Direct Estimation of Biological Growth Properties from Image Data Using the “GRID” Model

Abstract

Having acquired images of a growing organism, the question arises how they can be used to infer the properties of growth. We address this challenging image understanding problem by using the GRID (Growth as Random Iterated Diffeomorphisms) model for biological growth. In the GRID model, growth patterns are composed of smaller, local deformations, each resulting from elementary biological events (e.g.,cell division). A large number of such biological events, each occurring randomly and independently from one another, results in a visible growth pattern or biological shape changes as seen in images. A biological transformation underlying observed shape changes is a solution to a GRID visible growth differential equation. We propose its automatic generation via direct estimation of the growth magnitudes from image data. The growth magnitude is a GRID parameter that characterizes a local expansion/contraction rate throughout the organism’s domain. The estimation algorithm is based on the unconstrained optimal control problem formulation expressed in Darcyan coordinates of the organism’s domain and consequent application of Polak-Ribiere minimization routine. We demonstrate the proposed inference method using confocal micrographs of the Drosophila wing disc at larval stage of development.