Traction force microscopy is a methodology that enables to estimate cellular forces from the measurement of the displacement field of an extracellular matrix (ECM)-mimicking hydrogel that a cell is mechanically interacting with. In this paper, a new inverse and physically-consistent methodology is developed and implemented in the context of 3D nonlinear elasticity. The proposed method searches for a displacement field that approximates the measured one, through the imposition of fulfillment of equilibrium with real and known forces acting in the hydrogel. The overall mathematical formulation leads to a constrained optimisation problem that is treated through a Lagrange operator and that is solved numerically by means of a nonlinear finite element framework. In order to illustrate the potential and enhanced accuracy of the proposed inverse method, it is applied to a total of 5 different real cases of cells cultured in a 3D hydrogel that is considered to behave as a nonlinear elastic material. Different error indicators are defined in order to compare ground truth simulated displacements and tractions to the ones recovered by the new inverse as well as by the forward method. Results indicate that the evaluation of displacement gradients leads to errors, in terms of recovered tractions, that are more than three times lower (on average) for the inverse method compared to the forward method. They highlight the enhanced accuracy of the developed methodology and the importance of appropriate inverse methods that impose physical constraints to traction and stress recovery in the context of traction force microscopy.

中文翻译：

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牵引力显微镜框架下基于3D非线性物理约束最小化的逆方法

牵引力显微镜是一种方法，它能够通过测量与细胞机械相互作用的细胞外基质 (ECM) 模拟水凝胶的位移场来估计细胞力。在本文中，在 3D 非线性弹性的背景下开发并实施了一种新的逆向和物理一致的方法。所提出的方法通过使用作用在水凝胶中的真实和已知力来实现平衡来搜索与测量值近似的位移场。整体数学公式导致约束优化问题通过拉格朗日算子进行处理，并通过非线性有限元框架进行数值求解。为了说明所提出的逆方法的潜力和提高的准确性，它应用于总共 5 个不同的真实细胞案例，在 3D 水凝胶中培养，该水凝胶被认为表现为非线性弹性材料。定义了不同的误差指标，以便将地面实况模拟位移和牵引力与通过新逆方法和前向方法恢复的位移和牵引力进行比较。结果表明，位移梯度的评估导致在恢复牵引力方面的误差，与正向方法相比，反向方法的误差（平均）低三倍以上。他们强调了所开发方法的准确性提高，以及在牵引力显微镜下对牵引和应力恢复施加物理约束的适当逆方法的重要性。