Multiscale modeling of vascularized tissues via nonmatching immersed methods

Heltai, Luca; Caiazzo, Alfonso

doi:10.1002/cnm.3264

Cited by 17 publications

(16 citation statements)

References 32 publications

(75 reference statements)

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“…To this aim, we extend the original problem with a fictitious problem in . As described in, 9 we seek for the solution of a problem of the following form:…”

Section: Methodsmentioning

confidence: 99%

“…At each , we then seek such that The singular source term is defined in such a way to enforce, for each , the correct value of the normal stresses across depending on the fluid pressure on the vessel boundary . For the detailed derivation of the forcing term, we refer to, 9 which is briefly summarized below.…”

Section: Methodsmentioning

confidence: 99%

“…to the diameter of the sample domain . As detailed in, 9 this is achieved by computing the singular source terms that would be required to solve the problem exactly on a large domain with a single embedded vessel. A similar strategy is possible in the non-linear case, provided that one resorts to Lagrange multipliers, as in.…”

Section: Methodsmentioning

confidence: 99%

“…With the purpose of enhancing the quality and the outreach of MRE analysis, this paper addresses the issues of developing a computational multiscale model that can account for an arbitrary complexity of the (microscale) fluid vasculature and, at the same time, be efficiently upscaled for the application in the context of (coarse) MRE data. To this aim, we extend the immersed multiscale framework recently proposed in, 9 , 10 based on describing the tissue as an elastic material, taking into account the presence of the fluid network as a singular forcing term. In particular, we address the coupling between the three-dimensional elastic matrix and a finite volume one-dimensional blood flow model that can efficiently handle complex vasculature networks.…”

Section: Introductionmentioning

confidence: 99%

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Multiscale Coupling of One-dimensional Vascular Models and Elastic Tissues

2021

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We present a computational multiscale model for the efficient simulation of vascularized tissues, composed of an elastic three-dimensional matrix and a vascular network. The effect of blood vessel pressure on the elastic tissue is surrogated via hyper-singular forcing terms in the elasticity equations, which depend on the fluid pressure. In turn, the blood flow in vessels is treated as a one-dimensional network. Intravascular pressure and velocity are simulated using a high-order finite volume scheme, while the elasticity equations for the tissue are solved using a finite element method. This work addresses the feasibility and the potential of the proposed coupled multiscale model. In particular, we assess whether the multiscale model is able to reproduce the tissue response at the effective scale (of the order of millimeters) while modeling the vasculature at the microscale. We validate the multiscale method against a full scale (three-dimensional) model, where the fluid/tissue interface is fully discretized and treated as a Neumann boundary for the elasticity equation. Next, we present simulation results obtained with the proposed approach in a realistic scenario, demonstrating that the method can robustly and efficiently handle the one-way coupling between complex fluid microstructures and the elastic matrix.

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“…To this aim, we extend the original problem with a fictitious problem in . As described in, 9 we seek for the solution of a problem of the following form:…”

Section: Methodsmentioning

confidence: 99%

Section: Methodsmentioning

confidence: 99%

Section: Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Multiscale Coupling of One-dimensional Vascular Models and Elastic Tissues

2021

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“…The general results presented in the previous section can be applied immediately to immersed interface and immersed boundary methods [2,15,16,18]. In this section, we consider an interface problem whose variational formulation can be written as in the 21, and we shall consider its finite element approximation using the regularization of the forcing data given by the application of Definition 1 to functions in negative Sobolev spaces.…”

Section: Application To Immersed Methodsmentioning

confidence: 99%

A priori error estimates of regularized elliptic problems

Heltai

Lei

2020

Numer. Math.

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Approximations of the Dirac delta distribution are commonly used to create sequences of smooth functions approximating nonsmooth (generalized) functions, via convolution. In this work we show a-priori rates of convergence of this approximation process in standard Sobolev norms, with minimal regularity assumptions on the approximation of the Dirac delta distribution. The application of these estimates to the numerical solution of elliptic problems with singularly supported forcing terms allows us to provide sharp $$H^1$$ H 1 and $$L^2$$ L 2 error estimates for the corresponding regularized problem. As an application, we show how finite element approximations of a regularized immersed interface method results in the same rates of convergence of its non-regularized counterpart, provided that the support of the Dirac delta approximation is set to a multiple of the mesh size, at a fraction of the implementation complexity. Numerical experiments are provided to support our theories.

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A PDE-constrained optimization method for 3D-1D coupled problems with discontinuous solutions

2023

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A numerical method for coupled 3D-1D problems with discontinuous solutions at the interfaces is derived and discussed. This extends a previous work on the subject where only continuous solutions were considered. Thanks to properly defined function spaces a well posed 3D-1D problem is obtained from the original fully 3D problem and the solution is then found by a PDE-constrained optimization reformulation. This is a domain decomposition strategy in which unknown interface variables are introduced and a suitably defined cost functional, expressing the error in fulfilling interface conditions, is minimized constrained by the constitutive equations on the subdomains. The resulting discrete problem is robust with respect to geometrical complexity thanks to the use of independent discretizations on the various subdomains. Meshes of different sizes can be used without affecting the conditioning of the discrete linear system, and this is a peculiar aspect of the considered formulation. An efficient solving strategy is further proposed, based on the use of a gradient based solver and yielding a method ready for parallel implementation. A numerical experiment on a problem with known analytical solution shows the accuracy of the method, and two examples on more complex configurations are proposed to address the applicability of the approach to practical problems.

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Multiscale modeling of vascularized tissues via nonmatching immersed methods

Cited by 17 publications

References 32 publications

Multiscale Coupling of One-dimensional Vascular Models and Elastic Tissues

Multiscale Coupling of One-dimensional Vascular Models and Elastic Tissues

A priori error estimates of regularized elliptic problems

A PDE-constrained optimization method for 3D-1D coupled problems with discontinuous solutions

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