69 research outputs found
Mathematical modeling of local perfusion in large distensible microvascular networks
Microvessels -blood vessels with diameter less than 200 microns- form large,
intricate networks organized into arterioles, capillaries and venules. In these
networks, the distribution of flow and pressure drop is a highly interlaced
function of single vessel resistances and mutual vessel interactions. In this
paper we propose a mathematical and computational model to study the behavior
of microcirculatory networks subjected to different conditions. The network
geometry is composed of a graph of connected straight cylinders, each one
representing a vessel. The blood flow and pressure drop across the single
vessel, further split into smaller elements, are related through a generalized
Ohm's law featuring a conductivity parameter, function of the vessel cross
section area and geometry, which undergo deformations under pressure loads. The
membrane theory is used to describe the deformation of vessel lumina, tailored
to the structure of thick-walled arterioles and thin-walled venules. In
addition, since venules can possibly experience negative transmural pressures,
a buckling model is also included to represent vessel collapse. The complete
model including arterioles, capillaries and venules represents a nonlinear
system of PDEs, which is approached numerically by finite element
discretization and linearization techniques. We use the model to simulate flow
in the microcirculation of the human eye retina, a terminal system with a
single inlet and outlet. After a phase of validation against experimental
measurements, we simulate the network response to different interstitial
pressure values. Such a study is carried out both for global and localized
variations of the interstitial pressure. In both cases, significant
redistributions of the blood flow in the network arise, highlighting the
importance of considering the single vessel behavior along with its position
and connectivity in the network
A Computational Model for Biomass Growth Simulation in Tissue Engineering
This article deals with computational modeling of tissue growth under interstitial perfusion inside a polymeric scaffold-based bioreactor. The mathematical model is the result of the application of the volume averaging technique to the fluid, nutrient and cellular subsystems, and is capable to account for the temporal evolution of local matrix porosity, as the sum of a time-invariant component (the porosity of the uncellularized polymer scaffold) and a time-dependent component (due to the growing biomass). The solution algorithm is based on a block Gauss-Seidel iteration procedure that allows to reduce each time level of the simulated culture period into the successive solution of linearized subproblems, whose numerical approximation is carried out using stable and convergent finite elements. Numerical simulations are carried out to investigate the role of the design porosity of the scaffold on nutrient delivery and biomass production
A Learned-SVD approach for Regularization in Diffuse Optical Tomography
Diffuse Optical Tomography (DOT) is an emerging technology in medical imaging
which employs light in the NIR spectrum to estimate the distribution of optical
coefficients in biological tissues for diagnostic and monitoring purposes. DOT
reconstruction implies the solution of a severely ill-posed inverse problem,
for which regularization techniques are mandatory in order to achieve
reasonable results. Traditionally, regularization techniques put a variance
prior on the desired solution/gradient via regularization parameters, whose
choice requires a fine tuning, specific for each case. In this work we explore
deep learning techniques in a fully data-driven approach, able of
reconstructing the generating signal (optical absorption coefficient) in an
automated way. We base our approach on the so-called Learned Singular Value
Decomposition, which has been proposed for general inverse problems, and we
tailor it to the DOT application. We perform tests with increasing levels of
noise on the measure, and compare it with standard variational approaches
Mathematical methods for modeling the microcirculation
The microcirculation plays a major role in maintaining homeostasis in the body. Alterations or dysfunctions of the microcirculation can lead to several types of serious diseases. It is not surprising, then, that the microcirculation has been an object of intense theoretical and experimental study over the past few decades. Mathematical approaches offer a valuable method for quantifying the relationships between various mechanical, hemodynamic, and regulatory factors of the microcirculation and the pathophysiology of numerous diseases. This work provides an overview of several mathematical models that describe and investigate the many different aspects of the microcirculation, including geometry of the vascular bed, blood flow in the vascular networks, solute transport and delivery to the surrounding tissue, and vessel wall mechanics under passive and active stimuli. Representing relevant phenomena across multiple spatial scales remains a major challenge in modeling the microcirculation. Nevertheless, the depth and breadth of mathematical modeling with applications in the microcirculation is demonstrated in this work. A special emphasis is placed on models of the retinal circulation, including models that predict the influence of ocular hemodynamic alterations with the progression of ocular diseases such as glaucoma
Flux-Upwind Stabilization of the Discontinuous Petrov--Galerkin Formulation with Lagrangian Multipliers for Advection-Diffusion Problems
In this work we consider the dual-primal Discontinuous Petrov-Galerkin (DPG) method for the advection-diffusion model problem. Since in the DPG method both mixed internal variables are discontinuous, a static condensation procedure can be carried out, leading to a single-field nonconforming discretization scheme. For this latter formulation, we propose a flux-upwind stabilization technique to deal with the advection-dominated case. The resulting scheme is conservative and satisfies a discrete maximum principle under standard geometrical assumptions on the computational grid. A convergence analysis is developed, proving first-order accuracy of the method in a discrete H^1-norm, and the numerical performance of the scheme is validated on benchmark problems with sharp internal and boundary layers
Flux-Upwind Stabilization of the Discontinuous Petrov--Galerkin Formulation with Lagrangian Multipliers for Advection-Diffusion Problems
In this work we consider the dual-primal Discontinuous Petrov-Galerkin (DPG) method for the advection-diffusion model problem. Since in the DPG method both mixed internal variables are discontinuous, a static condensation procedure can be carried out, leading to a single-field nonconforming discretization scheme. For this latter formulation, we propose a flux-upwind stabilization technique to deal with the advection-dominated case. The resulting scheme is conservative and satisfies a discrete maximum principle under standard geometrical assumptions on the computational grid. A convergence analysis is developed, proving first-order accuracy of the method in a discrete H^1-norm, and the numerical performance of the scheme is validated on benchmark problems with sharp internal and boundary layers
Autocatalytic Loop, Amplification and Diffusion: A Mathematical and Computational Model of Cell Polarization in Neural Chemotaxis
The chemotactic response of cells to graded fields of chemical cues is a complex process that requires the coordination of several intracellular activities. Fundamental steps to obtain a front vs. back differentiation in the cell are the localized distribution of internal molecules and the amplification of the external signal. The goal of this work is to develop a mathematical and computational model for the quantitative study of such phenomena in the context of axon chemotactic pathfinding in neural development. In order to perform turning decisions, axons develop front-back polarization in their distal structure, the growth cone. Starting from the recent experimental findings of the biased redistribution of receptors on the growth cone membrane, driven by the interaction with the cytoskeleton, we propose a model to investigate the significance of this process. Our main contribution is to quantitatively demonstrate that the autocatalytic loop involving receptors, cytoplasmic species and cytoskeleton is adequate to give rise to the chemotactic behavior of neural cells. We assess the fact that spatial bias in receptors is a precursory key event for chemotactic response, establishing the necessity of a tight link between upstream gradient sensing and downstream cytoskeleton dynamics. We analyze further crosslinked effects and, among others, the contribution to polarization of internal enzymatic reactions, which entail the production of molecules with a one-to-more factor. The model shows that the enzymatic efficiency of such reactions must overcome a threshold in order to give rise to a sufficient amplification, another fundamental precursory step for obtaining polarization. Eventually, we address the characteristic behavior of the attraction/repulsion of axons subjected to the same cue, providing a quantitative indicator of the parameters which more critically determine this nontrivial chemotactic response
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