Diffeomorphic shape evolution coupled with a reaction-diffusion PDE on a growth potential
Authors:
Dai-Ni Hsieh, Sylvain Arguillère, Nicolas Charon and Laurent Younes
Journal:
Quart. Appl. Math. 80 (2022), 23-52
MSC (2020):
Primary 35R37, 35K57, 35Q92
DOI:
https://doi.org/10.1090/qam/1600
Published electronically:
August 24, 2021
MathSciNet review:
4360548
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Abstract: 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.
References
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- Milana Trifkovic, Mehdi Sheikhzadeh, and Sohrab Rohani, Multivariable real-time optimal control of a cooling and antisolvent semibatch crystallization process, AIChE Journal 55 (2009), no. 10, 2591–2602., DOI 10.1002/aic.11868
- Alain Trouvé, An approach of pattern recognition through infinite dimensional group action, Rapport de recherche du LMENS (1995).
- Laurent Younes, Constrained diffeomorphic shape evolution, Found. Comput. Math. 12 (2012), no. 3, 295–325. MR 2915564, DOI 10.1007/s10208-011-9108-2
- Laurent Younes, Gaussian diffeons for surface and image matching within a Lagrangian framework, Geom. Imaging Comput. 1 (2014), no. 1, 141–171. MR 3392141, DOI 10.4310/GIC.2014.v1.n1.a3
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References
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- Sylvain Arguillère, Emmanuel Trélat, Alain Trouvé, and Laurent Younès, Shape deformation and optimal control, Congrès SMAI 2013, ESAIM Proc. Surveys, vol. 45, EDP Sci., Les Ulis, 2014, pp. 300–307 (English, with English and French summaries). MR 3451842, DOI 10.1051/proc/201445031
- N. Aronszajn, Theory of reproducing kernels, Trans. Amer. Math. Soc. 68 (1950), 337–404. MR 51437, DOI 10.2307/1990404
- Naim Bajcinca, Analytic solutions to optimal control problems in crystal growth processes, Journal of Process Control 23 (2013-02), no. 2, 224–241.
- M Faisal Beg, Michael I Miller, Alain Trouvé, and Laurent Younes, Computing large deformation metric mappings via geodesic flows of diffeomorphisms, International journal of computer vision 61 (2005), no. 2, 139–157.
- Martin K. Bernauer and Roland Herzog, Optimal control of the classical two-phase Stefan problem in level set formulation, SIAM J. Sci. Comput. 33 (2011), no. 1, 342–363. MR 2783198, DOI 10.1137/100783327
- Alberto Bressan and Marta Lewicka, A model of controlled growth, Arch. Ration. Mech. Anal. 227 (2018), no. 3, 1223–1266. MR 3744385, DOI 10.1007/s00205-017-1183-3
- Martins Bruveris and François-Xavier Vialard, On completeness of groups of diffeomorphisms, J. Eur. Math. Soc. (JEMS) 19 (2017), no. 5, 1507–1544. MR 3635359, DOI 10.4171/JEMS/698
- Chris Burdzy, Zhen-Qing Chen, and John Sylvester, The heat equation in time dependent domains with insulated boundaries, J. Math. Anal. Appl. 294 (2004), no. 2, 581–595. MR 2061344, DOI 10.1016/j.jmaa.2004.02.032
- Nicolas Charon and Alain Trouvé, The varifold representation of nonoriented shapes for diffeomorphic registration, SIAM J. Imaging Sci. 6 (2013), no. 4, 2547–2580. MR 3138101, DOI 10.1137/130918885
- Philippe G Ciarlet, Three-dimensional elasticity, Vol. 20, Elsevier, 1988.
- Paul Dupuis, Ulf Grenander, and Michael I. Miller, Variational problems on flows of diffeomorphisms for image matching, Quart. Appl. Math. 56 (1998), no. 3, 587–600. MR 1632326, DOI 10.1090/qam/1632326
- Alain Goriely, The mathematics and mechanics of biological growth, Interdisciplinary Applied Mathematics, vol. 45, Springer, New York, 2017. MR 3585488, DOI 10.1007/978-0-387-87710-5
- Thierry Goudon and Alexis Vasseur, Regularity analysis for systems of reaction-diffusion equations, Ann. Sci. Éc. Norm. Supér. (4) 43 (2010), no. 1, 117–142 (English, with English and French summaries). MR 2583266, DOI 10.24033/asens.2117
- Barbara Gris, Stanley Durrleman, and Alain Trouvé, A sub-Riemannian modular framework for diffeomorphism-based analysis of shape ensembles, SIAM J. Imaging Sci. 11 (2018), no. 1, 802–833. MR 3775129, DOI 10.1137/16M1076733
- Dai-Ni Hsieh, On model-based diffeomorphic shape evolution and diffeomorphic shape registration, Ph.D. Thesis, 2021.
- Dai-Ni Hsieh, Sylvain Arguillère, Nicolas Charon, Michael I. Miller, and Laurent Younes, A model for elastic evolution on foliated shapes, Information processing in medical imaging, 2019, pp. 644–655.
