Ph.D Student | Safadi Mahmoud |
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Subject | An Eulerian Theoretical Structure for Modeling Growth, Remodeling and Morphogenesis of Soft Tissues |

Department | Department of Mechanical Engineering |

Supervisor | Professor Miles Rubin |

Full Thesis text |

Eulerian constitutive equations for modeling growth, remodeling and morphogenesis in soft tissues are proposed. The model assumes a simple continuum with a single velocity field. In contrast with many other formulations, evolution equations are proposed directly for a scalar measure of elastic dilatation and a tensorial measure of elastic distortion. The evolution equation for elastic dilatation includes a rate of mass supply or removal that controls volumetric growth and causes the elastic dilatation to evolve towards its homeostatic value. Similarly, the evolution equation for the elastic distortion includes a rate of growth that causes the elastic distortional deformation tensor to evolve towards its homeostatic value. Since the rate of growth appears in the evolution equations and not a growth tensor, it is possible to model the combined effects of multiple growth mechanisms simultaneously. Here, the constitutive equations for stress are hyperelastic in the sense that the they are determined by derivatives of the Helmholtz free energy function. Also, a robust, strongly objective, numerical algorithm is developed to integrate the nonlinear evolution equations. One advantage of this new theory is that it does not depend on arbitrary choices of a reference configuration, an unstressed intermediate configuration, a total deformation measure, or a growth deformation measure.

Remodeling in arteries is modeled using this new Eulerian approach. In particular, most analysis consider a loaded artery as a circular cylindrical tube which can be cut transmurally and unloaded elastically to an open sector of a circular cylindrical ring that is stress-free pointwise. The opening angle determines the residual stresses in the unloaded intact artery. The theory treats remodeling as an inelastic process which causes the artery to attain its homeostatic state. Compatible elastic deformations from this loaded homeostatic state determine the stress distribution in the loaded artery. The results indicate that when inelasticity of remodeling is included in the analysis, the stress distribution in the intact artery is no longer uniquely determined by the geometry of the open ring, which is also not stress-free pointwise.

This new model is also used to propose specific constitutive equations for early heart morphogenesis. In particular, an orthonormal triad of vectors is introduced to characterize the deformations of material fibers and surfaces. The idea of growth at zero stress is adopted to specify the homeostasis rate and the corresponding homeostatic values. The strain energy function of the tissue depends on the elastic dilatation, a scalar measure of elastic distortion, elastic strains of material line elements, elastic shearing strain and an elastic measure of area strain. Examples are presented related to different stages of cardiac c-looping associated with a simplified cardiac tube model.