Divergence-Free Fitting-Based Incompressible Deformation Quantification of Liver

Liver is an incompressible organ that maintains its volume during the respiration-induced deformation. Quantifying this deformation with the incompressible constraint is significant for liver tracking. The constraint can be accomplished with retaining the divergence-free field obtained by the deformation decomposition. However, the decomposition process is time-consuming, and the removal of non-divergence-free field weakens the deformation. In this study, a divergence-free fitting-based registration method is proposed to quantify the incompressible deformation rapidly and accurately. First, the deformation to be estimated is mapped to the velocity in a diffeomorphic space. Then, this velocity is decomposed by a fast Fourier-based Hodge-Helmholtz decomposition to obtain the divergence-free, curl-free, and harmonic fields. The curl-free field is replaced and fitted by the obtained harmonic field with a translation field to generate a new divergence-free velocity. By optimizing this velocity, the final incompressible deformation is obtained. Moreover, a deep learning framework (DLF) is constructed to accelerate the incompressible deformation quantification. An incompressible respiratory motion model is built for the DLF by using the proposed registration method and is then used to augment the training data. An encoder-decoder network is introduced to learn appearance-velocity correlation at patch scale. In the experiment, we compare the proposed registration with three state-of-the-art methods. The results show that the proposed method can accurately achieve the incompressible registration of liver with a mean liver overlap ratio of 95.33%. Moreover, the time consumed by DLF is nearly 15 times shorter than that by other methods.