Multi-Objective Optimisation of Growth-Mechanics Problem in the Developing Mammalian Heart

Abstract

The motivation behind this study is to investigate the early stages of heart formation. In the linear heart tube, even after a noticeable anatomical shape change, the tubular structure persists. This work aims to develop a continuum mechanics mouse model for cardiac looping in various heart formation stages. The geometry of the heart has been segmented in a number of samples produced from 2D light microscopy images. Kinematics of the developing heart have been developed through identifying and tracking a number of anatomical regions, which have been used to register the samples throughout the stages. The mathematical description of this mapping throughout the developmental stages is in the form of the deformation gradient tensor. In order to understand the mechanics of the heart development, the forward dynamic problem is being solved under the combined effects of growth and elasticity. The differential fibre growth model is seen in different hyperelastic behaviours. As the growth model varies spatially and temporally during the early developmental stages, anisotropic growth with spatial distribution has been used to drive the mesh to the grown state. Due to large deformation in this soft tissue growth-mechanics problem, a number of quasi-static simulations are performed to achieve the proper temporally-changing rates over time. Heuristic optimisation methods are used to solve the inverse dynamic problem and bring the optimum rates of the fibre growth model. To this goal, an optimisation framework is established and tested via standard cylindrical mesh samples. An estimation of the gradient of the growth field can be made according to the gradient of the deformation gradient tensor which leads to principal stretches. Therefore, the objective function, which was originally only measuring the t of the grown tissue with the target, becomes penalised with a smoothing term to make the pattern of the growth field similar to the gradient of a field of growth rates coming from the deformation gradient tensor in the stages. For the real geometry, growth model is inspired from the pattern of the deformation gradient field between stages. During this study, an examination of the effect of boundary conditions on the dorsal mesocardium is undertaken. It was found that a number of factors are important in modelling this growth- biomechanics problem, such as implementing essential boundary conditions, having compatibility between the kinematics factors and growth-dependent factors, and decreasing the dependency to the elastic behaviour. Additionally, in order to achieve reliable results, both understanding the nature of the growth rates as a target field for optimisation, and targeting a proper objective are necessary. However, the dynamics of the growing heart are purely investigated at the tissue scale, and there is a need for a multi-scale model to create a link between the cell scale and the tissue scale.