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

This paper studies a longitudinal shape transformation model in which shapes are deformed in response to an internal growth potential that evolves according to an advection reaction diffusion process. This model extends prior works that considered a static growth potential, i.e., the initial growth potential is only advected by diffeomorphisms. We focus on the mathematical study of the corresponding system of coupled PDEs describing the joint dynamics of the diffeomorphic transformation together with the growth potential on the moving domain. Specifically, we prove the uniqueness and long time existence of solutions to this system with reasonable initial and boundary conditions as well as regularization on deformation fields. In addition, we provide a few simple simulations of this model in the case of isotropic elastic materials in 2D.

[1]  Alain Trouvé,et al.  Computing Large Deformation Metric Mappings via Geodesic Flows of Diffeomorphisms , 2005, International Journal of Computer Vision.

[2]  Alain Trouvé,et al.  The Varifold Representation of Nonoriented Shapes for Diffeomorphic Registration , 2013, SIAM J. Imaging Sci..

[3]  A. Trouvé.  An Approach of Pattern Recognition through Infinite Dimensional Group Action , 1995 .

[4]  L. Younes.  Gaussian diffeons for surface and image matching within a Lagrangian framework , 2014 .

[5]  Alain Trouvé,et al.  A Sub-Riemannian Modular Framework for Diffeomorphism-Based Analysis of Shape Ensembles , 2018, SIAM J. Imaging Sci..

[6]  N. Aronszajn.  Theory of Reproducing Kernels. , 1950 .

[7]  Michael I. Miller,et al.  A Model for Elastic Evolution on Foliated Shapes , 2018, IPMI.

[8]  L. Younes,et al.  Shape deformation and optimal control , 2014 .

[9]  A. Bressan,et al.  A Model of Controlled Growth , 2016, 1608.08645.

[10]  Michael I. Miller,et al.  Landmark matching via large deformation diffeomorphisms , 2000, IEEE Trans. Image Process..

[11]  J D Humphrey,et al.  Perspectives on biological growth and remodeling. , 2011, Journal of the mechanics and physics of solids.

[12]  A. Trouvé,et al.  Reaction-Diffusion Model of Cortical Atrophy Spread during Early Stages of Alzheimer’s Disease , 2020 .

[13]  Alain Trouvé,et al.  Sub-Riemannian Methods in Shape Analysis , 2020 .

[14]  A. Goriely.  The Mathematics and Mechanics of Biological Growth , 2017 .

[15]  O. Ladyženskaja.  Linear and Quasilinear Equations of Parabolic Type , 1968 .

[16]  A. McCulloch,et al.  Stress-dependent finite growth in soft elastic tissues. , 1994, Journal of biomechanics.

[17]  N. Bajçinca.  Analytic solutions to optimal control problems in crystal growth processes , 2013 .

[18]  O. A. Ladyzhenskai︠a︡,et al.  Linear and Quasi-linear Equations of Parabolic Type , 1995 .

[19]  Paul Dupuis,et al.  Variational problems on ows of di eomorphisms for image matching , 1998 .

[20]  T. Goudon,et al.  Regularity analysis for systems of reaction-diffusion equations , 2010 .

[21]  T. R. Hughes,et al.  Mathematical foundations of elasticity , 1982 .

[22]  Sohrab Rohani,et al.  Multivariable Real-Time Optimal Control of a Cooling and Antisolvent Semibatch Crystallization Process , 2009 .

[23]  Martins Bruveris,et al.  On Completeness of Groups of Diffeomorphisms , 2014, 1403.2089.

[24]  P. G. Ciarlet,et al.  Three-dimensional elasticity , 1988 .

[25]  Laurent Younes,et al.  Mechanistic Modeling of Longitudinal Shape Changes: Equations of Motion and Inverse Problems , 2022, SIAM Journal on Applied Dynamical Systems.

[26]  L. Mahadevan,et al.  The Föppl-von Kármán equations for plates with incompatible strains , 2011, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[27]  John Sylvester,et al.  The heat equation in time dependent domains with insulated boundaries , 2004 .

[28]  L. Younes.  Shapes and Diffeomorphisms , 2010 .

[29]  J. Lions,et al.  Non-homogeneous boundary value problems and applications , 1972 .

[30]  Roland Herzog,et al.  Optimal Control of the Classical Two-Phase Stefan Problem in Level Set Formulation , 2011, SIAM J. Sci. Comput..

[31]  Jay D. Humphrey,et al.  Review Paper: Continuum biomechanics of soft biological tissues , 2003, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[32]  A. Mcguire,et al.  What is it to be a model? , 2000, HEPAC Health Economics in Prevention and Care.

[33]  Ellen Kuhl,et al.  Frontiers in growth and remodeling. , 2012, Mechanics research communications.

[34]  Laurent Younes,et al.  Constrained Diffeomorphic Shape Evolution , 2012, Found. Comput. Math..