Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userComplex Far-Field Geometries Determine the Stability of Solid Tumor Growth with Chemotaxis
95
0
0.0
(
0
)
Added by
Min-Jhe Lu
Publication date
2019
fields
Informatics Engineering
and research's language is
English
Ask ChatGPT about the research
No Arabic abstract
In this paper, we develop a sharp interface tumor growth model to study the effect of the tumor microenvironment using a complex far-field geometry that mimics a heterogeneous distribution of vasculature. Together with different nutrient uptake rates inside and outside the tumor, this introduces variability in spatial diffusion gradients. Linear stability analysis suggests that the uptake rate in the tumor microenvironment, together with chemotaxis, may induce unstable growth, especially when the nutrient gradients are large. We investigate the fully nonlinear dynamics using a spectrally accurate boundary integral method. Our nonlinear simulations reveal that vascular heterogeneity plays an important role in the development of morphological instabilities that range from fingering and chain-like morphologies to compact, plate-like shapes in two-dimensions.
rate research
Read More
In this paper, we develop a sharp interface tumor growth model to study the effect of both the intratumoral structure using a controlled necrotic core and the extratumoral nutrient supply from vasculature on tumor morphology. We first show that our model extends the benchmark results in the literature using linear stability analysis. Then we solve this generalized model numerically using a spectrally accurate boundary integral method in an evolving annular domain, not only with a Robin boundary condition on the outer boundary for the nutrient field which models tumor vasculature, but also with a static boundary condition on the inner boundary for pressure field which models the control of tumor necrosis. The discretized linear systems for both pressure and nutrient fields are shown to be well-conditioned through tracing GMRES iteration numbers. Our nonlinear simulations reveal the stabilizing effects of angiogenesis and the destabilizing ones of chemotaxis and necrosis in the development of tumor morphological instabilities if the necrotic core is fixed in a circular shape. When the necrotic core is controlled in a non-circular shape, the stabilizing effects of proliferation and the destabilizing ones of apoptosis are observed. Finally, the values of the nutrient concentration with its fluxes and the pressure level with its normal derivatives, which are solved accurately at the boundaries, help us to characterize the corresponding tumor morphology and the level of the biophysical quantities on interfaces required in keeping various shapes of the necrotic region of the tumor. Interestingly, we notice that when the necrotic region is fixed in a 3-fold non-circular shape, even if the initial shape of the tumor is circular, the tumor will evolve into a shape corresponding to the 3-fold symmetry of the shape of the fixed necrotic region.
We present a general -- i.e., independent of the underlying equation -- registration procedure for parameterized model order reduction. Given the spatial domain $Omega subset mathbb{R}^2$ and the manifold $mathcal{M}= { u_{mu} : mu in mathcal{P} }$ associated with the parameter domain $mathcal{P} subset mathbb{R}^P$ and the parametric field $mu mapsto u_{mu} in L^2(Omega)$, our approach takes as input a set of snapshots ${ u^k }_{k=1}^{n_{rm train}} subset mathcal{M}$ and returns a parameter-dependent bijective mapping ${Phi}: Omega times mathcal{P} to mathbb{R}^2$: the mapping is designed to make the mapped manifold ${ u_{mu} circ {Phi}_{mu}: , mu in mathcal{P} }$ more amenable for linear compression methods. In this work, we extend and further analyze the registration approach proposed in [Taddei, SISC, 2020]. The contributions of the present work are twofold. First, we extend the approach to deal with annular domains by introducing a suitable transformation of the coordinate system. Second, we discuss the extension to general two-dimensional geometries: towards this end, we introduce a spectral element approximation, which relies on a partition ${ Omega_{q} }_{q=1} ^{N_{rm dd}}$ of the domain $Omega$ such that $Omega_1,ldots,Omega_{N_{rm dd}}$ are isomorphic to the unit square. We further show that our spectral element approximation can cope with parameterized geometries. We present rigorous mathematical analysis to justify our proposal; furthermore, we present numerical results for a heat-transfer problem in an annular domain, a potential flow past a rotating symmetric airfoil, and an inviscid transonic compressible flow past a non-symmetric airfoil, to demonstrate the effectiveness of our method.
The inverse problem we consider is to reconstruct the location and shape of buried obstacles in the lower half-space of an unbounded two-layered medium in two dimensions from phaseless far-field data. A main difficulty of this problem is that the translation invariance property of the modulus of the far field pattern is unavoidable, which is similar to the homogenous background medium case. Based on the idea of using superpositions of two plane waves with different directions as the incident fields, we first develop a direct imaging method to locate the position of small anomalies and give a theoretical analysis of the algorithm. Then a recursive Newton-type iteration algorithm in frequencies is proposed to reconstruct extended obstacles. Finally, numerical experiments are presented to illustrate the feasibility of our algorithms.
Radial basis function generated finite difference (RBF-FD) methods for PDEs require a set of interpolation points which conform to the computational domain $Omega$. One of the requirements leading to approximation robustness is to place the interpolation points with a locally uniform distance around the boundary of $Omega$. However generating interpolation points with such properties is a cumbersome problem. Instead, the interpolation points can be extended over the boundary and as such completely decoupled from the shape of $Omega$. In this paper we present a modification to the least-squares RBF-FD method which allows the interpolation points to be placed in a box that encapsulates $Omega$. This way, the node placement over a complex domain in 2D and 3D is greatly simplified. Numerical experiments on solving an elliptic model PDE over complex 2D geometries show that our approach is robust. Furthermore it performs better in terms of the approximation error and the runtime vs. error compared with the classic RBF-FD methods. It is also possible to use our approach in 3D, which we indicate by providing convergence results of a solution over a thoracic diaphragm.
In this paper, we propose and analyze a first-order and a second-order time-stepping schemes for the anisotropic phase-field dendritic crystal growth model. The proposed schemes are based on an auxiliary variable approach for the Allen-Cahn equation and delicate treatment of the terms coupling the Allen-Cahn equation and temperature equation. The idea of the former is to introduce suitable auxiliary variables to facilitate construction of high order stable schemes for a large class of gradient flows. We propose a new technique to treat the coupling terms involved in the crystal growth model and introduce suitable stabilization terms to result in totally decoupled schemes, which satisfy a discrete energy law without affecting the convergence order. A delicate implementation demonstrates that the proposed schemes can be realized in a very efficient way. That is, it only requires solving four linear elliptic equations and a simple algebraic equation at each time step. A detailed comparison with existing schemes is given, and the advantage of the new schemes are emphasized. As far as we know this is the first second-order scheme that is totally decoupled, linear, unconditionally stable for the dendritic crystal growth model with variable mobility parameter.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
