THREE-DIMENSIONAL COMPUTATIONAL MODEL OF EARLY UPPER LIMB DEVELOPMENT

2021 ◽  
Author(s):  
LUIS FELIPE MILLAN CLARO ◽  
KALENIA MÁRQUEZ FLÓREZ ◽  
CARLOS A. DUQUE-DAZA ◽  
DIEGO A. GARZÓN-ALVARADO
Keyword(s):  
Computational Model ◽  
Upper Limb ◽  
Limb Development ◽  
Three Dimensional ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Mesenchymal Condensation ◽  
The One ◽  
Constituent Elements ◽  
Early Upper

Limb development begins during embryogenesis when a series of biochemical interactions are triggered between a particular region of the mesoderm and the ectoderm. These processes affect the morphogenesis and growth of bones, joints, and all the other constituent elements of limbs; nevertheless, how the biochemical regulation affects mesenchymal condensation is not entirely clear. In this study, a three-dimensional computational model is designed to predict the appearance and location of the mesenchymal condensation in the stylopod and zeugopod; the biochemical events were described with reaction–diffusion equations that were solved using the finite elements method. The result of the gene expression in our model was consistent with the one reported in literature; the obtained patterns of Fgf8, Fgf10, and Wnt3a can predict the shape of the mesenchymal condensation of early upper limb development; the simple diffusive patterns of molecules were suitable to explain the areas where sox9 is expressed. Furthermore, our results suggest that the expression of Tgf-[Formula: see text] in the upper limb could be due to the inhibition of retinoic acid. These results suggest the importance of building computational scenarios where pathologies may be comprehensively examined.

scholarly journals 3D Unsteady Diffusion and Reaction-Diffusion with Singularities by GFEM with 27-Node Hexahedrons

2014 ◽  
Vol 2014 ◽  
pp. 1-12 ◽  
Author(s):  
Estaner Claro Romão
Keyword(s):  
Finite Element ◽  
Variable Temperature ◽  
Three Dimensional ◽  
Reaction Diffusion ◽  
Third Order ◽  
Order Of Accuracy ◽  
Finite Element Approximations ◽  
Temperature Solution ◽  
The One ◽  
Primary Variable

The Galerkin Finite Element Method (GFEM) with 8- and 27-node hexahedrons elements is used for solving diffusion and transient three-dimensional reaction-diffusion with singularities. Besides analyzing the results from the primary variable (temperature), the finite element approximations were used to find the derivative of the temperature in all three directions. This technique does not provide an order of accuracy compatible with the one found in the temperature solution; thereto, a calculation from the third order finite differences is proposed here, which provide the best results, as demonstrated by the first two applications proposed in this paper. Lastly, the presentation and the discussion of a real application with two cases of boundary conditions with singularities are proposed.

A PARALLEL SOLVER FOR REACTION–DIFFUSION SYSTEMS IN COMPUTATIONAL ELECTROCARDIOLOGY

2004 ◽  
Vol 14 (06) ◽  
pp. 883-911 ◽  
Author(s):  
PIERO COLLI FRANZONE ◽  
LUCA F. PAVARINO
Keyword(s):  
Large Scale ◽  
Three Dimensional ◽  
Reaction Diffusion ◽  
Ionic Currents ◽  
Reaction Diffusion Equations ◽  
Reaction Diffusion Equation ◽  
Fiber Direction ◽  
Axially Symmetric ◽  
Reaction Diffusion Systems ◽  
Cardiac Model

In this work, a parallel three-dimensional solver for numerical simulations in computational electrocardiology is introduced and studied. The solver is based on the anisotropic Bidomain cardiac model, consisting of a system of two degenerate parabolic reaction–diffusion equations describing the intra and extracellular potentials of the myocardial tissue. This model includes intramural fiber rotation and anisotropic conductivity coefficients that can be fully orthotropic or axially symmetric around the fiber direction. The solver also includes the simpler anisotropic Monodomain model, consisting of only one reaction–diffusion equation. These cardiac models are coupled with a membrane model for the ionic currents, consisting of a system of ordinary differential equations that can vary from the simple FitzHugh–Nagumo (FHN) model to the more complex phase-I Luo–Rudy model (LR1). The solver employs structured isoparametric Q1finite elements in space and a semi-implicit adaptive method in time. Parallelization and portability are based on the PETSc parallel library. Large-scale computations with up to O(107) unknowns have been run on parallel computers, simulating excitation and repolarization phenomena in three-dimensional domains.