- Dai-Ni Hsieh, Sylvain Arguillère, Nicolas Charon, and Laurent Younes, Mechanistic Modeling of Longitudinal Shape Changes: equations of motion and inverse problems, arXiv:2003.05512 [math] (2020-03).
- J. D. Humphrey, Continuum biomechanics of soft biological tissues, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 459 (2003), no. 2029, 3–46. MR 1993342, DOI 10.1098/rspa.2002.1060
- Sarang C. Joshi and Michael I. Miller, Landmark matching via large deformation diffeomorphisms, IEEE Trans. Image Process. 9 (2000), no. 8, 1357–1370. MR 1808275, DOI 10.1109/83.855431
- Sue Kulason, Michael I. Miller, Alain Trouvé, and Alzheimer’s Disease Neuroimaging Initiative, Reaction-Diffusion Model of Cortical Atrophy Spread during Early Stages of Alzheimer’s Disease, bioRxiv (2020-11), 2020.11.02.362855.
- Olga A Ladyženskaja, Vsevolod Alekseevich Solonnikov, and Nina N Uralceva, Linear and quasi-linear equations of parabolic type, Translations of Mathematical Monographs, vol. 23, American Mathematical Society, Providence, RI, 1988.
- Marta Lewicka, L. Mahadevan, and Mohammad Reza Pakzad, The Föppl-von Kármán equations for plates with incompatible strains, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 467 (2011), no. 2126, 402–426. With supplementary data available online. MR 2748099, DOI 10.1098/rspa.2010.0138
- J.-L. Lions and E. Magenes, Non-homogeneous boundary value problems and applications. Vol. II, Die Grundlehren der mathematischen Wissenschaften, Band 182, Springer-Verlag, New York-Heidelberg, 1972. Translated from the French by P. Kenneth. MR 0350178
- Jerrold E. Marsden and Thomas J. R. Hughes, Mathematical foundations of elasticity, Dover Publications, Inc., New York, 1994. Corrected reprint of the 1983 original. MR 1262126
- Andreas Menzel and Ellen Kuhl, Frontiers in growth and remodeling, Mechanics Research Communications 42 (2012-06), 1–14 (en).
- Edward K Rodriguez, Anne Hoger, and Andrew D McCulloch, Stress-dependent finite growth in soft elastic tissues, Journal of biomechanics 27 (1994), no. 4, 455–467.
- Milana Trifkovic, Mehdi Sheikhzadeh, and Sohrab Rohani, Multivariable real-time optimal control of a cooling and antisolvent semibatch crystallization process, AIChE Journal 55 (2009), no. 10, 2591–2602.
- Alain Trouvé, An approach of pattern recognition through infinite dimensional group action, Rapport de recherche du LMENS (1995).
- Laurent Younes, Constrained diffeomorphic shape evolution, Found. Comput. Math. 12 (2012), no. 3, 295–325. MR 2915564, DOI 10.1007/s10208-011-9108-2
- Laurent Younes, Gaussian diffeons for surface and image matching within a Lagrangian framework, Geom. Imaging Comput. 1 (2014), no. 1, 141–171. MR 3392141, DOI 10.4310/GIC.2014.v1.n1.a3
- Laurent Younes, Shapes and diffeomorphisms, 2nd ed., Applied Mathematical Sciences, vol. 171, Springer, Berlin, 2019. MR 3932112, DOI 10.1007/978-3-662-58496-5
- Laurent Younes, Barbara Gris, and Alain Trouvé, Sub-Riemannian Methods in Shape Analysis, Handbook of Variational Methods for Nonlinear Geometric Data, 2020, pp. 463–495.
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Additional Information
Dai-Ni Hsieh
Affiliation:
Department of Applied Mathematics and Statistics, Johns Hopkins University, Baltimore, MD 21218
MR Author ID:
1036115
ORCID:
0000-0002-1365-4877
Email:
dnhsieh@jhu.edu
Sylvain Arguillère
Affiliation:
Laboratoire Paul Painlevé, University of Lille, France
ORCID:
0000-0002-8916-0291
Email:
sylvain.arguillere@univ-lille.fr
Nicolas Charon
Affiliation:
Department of Applied Mathematics and Statistics, Johns Hopkins University, Baltimore, MD 21218
MR Author ID:
1045405
ORCID:
0000-0002-6032-247X
Email:
charon@cis.jhu.edu
Laurent Younes
Affiliation:
Department of Applied Mathematics and Statistics, Johns Hopkins University, Baltimore, MD 21218
MR Author ID:
238511
Email:
laurent.younes@jhu.edu
Received by editor(s):
April 7, 2021
Received by editor(s) in revised form:
July 12, 2021
Published electronically:
August 24, 2021
Additional Notes:
The third author acknowledges the support of the NSF through the grant DMS-1945224.
Article copyright:
© Copyright 2021
Brown University