A healthy dose of chaos: Using fractal frameworks for engineering higher-fidelity biomedical systems

Abstract

Optimal levels of chaos and fractality are distinctly associated with physiological health and function in natural systems. Chaos is a type of nonlinear dynamics that tends to exhibit seemingly random structures, whereas fractality is a measure of the extent of organization underlying such structures. Growing bodies of work are demonstrating both the importance of chaotic dynamics for proper function of natural systems, as well as the suitability of fractal mathematics for characterizing these systems. Here, we review how measures of fractality that quantify the dose of chaos may reflect the state of health across various biological systems, including: brain, skeletal muscle, eyes and vision, lungs, kidneys, tumours, cell regulation, skin and wound repair, bone, vasculature, and the heart. We compare how reports of either too little or too much chaos and fractal complexity can be damaging to normal biological function, and suggest that aiming for the healthy dose of chaos may be an effective strategy for various biomedical applications. We also discuss rising examples of the implementation of fractal theory in designing novel materials, biomedical devices, diagnostics, and clinical therapies. Finally, we explain important mathematical concepts of fractals and chaos, such as fractal dimension, criticality, bifurcation, and iteration, and how they are related to biology. Overall, we promote the effectiveness of fractals in characterizing natural systems, and suggest moving towards using fractal frameworks as a basis for the research and development of better tools for the future of biomedical engineering.

Keywords: Biomedical engineering; Chaos; Fractals; Health; Pathophysiology; Tissue engineering.

Fig. 1:. The Coastline illustration of the concept of fractal scaling.

(A) Breakdown of Great Britain’s self-similar features in the coastline. (B) Measured length of the fractal coast increases as the scale of measurement, or compass setting, decreases. By contrast, the length of a non-fractal circle’s circumference has a converging value. (C) The scaling of a fractal structure follows a power law from which the fractal dimension may be approximated. Measurement precision is defined as the inverse of the compass setting where for a small setting s the precision 1/s is large. Figure reproduced with permission from [].

Fig. 2.. A healthy dose of chaos.

Different physiological systems and the phenomena that occur when the fractal dimension characterizing the system becomes too low or too high. (A) Cell outline of microglia in different brain regions following experimental diffuse brain injury [reproduced with permission from]. (B) MRI images of grade III and grade IV gliomas [reproduced with permission from]. (C) Images of silicone rubber casts of normal and asthmatic lungs [reproduced with permission from]. (D) Computed tomography images of normal and idiopathic pulmonary fibrotic lungs [reproduced with permission from]. (E) Skin lesions that are benign or cancerous [reproduced with permission from]. (F) Collagen staining of unwounded or scarred dermis [reproduced with permission from]. (G) Sparser maternal retinal vasculature was associated with reduced fetal growth during pregnancy [reproduced with permission from]. (H) Fundus camera images of the retina and skeletonized traces of the retinal vasculature in normal and chronic kidney disease eyes [reproduced with permission from].

Fig. 3:. Hierarchical organization in tissues.

A) self-similar brain convolutions during gestational development [reproduced with permission from]. B) Lung branching in embryonic chicken forms at precise locations that scale relative to the size of the organ [reproduced with permission from]. C) Collagen fibrils and bundles follow self-similar fibrous structure across scales [reproduced with permission from]. D) Bone tissue with up to 12 (I to XII) levels of fractal organization based on collagen fibrils and mineral aggregates [reproduced with permission from]. E) Hierarchical fibrillar structure of cardiac tissue from actin-myosin filaments, myofilaments, myofibrils, myocytes, and muscle fibers [reproduced with permission from].

Fig. 4.. Diagnostic applications of fractal variability data analysis.

(A) (i) Images of H&E-stained hyperplastic/benign, biphasic and epithelioid tumor samples, with (ii) fractal dimension analysis revealing significant differences between benign and malignant groups [reproduced with permission from]. (B) (i) Images of retina and skeletonized vasculature, and (ii) associated graph of fractal dimension vs hazard ratio for 14-year coronary heart disease mortality, where extreme values of fractal dimension (1st and 4th quartile ranges) were associated with elevated risk [reproduced and modified with permission from]. (C) Risk ratio for the univariate prediction of cardiovascular disease mortality based on the short-term fractal scaling exponent (DFA1). Low values of DFA1 reflect reduced correlation in heart rate variability signals, and increased risk of mortality [reproduced and modified with permission from]. (D) Recording of patient fractal dimension variation for EEG trace over 3 minutes, showing onset and end of epileptic seizure at 50s and 120s respectively [reproduced with permission from].

Fig. 5:. Fractal design approaches for novel functional materials.

(A) Silk fibroin aggregates into defined fractal forms [reproduced with permission from]. (B) Partially-ordered-polypeptide materials form hierarchical fractal structures based on collective, scaling interactions among ordered and disordered domains, with tunable hysteresis and tissue integration properties [reproduced with permission from]. (C) Minimal matrix model of scar tissue was fabricated by heterogeneously embedding collagen into gel resulting in fractal stiffness properties [reproduced with permission from]. (D) Fractal coiling designs of wires allows conformal mounting of metal sensor devices on soft skin while providing increased stretchability and delayed elastic-to-plastic material transition properties [reproduced with permission from]. (E) Stretchable device made by kirigami-based fractal hierarchical cutting [reproduced with permission from]. (F) Fractal electrodes improve excitability in simulation of neural integration for retinal therapy [reproduced with permission from]. (G) Fractal sensors increase surface area in a diagnostic chip to drastically reduce time-to-readout and increase signal gain by several hundred percent [reproduced with permission from].

Fig. 6:. Therapeutic applications enhanced with variable patterning.

A) Variable frequency trains in functional electrical stimulators (FES) can support improved gait [reproduced with permission from]. B) Chaotic pacing was more effective at stabilizing random arrhythmias than periodic pacing. Y-axis indicates beat-to-beat intervals and x-axis indicates beat number. Start and end positions of chaotic control and periodic control are indicated [reproduced with permission from]. C) Respiratory sinus arrhythmia (RSA) is an important component of normal breathing, where heart rate increases with inspiration and decreases with expiration, and is associated with ventilation/perfusion matching. Fractal ventilation (FV) patterns in a study of mechanical ventilators was shown to increase RSA compared to conventional ventilation (CV) [reproduced with permission from].

Fig. 7.. Transitions between order and chaos.

Here we draw parallels between (A) a mathematical system that transitions between order and chaos with various steady states, and (C) a physiological system that also experiences transitions between various states of health and functionality. (A) The typical bifurcation diagram for the quadratic iterator, with control parameter r on the x-axis, and stable state on the y-axis [reproduced with permission from]. (B) Markings of the different transition regions and phenomena for the bifurcation diagram in (A). (C) Cardiac rhythms for different cardiac conditions. (i) A simulated example of a sinusoidal heart rate pattern (bottom) and generated embedding (top). Sinusoidal heart rate patterns may be observed in various severe pathologies such as severe fetal anemia, asphyxia, acidosis^–. (ii) Ventricular tachy arrhythmia, (iii) normal, (iv) ventricular fibrillation conditions, showing the mapped attractor (top), cardiac rhythm time series (bottom left), and corresponding frequency plot/power spectrum (bottom right) [reproduced with permission from]. (D) Bifurcation diagrams for several other mathematical systems, showing the universality of transition properties such as bifurcations, self-similarity, and transitions between chaos and order. These universal properties may be useful to track in biological systems as well. (i) Hénon’s transformations [reproduced with permission from]; (ii) the quadratic map x_t+1 =a(2x_t −1)² [reproduced with permission from] (iii) Rössler’s system [reproduced with permission from].