A new approach to modeling early cardiac morphogenesis during c-looping

The objective of this paper is to formulate a theoretical structure for modeling early cardiac morphogenesis based on a recently developed Eulerian formulation of growth of soft tissues which is insensitive to unphysical arbitrary specifications of total deformation, growth deformation, a reference configuration and an intermediate stress-free configuration. In this theory, Eulerian evolution equations are proposed for an elastic dilatation Je and a second order elastic distortional deformation tensor Be′. These evolution equations depend on the velocity gradient and they include homeostasis, which is the inelastic rate process that causes a tendency for Je and Be′ to approach their homeostatic values. Specific new expressions are proposed for scalar measures of elastic strains of material line elements, the elastic shear strain between a pair of material fibers and an elastic measure of area strain, which model different stages of c-looping associated with a simplified heart tube model of cardiac morphogenesis. The strain energy function and stress depend on these elastic deformation measures. Also, robust strongly objective numerical algorithms are developed for integrating the evolution equations. Future work is planned to implement the model into a finite element code for calculations of more realistic heart geometry and for study of constitutive models of homeostasis and the homeostatic state.