Biological soft and fluid tissues, due to the high percentage of water, are nearly incompressible and consequently their velocity fields are nearly divergence-free. The two most commonly used types of vector field representation are piece-wise continuous representations, which are used in the finite element method (FEM), and discrete representations, which are used in the finite difference method (FDM). In both FEM and FDM frameworks divergence-free vector fields are approximated, i.e. they are not exactly divergence-free and both representation types require a relatively large number of degrees freedom. We showed that a continuous, divergence-free vector field model can effectively represent myocardial and blood velocity with a relatively small number of degrees of freedom [1]. This model, referred to as Divergence-free Interpolation (DFI), is illustrated in Fig. 1 on blood velocity measurements obtained from phase-velocity MRI.

Figure 1: Comparison of velocity fields obtained from the original phase-velocity MRI and from DFI of the velocity data. The location of the inferior vena cava jet-like pattern is properly resolved by DFI, while errant noise vectors are filtered out. The figure is from [1] and it is used with permission; Copyright © 2009 IEEE; All rights reserved.

References:

[1] Skrinjar, O., Bistoquet, A., Oshinski, J., Sundareswaran, K., Frakes, D., Yoganathan, A., "A Divergence-Free Vector Field Model for Imaging Applications", IEEE International Symposium on Biomedical Imaging, Boston, MA, pp. 891-894, June/July 2009. LINK