Fractality: An Adaptive Evolutionary Conserved Bioengineering Design of the Gas Exchangers

1. Introduction

Optimal levels of chaos and fractality are distinctly associated with physiological health and function in natural systems. [1]

Fractality or self-similarity is an elemental structural property of many, if not all, complex physical- and natural assemblages [2,3,4,5,6,7,8,9,10,11,12]. Fractal geometry (FG) has been designated as the ‘design of nature’ [3,4,9,10,11,12,13]. The physicist, polymath and information theorist Benoit B. Mandelbrot (1924-2010), whose broad interest was in the inexactness and stochasticity of natural forms, phenomena and processes, conceived the word ‘fractal’ from the Latin word ‘fractus’, meaning ‘fragmented’, ‘uneven’ and ‘broken’ [2,3]. FG is a branch of applied mathematics which characterizes complicated states and phenomena. It has been meaningfully applied to analyze complex structures and processes in diverse disciplines such as architecture, physics, engineering, chemistry, physiology-, pathology, pathogenesis of diseases, morphogenetic development and even in aspects such as human language, information technology, cognitive development, consciousness, urban growth and the dynamics of capital markets [14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29]. Among other properties, fractal forms and mechanisms exhibit well-ordered iteration, space-filling, self-similarity and scale invariance [2,3]. To various extents and levels of organization, practically all biological assemblages exhibit fractality [4,30,31,32,33,34,35]. Mathematically, fractalities are holistically expressed by power law functions, where the fractal dimensions (D_F) are non-integer exponents [2,3,4,36,37,38]: inclusively, D_F denotes the structural complexity of a self-similar natural assemblage.

Respiration is the mechanism by which molecular O₂ is acquired and at cellular level (in the mitochondria) utilized to generate energy in form of ATP (adenosine phosphate) which drives the metabolic processes: CO₂, a byproduct of the process, is eliminated [39,40,41]. It comprises highly coordinated central neural control, a battery of sensory input systems, functionally efficient ventilatory mechanisms and well-developed and effective gas exchangers, i.e., respiratory organs [42,43,44,45]. Animals will generally live for weeks without food, days without water and only minutes without molecular O₂ [46]. From its important role in driving metabolic processes, Max Kleiber [47] termed molecular O₂ as ‘the fire of life’. Adequate acquisition of O₂ was an evolutionary necessity to realization of the structural- and functional novelties which permitted animals to live and thrive under dynamically different environmental conditions and lead different lifestyles [43,48,49,50,51,52,53]. With trees being the most abundant and conspicuous examples, branched or dendritic structures preponderate in nature [54,55,56,57,58,59]. For the human lung, forming what is commonly termed ‘respiratory tree’, the branched bronchial (airway) morphology comprises a recursive assemblage of as many as twenty-three branches that terminate in a ‘crown = canopy’ of millions of the microscopic alveoli [38,60,61,62]. The regular branching of the bronchial- and the vascular systems (the pulmonary artery and vein) of the mammalian lung allow optimal flows of air and blood in the lung [32,63,64,65,66,67,68,69].

Here, the morphologies of various gas exchangers of phylogenetically diverse animal taxa are compared. It is shown that the principal structural components fundamentally share a branched structure. The design generates large respiratory surface area (RSA) in confined spaces while a close patterning and proximation of the passageways (airway- and the vascular systems) brings the respiratory fluid media, respectively air/water and blood, into close-proximity across thin tissue partitioning, a structural feature that enhances gas exchange by passive diffusion (60-62). In evolutionary terms, fractality is an important conserved structural design which importantly explains the layouts of the functional components of tissues, organs and organ systems.

2. Gas Exchangers

In different animal taxa, developed to various levels of structural complexity and functional efficiency, alongside other roles, gas exchangers (respiratory organs) are foremost involved in respiration, i.e., acquisition of O₂ and elimination of CO₂. Mammals and birds, the primary vertebrates which have evolved full endothermic-homeothermy, possess exceptionally morphologically complex and physiologically efficient gas exchangers [61,62,63,70,71] with the main structural- and functional systems comprising the airway- and the vascular (pulmonary artery and vein) systems. Anatomically, the parts track and pattern each other and closely proximate (Figure 1a-d; e-h). A constellation of gas exchangers has evolved in the animal life [43].