Travelling waves for a reaction–diffusion system in population dynamics and epidemiology

2005 ◽  
Vol 135 (4) ◽  
pp. 663-675 ◽  
Author(s):  
Shangbing Ai ◽  
Wenzhang Huang
Keyword(s):  
Population Dynamics ◽  
Travelling Waves ◽  
Three Dimensional ◽  
Diffusion Constant ◽  
Reaction Diffusion ◽  
Travelling Wave ◽  
Reaction Diffusion Equations ◽  
Dimensional System ◽  
Travelling Wave Solutions ◽  
Wave Solutions

The existence and uniqueness of travelling-wave solutions is investigated for a system of two reaction–diffusion equations where one diffusion constant vanishes. The system arises in population dynamics and epidemiology. Travelling-wave solutions satisfy a three-dimensional system about (u, u′, ν), whose equilibria lie on the u-axis. Our main result shows that, given any wave speed c > 0, the unstable manifold at any point (a, 0, 0) on the u-axis, where a ∈ (0, γ) and γ is a positive number, provides a travelling-wave solution connecting another point (b, 0, 0) on the u-axis, where b:= b(a) ∈ (γ, ∞), and furthermore, b(·): (0, γ) → (γ, ∞) is continuous and bijective

scholarly journals Three-dimensional stochastic simulation of chemoattractant-mediated excitability in cells

2021 ◽  
Vol 17 (7) ◽  
pp. e1008803
Author(s):  
Debojyoti Biswas ◽  
Peter N. Devreotes ◽  
Pablo A. Iglesias
Keyword(s):  
G Protein ◽  
Protein Interactions ◽  
Three Dimensional ◽  
Reaction Diffusion ◽  
Dose Dependence ◽  
Reaction Diffusion Equations ◽  
Tetrahedral Mesh ◽  
Inhibition Mechanism ◽  
External Signal ◽  
Excitable System

During the last decade, a consensus has emerged that the stochastic triggering of an excitable system drives pseudopod formation and subsequent migration of amoeboid cells. The presence of chemoattractant stimuli alters the threshold for triggering this activity and can bias the direction of migration. Though noise plays an important role in these behaviors, mathematical models have typically ignored its origin and merely introduced it as an external signal into a series of reaction-diffusion equations. Here we consider a more realistic description based on a reaction-diffusion master equation formalism to implement these networks. In this scheme, noise arises naturally from a stochastic description of the various reaction and diffusion terms. Working on a three-dimensional geometry in which separate compartments are divided into a tetrahedral mesh, we implement a modular description of the system, consisting of G-protein coupled receptor signaling (GPCR), a local excitation-global inhibition mechanism (LEGI), and signal transduction excitable network (STEN). Our models implement detailed biochemical descriptions whenever this information is available, such as in the GPCR and G-protein interactions. In contrast, where the biochemical entities are less certain, such as the LEGI mechanism, we consider various possible schemes and highlight the differences between them. Our stimulations show that even when the LEGI mechanism displays perfect adaptation in terms of the mean level of proteins, the variance shows a dose-dependence. This differs between the various models considered, suggesting a possible means for determining experimentally among the various potential networks. Overall, our simulations recreate temporal and spatial patterns observed experimentally in both wild-type and perturbed cells, providing further evidence for the excitable system paradigm. Moreover, because of the overall importance and ubiquity of the modules we consider, including GPCR signaling and adaptation, our results will be of interest beyond the field of directed migration.

scholarly journals Steady states of the reaction-diffusion equations. Part II: Uniqueness of solutions and some special cases

1983 ◽  
Vol 24 (4) ◽  
pp. 392-416 ◽  
Author(s):  
J. G. Burnell ◽  
A. A. Lacey ◽  
G. C. Wake
Keyword(s):  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Uniqueness Of Solutions ◽  
One Dimensional ◽  
Nonlinear Elliptic ◽  
Special Cases ◽  
Exothermic Chemical Reaction ◽  
Dimensional Shape ◽  
The One ◽  
Special Case

AbstractIn an earlier paper (Part I) the existence and some related properties of the solution to a coupled pair of nonlinear elliptic partial differential equations was considered. These equations arise when material is undergoing an exothermic chemical reaction which is sustained by the diffusion of a reactant. In this paper we consider the range of parameters for which the uniqueness of solution is assured and we also investigate the converse question of multiple solutions. The special case of the one dimensional shape of the infinite slab is investigated in full as this case admits to solution by integration.

