Model Order Reduction and Characterization for Soft Tissue Deformation Modelling

<p dir="ltr">Accurately modelling the deformations of biological tissues is essential for various medical applications, such as surgical simulation, disease diagnosis, palpation, surgical planning, and robot-assisted minimally invasive surgery. However, achieving both high physical fidelity and runtime computational efficiency is challenging, especially in real-time simulations. Although the finite element method provides high physical realism for soft tissue deformation modelling, its high computational cost makes it unsuitable for real-time applications.</p><p dir="ltr">To overcome this issue, this thesis proposes novel methods that combine model order reduction techniques with the Kalman filter to enable real-time and accurate modelling of nonlinear soft tissue deformation. Based on the model order reduction techniques, we formulate the deformable modelling problem as a reduced order filtering problem, accurately estimating deformation from local displacement measurements in real-time. The soft tissue deformation is spatially discretized using a hyperelasticity-based finite element method and further formulated as a state-space equation for filter estimation. The order of this state space equation was then reduced using a proper orthogonal decomposition (POD) to reduce the computational cost. Based on this reduced-order state space equation, an extended Kalman filter was constructed to calculate the non-linear behaviour of the physical deformation of the tissue online. However, POD-based modelling requires snapshots to construct the reduced basis for reduced-order modelling, but for some very complex systems, snapshots are difficult to obtain. To tackle this problem, a proper generalized decomposition (PGD) was developed. Therefore, we extend the POD-based approach to PGD reduced-order modelling to overcome this problem. The novelty of the PGD-based reduced-order modelling approach is that the solution to the reduced-order problem is expressed as a finite sum of separated representations, which is obtained by approximation using the enrichment process of the greedy algorithm. The reduced-order modelling of the problem is performed based on these separated representations without the need for a prior snapshot. Through a series of simulation analyses, the proposed PGD based reduction algorithms are shown to have good real-time performance and accuracy. We then introduce the dynamic mode decomposition (DMD) model order reduction technique, which models soft tissue deformation on top of the full model by using finite element modelling or experimental data, and then extracts and stores the most basic feature information of the model, which is used to construct a reduced-order model with smaller dimensions. Furthermore, compared to POD or PGD, the DMD based method does not require prior knowledge of the system governing equations and only requires a set of system snapshots. Simulation results show that the proposed DMD method cannot achieve only real-time, but also good simulation accuracy.</p><p dir="ltr">For realistic modelling of human soft tissues, it is also vital to consider complex material properties. Hence, we introduce a new approach that dynamically incorporates soft tissue characterization into the modelling process, enhancing accuracy. Our method addresses the non-linear deformation of soft tissues with unknown mechanical properties by treating it as a non-linear filter identification problem. This enables dynamic identification of mechanical properties and estimation of non-linear tissue behaviour. By combining maximum likelihood theory, non-linear filtering, and the non-linear finite element method (NFEM), we accurately model non-linear tissue deformation based on the dynamic identification of homogeneous tissue properties. Through NFEM-based discretization of tissue deformation and the establishment of non-linear state space equations for dynamic filtering, we capture the deformation process effectively. Furthermore, we have developed a maximum likelihood algorithm and an extended Kalman filter to dynamically identify and estimate tissue mechanical properties and non-linear deformation. Simulation and experimental analysis demonstrate that our method overcomes NFEM's computational limitations while maintaining high accuracy in modelling the deformation of homogeneous tissues. Moreover, our method effectively identifies tissue mechanical properties during the deformation modelling process.</p>