failure behavior. However, existing nonlinear models applied to trabecular bone (TB) have largely been confined to micro-computed tomography (μCT) and, more recently, HR peripheral quantitative computed tomography (HR-pQCT) images, and typically have ignored evaluation of the post-yield behavior. The primary purpose of this work was threefold: (1) to provide an improved algorithm and program to assess TB yield as well as post-yield properties; (2) to explore the potential benefits of nonlinear μFEA beyond its linear counterpart; and (3) to assess the feasibility and practicality of performing nonlinear analysis on desktop computers on the basis of micro-magnetic resonance (μMR) images obtained in vivo in patients.
A method for nonlinear μFE modeling of TB yield as well as post-yield behavior has been designed where material nonlinearity is captured by adjusting the tissue modulus iteratively according to the tissue-level effective strain obtained from linear analysis using a computationally optimized algorithm. The software allows for images at in vivo μMRI resolution as input with retention of grayscale information. Associations between axial stiffness estimated from linear analysis and yield as well as post-yield parameters from nonlinear analysis were investigated from in vivo μMR images of the distal tibia (N = 20; ages: 58-84) and radius (N = 20; ages: 50-75).
All simulations were completed in 1 h or less for 61 strain levels using a desktop computer (dual quad-core Xeon 3.16 GHz CPUs equipped with 40 GB of RAM). Although yield stress and ultimate stress correlated strongly (R(2) > 0.95, p < 0.001) with axial stiffness, toughness correlated moderately at the distal tibia (R(2) = 0.81, p < 0.001) and only weakly at the distal radius (R(2) = 0.34, p = 0.007). Further, toughness was found to vary by up to 16% for bone of very similar axial stiffness (<2%).
The work demonstrates the practicality of nonlinear μFE simulations at in vivo μMRI resolution, as well as its potential for providing additional information beyond that obtainable from linear analysis. The data suggest that a direct assessment of toughness may provide information not captured by stiffness.
骨强度是影响骨折风险的关键因素。即使在屈服点之外发生失效，即超出线性范围，也主要依赖于高分辨率（HR）图像的线性微有限元分析（μFEA）来评估骨强度。因此，非线性 μFEA 可能更有助于预测失效行为。然而，应用于小梁骨（TB）的现有非线性模型主要局限于微计算机断层扫描（μCT）和最近的 HR 外周定量计算机断层扫描（HR-pQCT）图像，并且通常忽略了对屈服后行为的评估。这项工作的主要目的有三个：（1）提供一种改进的算法和程序来评估 TB 的屈服以及屈服后的性能；（2）探索非线性 μFEA 相对于其线性对应物的潜在优势；（3）评估基于从患者体内获得的微观磁共振（μMR）图像在台式计算机上进行非线性分析的可行性和实用性。
设计了一种用于 TB 屈服以及屈服后行为的非线性 μFE 建模方法，其中通过根据线性分析中根据组织水平有效应变迭代调整组织模量来捕获材料非线性，该线性分析使用计算优化的算法。该软件允许以体内 μMRI 分辨率的图像作为输入，同时保留灰度信息。从远端胫骨（N=20；年龄：58-84）和桡骨（N=20；年龄：50-75）的体内 μMR 图像中研究了从线性分析估计的轴向刚度与非线性分析中的屈服和屈服后参数之间的相关性。
使用台式计算机（配备 40GB 内存的双四核 Xeon 3.16GHz CPU）在 61 个应变水平下，所有模拟在 1 小时或更短时间内完成。尽管屈服应力和极限应力与轴向刚度高度相关（R²>0.95，p<0.001），但韧性在远端胫骨处中度相关（R²=0.81，p<0.001），在远端桡骨处仅弱相关（R²=0.34，p=0.007）。此外，对于轴向刚度非常相似（<2%）的骨，韧性可变化高达 16%。
这项工作证明了在体内 μMRI 分辨率下进行非线性 μFE 模拟的实用性，以及它提供线性分析之外的潜在信息的能力。数据表明，韧性的直接评估可能提供刚度无法捕捉的信息。
