Computational Multiscale Methods for Inverse Estimation of Effective Properties of Poroelastic Tissues
Final Report Abstract
Tissue imaging techniques, such as magnetic resonance elastography (MRE), are rapidly evolving into clinical tools to support non-invasive diagnosis and staging of diseases such as cancer. However, realising their full potential requires mathematical models that link image-scale mechanical properties to sub-resolution microvascular processes that drive pathology. The aim of this project was therefore to develop and implement novel multiscale computational models and algorithms that bridge the gap between macroscopic tissue mechanics and microscale fluid flow in complex vascular networks, making biological tissue simulations more accurate and efficient, thus improving the solution of their associated inverse problems in tissue imaging. The project achieved three notable advances. Firstly, we formulated and implemented a mixeddimensional model that couples a three-dimensional elastic tissue matrix with one-dimensional vascular networks embedded within it. This model captures the two-way interaction between tissue deformation and microvascular flow. Secondly, building on this, we developed a family of efficient and robust finite element methods for this class of mixed-dimensional problems. The proposed approach allows us to efficiently simulate elastic tissues with arbitrary vascular networks, providing an efficient computational tool to study how vascular processes influence macroscopic mechanics. Finally, integrating a Localized Orthogonal Decomposition (LOD) model-order reduction enabled simulations while retaining accuracy in key mechanical fields. The project was carried out in close collaboration between the Weierstrass Institute for Applied Analysis and Stochastics (WIAS) in Berlin and the University of Augsburg. This partnership brought together complementary expertise enabling significant steps towards multiscale tissue modelling, paving the way for the participation in a subsequent project, "dealii-X: An Exascale Framework for Digital Twins of the Human Body', in which our mixed-dimensional modelling framework will form the basis for creating patient-specific digital twins of liver tissue.
Publications
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Mathematical Modeling of Blood Flow in the Cardiovascular System. Quantification of Biophysical Parameters in Medical Imaging, 39-61. Springer International Publishing.
Caiazzo, Alfonso; Heltai, Luca & Vignon-Clementel, Irene E.
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Reduced Lagrange multiplier approach for non-matching coupled problems in multiscale elasticity
C. Belponer, A. Caiazzo & L. Heltai
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Multiscale and homogenized modelling of vascular tissues, in: 7th International Conference on Computational & Mathematical Biomedical Engineering (CMBE22), 27th-29th June, 2022, Milan, Italy, P. Nithiarasu, C. Vergara, eds., 1, CMBE, Cardiff, UK, 2022, pp. 29–31
C. Belponer, A. Caiazzo, L. Heltai, L.O. Müller & D. Peterseim
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Reduced Lagrangian Multipliers
L. Heltai
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SLOD for Linear Elasticity
C. Belponer
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Super-Localized Orthogonal Decomposition Method for Heterogeneous Linear Elasticity. Computational Methods in Applied Mathematics, 25(3), 561-579.
Belponer, Camilla; Garay, José C.; Munch, Peter & Peterseim, Daniel