Quasi-Optimized Overlapping Schwarz Waveform Relaxation Algorithm for PDEs with Time-Delay

2013 ◽  
Vol 14 (3) ◽  
pp. 780-800 ◽  
Author(s):  
Shu-Lin Wu ◽  
Ting-Zhu Huang
Keyword(s):  
Time Delay ◽  
Reaction Diffusion ◽  
Transmission Condition ◽  
Reaction Diffusion Equations ◽  
Complete Analysis ◽  
Waveform Relaxation ◽  
Schwarz Waveform Relaxation ◽  
Regular Time ◽  
The One ◽  
Time Dependent Problems

AbstractSchwarz waveform relaxation (SWR) algorithm has been investigated deeply and widely for regular time dependent problems. But for time delay problems, complete analysis of the algorithm is rare. In this paper, by using the reaction diffusion equations with a constant discrete delay as the underlying model problem, we investigate the convergence behavior of the overlapping SWR algorithm with Robin transmission condition. The key point of using this transmission condition is to determine a free parameter as better as possible and it is shown that the best choice of the parameter is determined by the solution of a min-max problem, which is more complex than the one arising for regular problems without delay. We propose new notion to solve the min-max problem and obtain a quasi-optimized choice of the parameter, which is shown efficient to accelerate the convergence of the SWR algorithm. Numerical results are provided to validate the theoretical conclusions.

scholarly journals Microenvironmental factors in cell segregation and heterogeneity in breast cancer development

2021 ◽  
Author(s):  
J. Roberto Romero-Arias ◽  
Carlos A. González-Castro ◽  
Guillermo Ramírez-Santiago
Keyword(s):  
Breast Cancer ◽  
Three Dimensional ◽  
Reaction Diffusion ◽  
Quantitative Model ◽  
Reaction Diffusion Equations ◽  
Cancer Development ◽  
Breast Cancer Development ◽  
Spatial Gradients ◽  
Microenvironmental Factors ◽  
The Individual

ABSTRACTWe analyzed a quantitative model that describes the epigenetic dynamics during the growth and evolution of an avascular tumor. A gene regulatory network (GRN) formed by a set of ten genes that are believed to play an important role in breast cancer development was kinetically coupled to the microenvironmental agents: glucose, estrogens and oxygen. The dynamics of spontaneous mutations was described by a Yule-Furry master equation whose solution represents the probability that a given cell in the tissue undergoes a certain number of mutations at a given time. We assumed that mutations rate is modified by nutrients spatial gradients. The tumor mass was grown by means of a cellular automata supplemented with a set of reaction diffusion equations that described the transport of the microenvironmental agents. By analyzing the epigenetic states space described by the GRN dynamics, we found three attractors that were identified with the cellular epigenetic states: normal, precancer and cancer. For two-dimensional (2D) and three-dimensional (3D) tumors we calculated the spatial distributions of the following quantities: (i) number of mutations, (ii) mutations of each gene and, (iii) phenotypes. Using estrogens as the principal microenvironmental agent that regulates cells proliferation process, we obtained the tumor shapes for different values of the estrogen consumption and supply rates. It was found that he majority of mutations occurred in cells that were located close to the 2D tumor perimeter or close to the 3D tumor surface. Also It was found that the occurrence of different phenotypes in the tumor are controlled by the levels of estrogen concentration since they can change the individual cell threshold and gene expression levels. All the results were consistently observed for 2D and 3D tumors.

scholarly journals Multidimensional stability of pyramidal traveling fronts in degenerate Fisher-KPP monostable and combustion equations

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Denghui Wu ◽  
Zhen-Hui Bu
Keyword(s):  
Comparison Principle ◽  
Initial Perturbation ◽  
Three Dimensional ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Asymptotically Stable ◽  
Traveling Fronts ◽  
Bounded Perturbations ◽  
Combustion Equations

<p style='text-indent:20px;'>In this paper, multidimensional stability of pyramidal traveling fronts are studied to the reaction-diffusion equations with degenerate Fisher-KPP monostable and combustion nonlinearities. By constructing supersolutions and subsolutions coupled with the comparison principle, we firstly prove that under any initial perturbation (possibly large) decaying at space infinity, the three-dimensional pyramidal traveling fronts are asymptotically stable in weighted <inline-formula><tex-math id="M1">\begin{document}$ L^{\infty} $\end{document}</tex-math></inline-formula> spaces on <inline-formula><tex-math id="M2">\begin{document}$ \mathbb{R}^{n}\; (n\geq4) $\end{document}</tex-math></inline-formula>. Secondly, we show that under general bounded perturbations (even very small), the pyramidal traveling fronts are not asymptotically stable by constructing a solution which oscillates permanently between two three-dimensional pyramidal traveling fronts on <inline-formula><tex-math id="M3">\begin{document}$ \mathbb{R}^{4} $\end{document}</tex-math></inline-formula>.</p>

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