Abstract
We study a shape evolution framework in which the deformation of shapes from time t to \(t+dt\) is governed by a regularized anisotropic elasticity model. More precisely, we assume that at each time shapes are infinitesimally deformed from a stress-free state to an elastic equilibrium as a result of the application of a small force. The configuration of equilibrium then becomes the new resting state for subsequent evolution. The primary motivation of this work is the modeling of slow changes in biological shapes like atrophy, where a body force applied to the volume represents the location and impact of the disease. Our model uses an optimal control viewpoint with the time derivative of force interpreted as a control, deforming a shape gradually from its observed initial state to an observed final state. Furthermore, inspired by the layered organization of cortical volumes, we consider a special case of our model in which shapes can be decomposed into a family of layers (forming a “foliation”). Preliminary experiments on synthetic layered shapes in two and three dimensions are presented to demonstrate the effect of elasticity.
Laurent Younes partially supported by NIH 1R01DC016784-01.
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References
Amar, M.B., Goriely, A.: Growth and instability in elastic tissues. J. Mech. Phys. Solids 53(10), 2284–2319 (2005)
Beg, M.F., Miller, M.I., Trouvé, A., Younes, L.: Computing large deformation metric mappings via geodesic flows of diffeomorphisms. Int. J. Comput. Vis. 61(2), 139–157 (2005)
Braak, H., Braak, E.: Neuropathological stageing of Alzheimer-related changes. Acta Neuropathol. 82(4), 239–259 (1991)
Ciarlet, P.G.: Three-Dimensional Elasticity, vol. 20. Elsevier, North Holland (1988)
Das, S.R., Avants, B.B., Grossman, M., Gee, J.C.: Registration based cortical thickness measurement. Neuroimage 45(3), 867–879 (2009)
DiCarlo, A., Quiligotti, S.: Growth and balance. Mech. Res. Commun. 29(6), 449–456 (2002)
Durrleman, S., Pennec, X., Trouvé, A., Braga, J., Gerig, G., Ayache, N.: Toward a comprehensive framework for the spatiotemporal statistical analysis of longitudinal shape data. Int. J. Comput. Vis. 103(1), 22–59 (2013)
Grenander, U., Srivastava, A., Saini, S.: A pattern-theoretic characterization of biological growth. IEEE Trans. Med. Imaging 26(5), 648–659 (2007)
Gris, B., Durrleman, S., Trouvé, A.: A sub-riemannian modular approach for diffeomorphic deformations. In: Nielsen, F., Barbaresco, F. (eds.) GSI 2015. LNCS, vol. 9389, pp. 39–47. Springer, Cham (2015). https://doi.org/10.1007/978-3-319-25040-3_5
Gurtin, M.E.: The linear theory of elasticity. In: Truesdell, C. (ed.) Linear Theories of Elasticity and Thermoelasticity, pp. 1–295. Springer, Heidelberg (1973). https://doi.org/10.1007/978-3-662-39776-3_1
Jones, S.E., Buchbinder, B.R., Aharon, I.: Three-dimensional mapping of cortical thickness using Laplace’s equation. Hum. Brain Mapp. 11(1), 12–32 (2000)
Khanal, B., Lorenzi, M., Ayache, N., Pennec, X.: A biophysical model of brain deformation to simulate and analyze longitudinal MRIs of patients with Alzheimer’s disease. NeuroImage 134, 35–52 (2016)
Lubarda, V.A., Hoger, A.: On the mechanics of solids with a growing mass. Int. J. Solids Struct. 39(18), 4627–4664 (2002)
Marsden, J.E., Hughes, T.J.: Mathematical Foundations of Elasticity. Courier Corporation, North Chelmsford (1994)
Porumbescu, S.D., Budge, B., Feng, L., Joy, K.I.: Shell maps. ACM Trans. Graph. (TOG) 24(3), 626–633 (2005)
Qiu, A., Albert, M., Younes, L., Miller, M.I.: Time sequence diffeomorphic metric mapping and parallel transport track time-dependent shape changes. NeuroImage 45(1), S51–S60 (2009)
Ratnanather, J.T., Arguillère, S., Kutten, K.S., Hubka, P., Kral, A., Younes, L.: 3D normal coordinate systems for cortical areas. arXiv preprint arXiv:1806.11169 (2018)
Rodriguez, E.K., Hoger, A., McCulloch, A.D.: Stress-dependent finite growth in soft elastic tissues. J. Biomech. 27(4), 455–467 (1994)
Tallinen, T., Chung, J.Y., Rousseau, F., Girard, N., Lefèvre, J., Mahadevan, L.: On the growth and form of cortical convolutions. Nat. Phys. 12(6), 588 (2016)
Younes, L.: Shapes and Diffeomorphisms, vol. 171. Springer, Heidelberg (2010). https://doi.org/10.1007/978-3-642-12055-8
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Hsieh, DN., Arguillère, S., Charon, N., Miller, M.I., Younes, L. (2019). A Model for Elastic Evolution on Foliated Shapes. In: Chung, A., Gee, J., Yushkevich, P., Bao, S. (eds) Information Processing in Medical Imaging. IPMI 2019. Lecture Notes in Computer Science(), vol 11492. Springer, Cham. https://doi.org/10.1007/978-3-030-20351-1_50
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