Cell migration plays a particular important role in the initiation and progression of many physical processes and pathological conditions such as tumor invasion and metastasis. Three-dimensional traction force microscopy (TFM) of high resolution and high accuracy is being developed in an effort to unveil the underlying mechanical process of cell migration in a vivo-like environment. Linear elasticity-based TFM (LETM) as a mainstream approach relies on the Green’s function (that relates traction forces to matrix deformation), of which the inherent boundary conditions and geometry of the matrix could remarkably affect the result as suggested by previous 2D studies. In this study, we investigated this close linkage in 3D environment, via modeling of a cell sensing a close-by fixed boundary of a 3D matrix surrounding it, and comparing the reconstructed traction forces from three different solutions of the Green’s function, including a fully matching solution derived using the adapted Mindlin’s approach. To increase fidelity in the estimate of traction forces for extreme conditions such as a sparse sampling of deformation field or targeting small focal adhesions, we numerically solved the singularity problem of the Green’s function in a non-conventional way to avoid exclusion of singular point regions that could contain representative deformation indicators for such extreme conditions. A single case experimental study was conducted for a multi-cellular structure of endothelial cells that just penetrated into the gel at the early stage of angiogenesis.

Statement of Significance

This study focused on the fundamental issue regarding extension of linear elasticity-based TFM to deal with physically realistic matrices (where cells are encapsulated), which concerns determination of the Green’s function matching their geometry and boundary conditions.

To increase fidelity in the estimate of traction forces for extreme conditions such as a sparse sampling of deformation field or targeting small focal adhesions, we numerically solved the singularity problem of the Green’s function to avoid exclusion of singular point regions that could contain representative deformation indicators for such extreme conditions.

The proposed approach to adapting the Green’s function for the specific 3D cell culture situation was examined in a single case experimental study of endothelial cells in sprouting angiogenesis.

中文翻译：

---

基于细胞培养基质的几何和边界条件确定格林函数的三维牵引力重构函数

细胞迁移在许多物理过程和病理状况（例如肿瘤浸润和转移）的开始和进展中起着特别重要的作用。正在开发高分辨率和高精度的三维牵引力显微镜（TFM），以揭示在类似生物的环境中细胞迁移的潜在机械过程。基于线性弹性的TFM（LETM）作为一种主流方法依赖于格林函数（将牵引力与基质变形相关联），如先前的2D研究所建议的那样，其固有的边界条件和基质的几何形状可能会显着影响结果。在这项研究中，我们通过对3D环境中这种紧密联系进行了研究，方法是对一个单元进行建模，以感测围绕在其周围的3D矩阵的固定边界，并比较了格林函数的三个不同解的重构牵引力，其中包括采用改编的Mindlin方法得出的完全匹配解。为了增加对极端条件（例如，变形场的稀疏采样或针对小的粘着力）的极端条件下的牵引力估算的逼真度，我们以非常规方式数值解决了格林函数的奇异性问题，以避免排除奇点区域。可能包含此类极端条件下的代表性变形指标。对内皮细胞的多细胞结构进行了单例实验研究，该内皮细胞在血管生成的早期阶段才渗透到凝胶中。包括使用改编的Mindlin方法得出的完全匹配的解决方案。为了提高在极端条件下（例如，变形场的稀疏采样或针对小的粘着力）的牵引力估计值的逼真度，我们以非常规方式用数值方法解决了格林函数的奇异性问题，以避免排除奇点区域。可以包含此类极端条件下的代表性变形指标。对内皮细胞的多细胞结构进行了单例实验研究，该内皮细胞在血管生成的早期阶段才渗透到凝胶中。包括使用改编的Mindlin方法得出的完全匹配的解决方案。为了增加对极端条件（例如，变形场的稀疏采样或针对小的粘着力）的极端条件下的牵引力估算的逼真度，我们以非常规方式数值解决了格林函数的奇异性问题，以避免排除奇点区域。可以包含此类极端条件下的代表性变形指标。对内皮细胞的多细胞结构进行了单例实验研究，该内皮细胞在血管生成的早期阶段才渗透到凝胶中。我们用一种非常规的方式用数值方法解决了格林函数的奇异性问题，以避免排除可能包含此类极端条件下具有代表性的变形指标的奇异点区域。对内皮细胞的多细胞结构进行了单例实验研究，该内皮细胞在血管生成的早期阶段才渗透到凝胶中。我们以非常规方式用数值方法解决了格林函数的奇异性问题，以避免排除可能包含此类极端条件下具有代表性的变形指标的奇异点区域。对内皮细胞的多细胞结构进行了单例实验研究，该内皮细胞在血管生成的早期阶段才渗透到凝胶中。

重要声明

这项研究的重点是关于基于线性弹性的TFM扩展以处理物理上现实的矩阵（封装了细胞）的基本问题，该问题涉及确定格林函数与其几何形状和边界条件相匹配的功能。

为了增加对极端条件（例如，变形场的稀疏采样或针对小的粘着力）的牵引力估计值的保真度，我们在数值上解决了格林函数的奇异性问题，以避免排除可能包含具有代表性变形指标的奇异点区域这样的极端条件。

在针对内皮细胞发芽血管生成的单例实验研究中，对提出的使Green功能适应特定3D细胞培养情况的方法进行了研究。