173 research outputs found
Analysis of a geometrical multiscale blood flow model based on the coupling of ODEs and hyperbolic PDEs
For the numerical simulation of the circulatory system, geometrical multiscale models based on the coupling of systems of differential equations with different spatial dimensions are becoming common practice. In this paper we address the mathematical analysis of a coupled multiscale system involving a zero-dimensional (0D) model, describing the global characteristics of the circulatory system, and a one-dimensional (1D) model giving the pressure propagation along a straight vessel. We provide a local-in-time existence and uniqueness of classical solutions for this coupled problem. To this purpose we reformulate the original problem in a general abstract framework by splitting it into subproblems (the 0D system of ODEs and the 1D hyperbolic system of PDEs); then we use fixed-point techniques. The abstract result is then applied to the original blood flow case under very realistic hypotheses on the data
Newtonian Flow in Converging-Diverging Capillaries
The one-dimensional Navier-Stokes equations are used to derive analytical
expressions for the relation between pressure and volumetric flow rate in
capillaries of five different converging-diverging axisymmetric geometries for
Newtonian fluids. The results are compared to previously-derived expressions
for the same geometries using the lubrication approximation. The results of the
one-dimensional Navier-Stokes are identical to those obtained from the
lubrication approximation within a non-dimensional numerical factor. The
derived flow expressions have also been validated by comparison to numerical
solutions obtained from discretization with numerical integration. Moreover,
they have been certified by testing the convergence of solutions as the
converging-diverging geometries approach the limiting straight geometry.Comment: 23 pages, 5 figures, 1 table. This is an extended and improved
version. arXiv admin note: substantial text overlap with arXiv:1006.151
One-Dimensional Navier-Stokes Finite Element Flow Model
This technical report documents the theoretical, computational, and practical
aspects of the one-dimensional Navier-Stokes finite element flow model. The
document is particularly useful to those who are interested in implementing,
validating and utilizing this relatively-simple and widely-used model.Comment: 46 pages, 1 tabl
Numerical modelling of a peripheral arterial stenosis using dimensionally reduced models and kernel methods
In this work, we consider two kinds of model reduction techniques to simulate
blood flow through the largest systemic arteries, where a stenosis is located
in a peripheral artery i.e. in an artery that is located far away from the
heart. For our simulations we place the stenosis in one of the tibial arteries
belonging to the right lower leg (right post tibial artery). The model
reduction techniques that are used are on the one hand dimensionally reduced
models (1-D and 0-D models, the so-called mixed-dimension model) and on the
other hand surrogate models produced by kernel methods. Both methods are
combined in such a way that the mixed-dimension models yield training data for
the surrogate model, where the surrogate model is parametrised by the degree of
narrowing of the peripheral stenosis. By means of a well-trained surrogate
model, we show that simulation data can be reproduced with a satisfactory
accuracy and that parameter optimisation or state estimation problems can be
solved in a very efficient way. Furthermore it is demonstrated that a surrogate
model enables us to present after a very short simulation time the impact of a
varying degree of stenosis on blood flow, obtaining a speedup of several orders
over the full model
Multiscale constitutive framework of 1D blood flow modeling: Asymptotic limits and numerical methods
In this paper, a multiscale constitutive framework for one-dimensional blood
flow modeling is presented and discussed. By analyzing the asymptotic limits of
the proposed model, it is shown that different types of blood propagation
phenomena in arteries and veins can be described through an appropriate choice
of scaling parameters, which are related to distinct characterizations of the
fluid-structure interaction mechanism (whether elastic or viscoelastic) that
exist between vessel walls and blood flow. In these asymptotic limits,
well-known blood flow models from the literature are recovered. Additionally,
by analyzing the perturbation of the local elastic equilibrium of the system, a
new viscoelastic blood flow model is derived. The proposed approach is highly
flexible and suitable for studying the human cardiovascular system, which is
composed of vessels with high morphological and mechanical variability. The
resulting multiscale hyperbolic model of blood flow is solved using an
asymptotic-preserving Implicit-Explicit Runge-Kutta Finite Volume method, which
ensures the consistency of the numerical scheme with the different asymptotic
limits of the mathematical model without affecting the choice of the time step
by restrictions related to the smallness of the scaling parameters. Several
numerical tests confirm the validity of the proposed methodology, including a
case study investigating the hemodynamics of a thoracic aorta in the presence
of a stent
On the foundations of cancer modelling: selected topics, speculations, & perspectives
This paper presents a critical review of selected topics related to the modelling of cancer onset, evolution and growth, with the aim of illustrating, to a wide applied mathematical readership, some of the novel mathematical problems in the field. This review attempts to capture, from the appropriate literature, the main issues involved in the modelling of phenomena related to cancer dynamics at all scales which characterise this highly complex system: from the molecular scale up to that of tissue. The last part of the paper discusses the challenge of developing a mathematical biological theory of tumour onset and evolution
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