4D reconstruction of developmental trajectories using spherical harmonics

Abstract

Although the full embryonic development of species such as Drosophila and zebrafish can be 3D imaged in real time, this is not true for mammalian organs, as normal organogenesis cannot be recapitulated in vitro. Currently available 3D data is therefore ex vivo images which provide only a snap shot of development at discrete moments in time. Here we propose a computer-based approach to recreate the continuous evolution in time and space of developmental stages from 3D volumetric images. Our method uses the mathematical approach of spherical harmonics to re-map discrete shape data into a space in which facilitates a smooth interpolation over time. We tested our approach on mouse limb buds (from E10 to E12.5) and embryonic hearts (from 10 to 29 somites). A key advantage of the method is that the resulting 4D trajectory takes advantage of all the available data (i.e. it is not dominated by the choice of a few "ideal" images), while also being able to interpolate well through time intervals for which there is little or no data. This method not only provides a quantitative basis for validating predictive models, but it also increases our understanding of morphogenetic processes. We believe this is the first data-driven quantitative 4D description of limb morphogenesis.