This dissertation presents three treatments of mathematical modeling to two relevant areas of biological application.We apply foam physics to the morphometry of cell packings of notochords, elastic membrane physics to notochord biomechanics, and the theory of morphoelasticity to morphogenesis in the developing stomach.
The notochord is the defining feature of chordates, and during development it lengthens the embryo, provides structural support, and in many organisms acts as a template for spine development. It consists of inner, vacuolated chordocytes surrounded by an epithelial sheath of chordoblasts. Previous foam models have investigated howgeometric and physical ratios might control chordocyte packing within the notochord.We examine the interplay between these ratios in the two lowest order regular patterns, determine a functional relationship between them, and compare our results to the previous work. Further, we employ an elastic membrane model to probe how these two lowest order patterns might affect overall biomechanics of the structure, providing evidence for the structural importance of both packing pattern and pattern orientation. During development, the embryonic stomach anlage undergoes asymmetric radial cell rearrangement of the left mesoderm, giving rise to the curvature found in the mature organ. This rearrangement might occur by a number of mechanisms; in particular, we model two the hypotheses – radial thinning combined with volume-preserving, equal axial and circumferential elongation, and radial thinning combined with volume preserving axial elongation – using a morphoelastic framework. Morphoelasticity is an area of continuum mechanics which decomposes deformations into growth and elasticity components, allowing an investigation of large strains. Under an assumption that the time scale of growth is longer than that of residual
stress, this technique can be iterated.