Mechanistic Modeling of Longitudinal Shape Changes: equations of motion and inverse problems

Abstract

This paper examines a longitudinal shape evolution model in which a 3D volume progresses through a family of elastic equilibria in response to the time-derivative of an internal force, or yank, with an additional regularization to ensure diffeomorphic transformations. We consider two different models of yank and address the long time existence and uniqueness of solutions for the equations of motion in both models. In addition, we derive sufficient conditions for the existence of an optimal yank that best describes the change from an observed initial volume to an observed volume at a later time. The main motivation for this work is the understanding of processes such as growth and atrophy in anatomical structures, where the yank could be roughly interpreted as a metabolic event triggering morphological changes. We provide preliminary results on simple examples to illustrate, under this model, the retrievability of some attributes of such events.

Diffeomorphic shape evolution coupled with a reaction-diffusion PDE on a growth potential

“…This paper follows and completes Hsieh et al [2020] (see also Hsieh et al [2019]), which adopts a similar approach with a functional dependency of the growth potential on the diffeomorphic flow. This assumption is relaxed here, since the potential follows its own PDE, with an evolution coupled with that of the shape.…”

“…In this paper, following Hsieh et al [2020], we fix a weight ω > 0 to combine the LDDMM model and the deformable solid model, and define…”

“…It remains to model a yank j(•) that induces the deformation on M t . The model considered in Hsieh et al [2020] starts with a fixed positive function g 0 ∈ C 1 (M 0 , R), with negligible values on the boundary ∂M 0 . In the diseased tissue analogy, this function may be thought of as describing an initial physiological impact of a disease, for example the density of dying cells, or of some protein responsible for tissue remodeling.…”

“…In the diseased tissue analogy, this function may be thought of as describing an initial physiological impact of a disease, for example the density of dying cells, or of some protein responsible for tissue remodeling. Then, for a deformation ϕ ∈ Diff m (R d ), [Hsieh et al, 2020] defines the corresponding yank j ϕ along ϕ(M 0 ) to be the negative gradient of a some function Q : R + → R + of the transported function g 0 •ϕ −1 , so that the yank pulls the shape towards places where g 0 • ϕ −1 is highest. Formally, this gives:…”

“…It was studied in Hsieh et al [2020] and proved to have solutions over arbitrary time intervals. In the same paper, the issue of identifying g 0 among a parametrized family of candidates given the initial state M 0 and a deformed state at time T , M T = ϕ(T, M 0 ), was also considered.…”