scholarly journals A COMPUTATIONAL ANALYSIS OF BONE FORMATION IN THE CRANIAL VAULT USING A COUPLED REACTION–DIFFUSION-STRAIN MODEL

2017 ◽  
Vol 17 (04) ◽  
pp. 1750073 ◽  
Author(s):  
CHANYOUNG LEE ◽  
JOAN T. RICHTSMEIER ◽  
REUBEN H. KRAFT
Keyword(s):  
Bone Formation ◽  
Mechanical Strain ◽  
Mathematical Formulation ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Cranial Vault ◽  
Model Parameters ◽  
Computational Domain ◽  
Reaction Diffusion Model

Bones of the murine cranial vault are formed by differentiation of mesenchymal cells into osteoblasts, a process that is primarily understood to be controlled by a cascade of reactions between extracellular molecules and cells. We assume that the process can be modeled using Turing's reaction–diffusion equations, a mathematical model describing the pattern formation controlled by two interacting molecules (activator and inhibitor). In addition to the processes modeled by reaction–diffusion equations, we hypothesize that mechanical stimuli of the cells due to growth of the underlying brain contribute significantly to the process of cell differentiation in cranial vault development. Structural analysis of the surface of the brain was conducted to explore the effects of the mechanical strain on bone formation. We propose a mechanobiological model for the formation of cranial vault bones by coupling the reaction-–diffusion model with structural mechanics. The mathematical formulation was solved using the finite volume method. The computational domain and model parameters are determined using a large collection of experimental data that provide precise three-dimensional (3D) measures of murine cranial geometry and cranial vault bone formation for specific embryonic time points. The results of this study suggest that mechanical strain contributes information to specific aspects of bone formation. Our mechanobiological model predicts some key features of cranial vault bone formation that were verified by experimental observations including the relative location of ossification centers of individual vault bones, the pattern of cranial vault bone growth over time, and the position of cranial vault sutures.

A MODEL OF THE FORMATION OF THE CEREBRAL CORTEX THROUGH A MIXED APPROACH OF REACTION DIFFUSION EQUATIONS AND MECHANICAL STRAIN

2012 ◽  
Vol 12 (05) ◽  
pp. 1250090
Author(s):  
DIEGO A GARZÓN-ALVARADO
Keyword(s):  
Cerebral Cortex ◽  
Mechanical Strain ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Reaction Diffusion Model ◽  
Proposed Model ◽  
Mixed Approach ◽  
And Function ◽  
The Brain

During fetal development the morphology and function of the organs and tissues is determined. An example occurs with the formation of the cerebral cortex. On the external surface of the brain there are numerous folds (gyri, sulci, and fissures) that determine brain function. The exact cause for the formation of patterns of these folds is unknown. This article proposes a reaction-diffusion model in conjunction with a process of surface mechanical strain to explain the morphogenesis of the superficial structure of the brain.The model is solved using finite elements. There have been tests done on the formation of brain patterns through the reaction-diffusion equations with parameters in the space of Turing and by random mechanical strain. Several numerical examples have been developed that show an acceptable correlation between the results and clinical reality. With the model we can represent, qualitatively, the formation of the cerebral cortex by the proposed model. The model can approximate, and explain, lissencephaly and polymicrogyria, diseases that develop in the cerebral cortex and lead to medical complications to sufferers.

scholarly journals Spatiotemporal Complexity of the Nutrient-Phytoplankton Model

2015 ◽  
Vol 2015 ◽  
pp. 1-11
Author(s):  
Debing Mei ◽  
Min Zhao ◽  
Hengguo Yu ◽  
Chuanjun Dai
Keyword(s):  
Mathematical Formulation ◽  
Reaction Diffusion ◽  
Turing Instability ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Critical Conditions ◽  
Horizontal Movement ◽  
Advection Diffusion ◽  
Advection Term ◽  
Formulation Analysis

We consider the mathematical formulation, analysis, and numerical solution of a nonlinear system of nutrient-phytoplankton, which consists of a series of reaction-advection-diffusion equations. We derive the critical conditions for Turing instability without an advection term and define the range of Turing instability with the change of nutrient concentration. We show that horizontal movement of phytoplankton could influence the system and that it is unstable when the horizontal velocity exceeds a critical value. We also compare reaction-diffusion equations with reaction-advection-diffusion equations through simulations, with spotted, banded, and crenulate patterns produced from our model. We found that different spatial constructions could occur, impacted by the diffusion and sinking of nutrients and phytoplankton. The new model may help us better understand the dynamics of an aquatic community.

scholarly journals A contraction-reaction-diffusion model for circular pattern formation in embryogenesis

2021 ◽  
Author(s):  
Tiankai Zhao ◽  
Yubing Sun ◽  
Xin Li ◽  
Mehdi Baghaee ◽  
Yuenan Wang ◽  
...  
Keyword(s):  
Pattern Formation ◽  
Cell Fate ◽  
Diffusion Model ◽  
Early Stage ◽  
Reaction Diffusion ◽  
Diffusion Models ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Reaction Diffusion Model ◽  
Stress Dependent

Reaction-diffusion models have been widely used to elucidate pattern formation in developmental biology. More recently, they have also been applied in modeling cell fate patterning that mimic early-stage human development events utilizing geometrically confined pluripotent stem cells. However, the traditional reaction-diffusion equations could not satisfactorily explain the concentric ring distributions of various cell types, as they do not yield circular patterns even for circular domains. In previous mathematical models that yield ring patterns, certain conditions that lack biophysical understandings had been considered in the reaction-diffusion models. Here we hypothesize that the circular patterns are the results of the coupling of the mechanobiological factors with the traditional reaction-diffusion model. We propose two types of coupling scenarios: tissue tension-dependent diffusion flux and traction stress-dependent activation of signaling molecules. By coupling reaction-diffusion equations with the elasticity equations, we demonstrate computationally that the contraction-reaction-diffusion model can naturally yield the circular patterns.

scholarly journals Blow-up analyses in nonlocal reaction diffusion equations with time-dependent coefficients under Neumann boundary conditions

2020 ◽  
Vol 18 (1) ◽  
pp. 1552-1564
Author(s):  
Huimin Tian ◽  
Lingling Zhang
Keyword(s):  
Boundary Conditions ◽  
Blow Up ◽  
Sufficient Conditions ◽  
Neumann Boundary ◽  
Reaction Diffusion ◽  
Neumann Boundary Conditions ◽  
Reaction Diffusion Equations ◽  
Time Dependent ◽  
Diffusion Equations ◽  
Nonlocal Reaction

Abstract In this paper, the blow-up analyses in nonlocal reaction diffusion equations with time-dependent coefficients are investigated under Neumann boundary conditions. By constructing some suitable auxiliary functions and using differential inequality techniques, we show some sufficient conditions to ensure that the solution u ( x , t ) u(x,t) blows up at a finite time under appropriate measure sense. Furthermore, an upper and a lower bound on blow-up time are derived under some appropriate assumptions. At last, two examples are presented to illustrate the application of our main results.

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