On the stability of stationary solutions in diffusion models of oncological processes

Debbouche, Amar; Polovinkina, M. V.; Polovinkin, I. P.; Valentim, Carlos Alberto; David, Sérgio Adriani

doi:10.1140/epjp/s13360-020-01070-8

Cited by 10 publications

(5 citation statements)

References 24 publications

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“…The oncological application of anomalous diffusion is made by Debbouche et al (2021), who remark that the first mathematical tumor growth models were integer partial differential equation (IPDE) models taking into account tumor, normal and dead cells, nutrition, various inhibitory substances and immune system response. Cells of a healthy organism are mortal with apoptosis being the end of the life cycle, whereas the lack of apoptosis is a main feature of tumor cells.…”

Section: Om and Diffusionmentioning

confidence: 99%

The Fractal Tapestry of Life: II Entailment of Fractional Oncology by Physiology Networks

West

2022

Front. Netw. Physiol.

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This is an essay advocating the efficacy of using the (noninteger) fractional calculus (FC) for the modeling of complex dynamical systems, specifically those pertaining to biomedical phenomena in general and oncological phenomena in particular. Herein we describe how the integer calculus (IC) is often incapable of describing what were historically thought to be simple linear phenomena such as Newton’s law of cooling and Brownian motion. We demonstrate that even linear dynamical systems may be more accurately described by fractional rate equations (FREs) when the experimental datasets are inconsistent with models based on the IC. The Network Effect is introduced to explain how the collective dynamics of a complex network can transform a many-body noninear dynamical system modeled using the IC into a set of independent single-body fractional stochastic rate equations (FSREs). Note that this is not a mathematics paper, but rather a discussion focusing on the kinds of phenomena that have historically been approximately and improperly modeled using the IC and how a FC replacement of the model better explains the experimental results. This may be due to hidden effects that were not anticapated in the IC model, or to an effect that was acknowledged as possibly significant, but beyond the mathematical skills of the investigator to Incorporate into the original model. Whatever the reason we introduce the FRE used to describe mathematical oncology (MO) and review the quality of fit of such models to tumor growth data. The analytic results entailed in MO using ordinary diffusion as well as fractional diffusion are also briefly discussed. A connection is made between a time-dependent fractional-order derivative, technically called a distributed-order parameter, and the multifractality of time series, such that an observed multifractal time series can be modeled using a FRE with a distributed fractional-order derivative. This equivalence between multifractality and distributed fractional derivatives has not received the recognition in the applications literature we believe it warrants.

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Section: Om and Diffusionmentioning

confidence: 99%

The Fractal Tapestry of Life: II Entailment of Fractional Oncology by Physiology Networks

West

2022

Front. Netw. Physiol.

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“…A refined sufficient condition for the asymptotic stability of a stationary solution consists in the fulfillment of two inequalities [25]:…”

Section: Diffusion Models Of Oncological Processesmentioning

confidence: 99%

“…Condition (25) for the zero solution turns out to be superfluous, since it becomes a consequence of condition (27).…”

Section: Diffusion Models Of Oncological Processesmentioning

confidence: 99%

“…In the work [25], in order to prove the sufficiency of conditions ( 24), ( 25), the authors had to use a weighted version of the Steklov-Poincare-Friedrichs inequality because of the convective term −v∂q/∂x in the second equation. Another weighted variant is used in the work [26], where it is shown that the condition…”

Section: Diffusion Models Of Oncological Processesmentioning

confidence: 99%

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On the Stability of Stationary States in Diffusion Models in Biology and Humanities

Polovinkina

Polovinkin²

2022

Lobachevskii J Math

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We consider an initial-boundary value problem for the system of partial differential equations describing processes of growth and spread of substance in biology, sociology, economics and linguistics. We note from a general point of view that adding diffusion (migration) terms to ordinary differential equations, for example, to logistic ones, can in some cases improve sufficient conditions for the stability of a stationary solution. We give examples of models in which the addition of diffusion terms to ordinary differential equations changes the stability conditions of a stationary solution.

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“…There is a myriad of physiological processes in biological systems [1] that enable an organism to perform multiple activities. The modelling of these phenomena is a challenging task wherein different mathematical techniques capable of adequately describing such models have been employed [2][3][4][5][6]. The brain is composed of a complex network of actions and reactions working in a coordinated effort to control several processes in the whole body.…”

Section: Introductionmentioning

confidence: 99%

Fractal Methods and Power Spectral Density as Means to Explore EEG Patterns in Patients Undertaking Mental Tasks

2021

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Brain electrical activity recorded as electroencephalogram data provides relevant information that can contribute to a better understanding of pathologies and human behaviour. This study explores extant electroencephalogram (EEG) signals in search of patterns that could differentiate subjects undertaking mental tasks and reveals insights on said data. We estimated the power spectral density of the signals and found that the subjects showed stronger gamma brain waves during activity while presenting alpha waves at rest. We also found that subjects who performed better in those tasks seemed to present less power density in high-frequency ranges, which could imply decreased brain activity during tasks. In a time-domain analysis, we used Hall–Wood and Robust–Genton estimators along with the Hurst exponent by means of a detrented fluctuation analysis and found that the first two fractal measures are capable of better differentiating signals between the rest and activity datasets. The statistical results indicated that the brain region corresponding to Fp channels might be more suitable for analysing EEG data from patients conducting arithmetic tasks. In summary, both frequency- and time-based methods employed in the study provided useful insights and should be preferably used together in EEG analysis.

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On the stability of stationary solutions in diffusion models of oncological processes

Cited by 10 publications

References 24 publications

The Fractal Tapestry of Life: II Entailment of Fractional Oncology by Physiology Networks

The Fractal Tapestry of Life: II Entailment of Fractional Oncology by Physiology Networks

On the Stability of Stationary States in Diffusion Models in Biology and Humanities

Fractal Methods and Power Spectral Density as Means to Explore EEG Patterns in Patients Undertaking Mental Tasks

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