Non-Linear Behaviours in the Dynamics of Some Biostructures (original) (raw)
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Dynamics of Biostructures on a Fractal/Multifractal Space-Time Manifold
Progress in Relativity [Working Title]
A theory of space-time is built on a fractal/multifractal variety. Thus, considering that both the spatial coordinates and the time are fractal/multifractal, it is shown that both the energy and the non-differentiable mass of any biostructure depend on both the "state" of the biostructure and a speed limit of constant value. For the dynamics on Peano fractal/multifractal curves and Compton scale resolutions, it is shown that our results are reduced to those of Einstein relativity. In such a context, it has been shown that the "chameleon effect" of cholesterol corresponds to the HDL-LDL state transfer dictated by the spontaneous symmetry breaking through a fractal/multifractal tunnel effect. Then both HDL and LDL become distinct states of the same biostructure as in nuclear physics where proton and neutron are distinct states of the same nucleon.
Fractal Transport Phenomena through the Scale Relativity Model
… Physica Polonica A, 2009
A correspondence between Nottale's scale relativity model and Cresson's mathematical procedures is analyzed. It results that the "synchronization" of the movements at different scales (fractal scale, differential scale etc.) gives conductive type properties to the fractal fluid, while the absence of "synchronization" is inducing properties of convective type. The behavior of a conductive fractal fluid is illustrated through the numerical simulation of plasma diffusion that is generated by laser ablation. Rotational and irrotational convective behaviors of a fractal fluid are established. Particularly, at Compton spatial and temporal scales, the irrotational behavior implies the standard Schrödinger equation.
On the Non-Linear Dynamics in Biological Structures. Complementary Mathematical Aspects
2015
Assuming that the biological systems are fractal systems, a few aspects ofnatural dynamics in biological structures are studied. The "non-linear dynamics" analysis in an arbitrary space with constant fractal dimension, using an extended version of the Scale Relativity Theory, has been performed. Additionally, a dedicated mathematical model of biological non-linear system by association with stochastic Levi type processes was developed.
Fractal model for blood flow in cardiovascular system
Computers in Biology and Medicine, 2008
Blood flow in the cardiovascular system is the central point of experimental and theoretical investigation. The objective of the study is to determine the blood flow in the cardiovascular system using Darcy's law, Reynold's number and Poiseuille's equation. A possible way of modeling of self-similar biological tree-like structure is proposed. Special attention is paid to the blood vessel system, with elaboration on a model with certain spatial arrangement of the vessels and reasonable dependence of the blood pressure on the vessels diameter such that the organism has a homogeneous oxygen supply. Flow analysis in the above systems is analyzed by invasion percolation. The blood flow in the cardiovascular system has been numerically calculated for both normal and abnormal patients. A new algorithm has been introduced to visit the blood vessels in a robust manner which avoids loops and provides us the results in a simple manner.
Implications of an extended fractal hydrodynamic model
European Physical Journal D, 2010
Considering that the motions of the particles take place on continuous but non-differentiable curves, i.e. on fractals with constant fractal dimension, an extended scale relativity model in its hydrodynamic version is built. In this approach, static (particle in a box and harmonic oscillator) and time-dependent (free particle etc.) systems are analyzed. The static systems can be associated with a coherent fractal fluid (of superconductor or of super-fluid types behavior), whose particles are moving on stationary trajectories. The complex speed field of the fractal fluid proves to be essential: the zero value of the real (differentiable) part specifies the coherence of the fractal fluid, while the non-zero value of the imaginary (non-differentiable or fractal) part selects, through some “quantization” relations, the “stationary” trajectories (that may correspond to the observables from quantum mechanics) of the fractal fluid particles. Moreover, the momentum transfer in the fractal fluid is achieved only through the fractal component of the complex speed field. The free time-dependent systems can be associated with an incoherent fractal fluid, and both the differentiable and fractal components of complex speed field are inhomogeneous in fractal coordinates due to the action of a fractal potential. It exist momentum transfer on both speed components and the “observable” in the form of an uniform motion is generated through a specific mechanism of “vacuum” polarization induced by the same fractal potential. The analysis on the fractal fluid specifies conductive properties in the case of movements synchronization both on differentiable and fractal scales, and convective properties in the absence of synchronization.
Blood flow simulation through fractal models of circulatory system
Chaos Solitons & Fractals, 2006
The blood flow in human arteries has been analytically calculated according to PoiseuilleÕs equation. Geometry of the fractal arterial trees has been described in previous article [Gabryś E, Rybaczuk M, Kędzia A. Fractal model of circulatory system. Symmetrical and asymmetrical approach comparison. Chaos, Solitons & Fractals, in press]. Blood vessel trees are consisted of straight, rigid cylindrical tubes. In each bifurcation two new children segments appears according to Murray law. Blood flow in circulatory system is driven by the pressure differences at the two ends of the blood vessel. A mathematical analysis shows the continuous dependence of the solution on vessel tree parameters and boundary condition.
Flow of a Self-Similar Non-Newtonian Fluid Using Fractal Dimensions
Fractal and Fractional
In this paper, the study of the fully developed flow of a self-similar (fractal) power-law fluid is presented. The rheological way of behaving of the fluid is modeled utilizing the Ostwald–de Waele relationship (covering shear-thinning, Newtonian and shear-thickening fluids). A self-similar (fractal) fluid is depicted as a continuum in a noninteger dimensional space. Involving vector calculus for the instance of a noninteger dimensional space, we determine an analytical solution of the Cauchy equation for the instance of a non-Newtonian self-similar fluid flow in a cylindrical pipe. The plot of the velocity profile obtained shows that the rheological behavior of a non-Newtonian power-law fluid is essentially impacted by its self-similar structure. A self-similar shear thinning fluid and a self-similar Newtonian fluid take on a shear-thickening way of behaving, and a self-similar shear-thickening fluid becomes more shear thickening. This approach has many useful applications in indus...
Fractal structures in stenoses and aneurysms in blood vessels
Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2010
Recent advances in the field of chaotic advection provide the impetus to revisit the dynamics of particles transported by blood flow in the presence of vessel wall irregularities. The irregularity, being either a narrowing or expansion of the vessel, mimicking stenoses or aneurysms, generates abnormal flow patterns that lead to a peculiar filamentary distribution of advected particles, which, in the blood, would include platelets. Using a simple model, we show how the filamentary distribution depends on the size of the vessel wall irregularity, and how it varies under resting or exercise conditions. The particles transported by blood flow that spend a long time around a disturbance either stick to the vessel wall or reside on fractal filaments. We show that the faster flow associated with exercise creates widespread filaments where particles can get trapped for a longer time, thus allowing for the possible activation of such particles. We argue, based on previous results in the field of active processes in flows, that the non-trivial long-time distribution of transported particles has the potential to have major effects on biochemical processes occurring in blood flow, including the activation and deposition of platelets. One aspect of the generality of our approach is that it also applies to other relevant biological processes, an example being the coexistence of plankton species investigated previously.
2019
In the framework of Fractal Theory of Motion for the Scale Relativity Theory with arbitrary and constant fractal dimensions, dynamics in complex systems associated to the fractal-non-fractal transition are analyzed. Working with the assumption that these dynamics are described by means of fractal curves, Lorenz type behaviors become “operational” through a Galerkin method. Then Rayleigh and Prandtl effective numbers are specified both by means of classical kinetic coefficients and scale resolution while the dynamics variables act as the limit of a family of mathematical functions, non-differentiable for non-null scale resolution.