Soft tissue modelling plays a significant role in surgery simulation as well as surgical procedure planning and training. However, it is a challenging research task to satisfy both physical realism and real-time simulation for soft tissue deformation. The finite element method (FEM) is a representative strategy for modelling of soft tissue deformation with highly physical realism. However, it suffers from expensive computations, unable to meet the requirement of real-time simulation. This paper proposes a novel method by combining FEM with the Kalman filter for real-time and accurate modelling of soft tissue deformation. The novelty of this method is that soft tissue deformation is formulated as a filtering identification process to online estimate soft tissue deformation from local measurement of displacement. To construct the discrete system state equation for filtering estimation, soft tissue deformation is discretised based on elastic theory in the space domain by FEM and is further discretised in the time domain by using the Wilson- $\theta $ implicit integration to solve the dynamic equilibrium equation of FEM deformation modelling. Subsequently, a Kalman filter is developed for online estimation and analysis of soft tissue deformation according to local measurement of displacement. Interactive tool-tissue interaction with haptic feedback is also achieved for surgery simulation. The presented method significantly improves the computational performance of the traditional FEM, but still maintains a similar level of accuracy. It not only achieves the real-time performance, but also exhibits the similar deformation behaviours as the traditional FEM and enables the use of large time steps to improve the simulation efficiency.

中文翻译：

---

用于实时软组织建模的卡尔曼滤波器有限元方法

软组织建模在手术模拟以及手术程序规划和培训中起着重要作用。然而，同时满足软组织变形的物理真实性和实时模拟是一项具有挑战性的研究任务。有限元法 (FEM) 是一种具有高度物理真实感的软组织变形建模的代表性策略。但其计算量大，无法满足实时仿真的要求。本文提出了一种将 FEM 与卡尔曼滤波器相结合的新方法，用于实时准确地建模软组织变形。该方法的新颖之处在于将软组织变形表述为过滤识别过程，以根据位移的局部测量在线估计软组织变形。为了构建用于滤波估计的离散系统状态方程，软组织变形基于弹性理论在空间域通过 FEM 离散，并通过使用 Wilson-$\theta $ 隐式积分在时域进一步离散以求解动态平衡有限元变形建模方程。随后，根据位移的局部测量，开发了卡尔曼滤波器用于软组织变形的在线估计和分析。还实现了具有触觉反馈的交互式工具-组织交互用于手术模拟。所提出的方法显着提高了传统 FEM 的计算性能，但仍保持相似的精度水平。它不仅实现了实时性能，