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

Abstract

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.

The Fractal Tapestry of Life: III Multifractals Entail the Fractional Calculus

“…We closed prequel II [76] with a brief discussion of multifractals along with a promise to expand on its overlap with the FC. These earlier comments began with the observation of the need to replace the historical engineering paradigm of 'signal-plus-noise' with a model of biomedical time series having fractal statistics.…”

“…A number of physiologic time series were found to be characterized by a distribution of scaling parameters and therefore to belong to a broader class of complex processes known as multifractals. Such multifractal time series appear in the rich variability of healthy physiological networks, including but not restricted to human gait [32], cerebral blood flow (CBF) [77,78] and heart rate variability (HRV) [79][80][81], and changes in scaling behavior [82] reflect certain kinds of pathologies, as discussed in prequel I [2] of this series.…”

“…He enabled the more adventurous to step beyond the safeguards of geometry during that time of transition into the new and more encompassing world of the differential. It was speculated in prequel I that we have reached another epoch of transition [2]: . .…”

“…It is the statistics of physiological fluctuations that determine the spatial properties of the tree-like structures of the human lung, arterial and venous systems, and other ramified structures [5]. Statistical fractals [6,7] determine the probability density function (PDF) of time intervals in the beating human heart [2,[8][9][10], in respiration [11][12][13][14], synchronous response to a metronome [15], in human locomotion [16][17][18][19], in the human nevous sysem [20], in the dynamics of the brain [21][22][23][24], in the walking rehabilitation of the elderly [25,26], motor control [27,28], and interpersonal coordination [29,30], and in human cognition [31].…”

“…For example, such behavior is found in the firing of certain neurons [22,33,34], as well as in the time series of posture control [35], which determine the dynamic properties of physiologic networks having a large number of characteristic time scales. These and other fractal physiology processes have been the focus of interdisciplinary research on complex networks for more than a decade; see [2,[36][37][38].…”