Diagnosing Disease with Multifractality
Authors: Copling Sage 
Published: May 2022

The resting activity of the heart, without external sensory input, has provided novel information on the interactions that occur between biological and entropic systems in the body. The intersection between multifractality and disease diagnosis has been extensively worked on in the biophysical field, and yet, it is one that still has a lot of potential for new discoveries. In this paper, I will attempt to briefly describe the current literature on the use of multifractality on disease diagnosis, in addition to briefly comment on the future of this diagnostic methodology in the fight against cancer.

A fractal is described as a never-ending pattern, one that is infinitely complex and seems to repeat a process over and over in a loop. Fractals exhibit self-similarity, meaning they are patterns that are identical or near-identical on many scales, including time scales. In the context of this paper, fractals are visible patterns in the heartbeat 1[Formula: see text]s into a time series that will also be visible 1 day into a time series. This self-similarity is described by exponents. For example, monofractal processes only scale fractally in one manner, meaning that one exponent will help define them mathematically. On a graph of a power law over time, a monofractal state would present as a linear curve, as one exponent is defining it. Multifractality, on the other hand, is a term defining a spectrum of exponents used to help mathematically define a natural state. It would present as a nonlinear curve on the graph of a power law, as multiple exponents of multiple orders are describing its self-similarity over time. 1 A heartbeat time series, in this paper, will be defined as 1800 evenly-spaced measurements of heart rate from one patient.5 In addition, the term crucial renewal events, also called crucial events, will be defined as events in a heartbeat time series that store the long-term memory of the heartbeat, therefore impacting the future patterns of the heartbeat. Crucial events build upon each other, meaning that the occurrence of earlier crucial events will correlate to the occurrence of subsequent crucial events. Over time, a decrease in the correlation between crucial events would indicate the presence of Poisson-like events, which in this paper will be defined as a disturbance in the healthy physiological process of a heartbeat. 3

The concept that multifractality and crucial events may play a role in disease diagnosis has been presented in different ways in the past. The first method was through broad multifractal spectrum analysis, in which Ivanov et al. determined that a loss of multifractality occurs in a non-healthy state, specifically when they analyzed congestive heart failure. This finding suggested that the presence of pathology moved the heartbeat closer to a monofractal state, making the difference between healthy and pathological individuals easy to identify. 2 The second method, presented later on, presented evidence that healthy patients were less likely to have unrelated Poisson-like events than diseased or unhealthy patients. Crucially, West and Grigolini in 2017 were able to find an intersection between these two methods by proving that increasing the percentage of unrelated Poisson-like events occurring in a system would directly correlate to a narrower multifractal spectrum, connecting the two diagnostic methods and providing a view of multifractality such that it could have a drastic impact on potential diagnosis methodologies in the future. 3 , 4