2.1. Bronchioalveolar (Mammalian) Lung

The volume of the mammalian, i.e., bronchioalveoar, lung is determined by that of the thoracic cavity (chest) which is bordered by the thoracic vertebrae (dorsally), ribs (laterally) and the diaphragm (caudally): extensive RSA and large pulmonary capillary blood volume (PCBV) are optimized in a limited space. For a 74 kg body mass person, while the mean total lung volume is ~4.3 L, the RSA is 143 m² and the PCBV is 213 cm³ [72]. Although outstandingly large, as observed by Rao and Johncy [73], it is important to correct a statement which often appears especially in the popular literature and unfortunately in journal articles that the RSA of the human lung is equivalent to that of a tennis court. While the surface area of a tennis court, which most people are familiar with, is an informative example, the claim lacks scientific validity and it is a clear overstatement. The surface area of a double’s tennis court is ~262m² while that of a single’s one is 196m²: the RSA of the human lung does not meet either of the measurements.

The vast RSA of the vertebrate lung is achieved by the regular branching of the bronchial- and the vascular systems (Figure 1). For the human lung, illustratively, Weibel [74] compared the compacting of the bronchial- and the vascular systems in the human lung to folding a piece of A-4 size paper to fit into a thimble! At regular intervals, the bronchial- and the vascular systems branch at constant angles and decrease in diameter at consistent ratios [38,60,61,62] (Figure 2), structural properties which optimize air- and blood flows, making ventilatory- and perfusive functions cost-effective [71,75,76,77]. The morphological patterning and tight packing of the airways- and the blood vessels of the vertebrate lungs [58,60,61,62] (Figure 1) develops by a widely conserved processes termed branching morphogenesis [10,32,38,78,79,80,81,82,83] which is well-regulated spatiotemporally expression of various molecular cues or growth factors [10,32,38,78,79,80,81,82,83].

The bronchial system of the human lung comprises 23 branches that terminate in ~500 million alveoli [60,61,62,84,85,86,87,88]. Originating from the right ventricle, the pulmonary artery of which compared to the pulmonary vein branches and patterns relatively more closely to that of the bronchial tree [62], generates ~70 million precapillary arterioles, >250,000 arterioles and 280 billion blood capillary segments which contain ~200 cm³ of pulmonary capillary blood volume [72,89]. After blood is oxygenated in the lung, it is returned to the heart via the pulmonary vein [90,91]. From their morphologies, Lefevre [75] determined that the blood vessels of the human lung are characteristically fractal. For the pulmonary vascular systems of the pig- (Figure 1a-d), the human- (Figure 1e-h), the dog- and the cat lungs, the angles of bifurcation are constant across the generations and the diameters decrease at about a constant ratio [61,92,93,94,95,96,97,98] (Figure 2). In the vertebrate lungs, the branching diameter value of the airways is about the perfect measure which accords with the Hess-Murray’s Law of Hemodynamics [64]: the flow of air obeys what has been termed ‘principle of minimum work’ or ‘principle of adequate design’ [59,76,99].

The morphologies and morphometries of the bronchial- and the vascular systems of the pig- [98] (Figure 1 a-d) and the human lungs [38] (Fig. e-h) have been investigated by latex rubber cast preparations: the diameters and the lengths of the constitutive parts and their angles of bifurcation were determined. For the human lung, the diametric- and the length D_Fs of the three main structural components were determined by plotting the total branch diameters against the mean branch diameters and the total branch lengths against the mean branch lengths on double logarithmic graphic scales [38] (Figure 3). For the bronchial system, the pulmonary artery and the pulmonary vein, the diameter-based D_Fs were respectively 2.714, 2.882 and 2.334 while the respective length-based D_Fs were 3.098, 3.916 and 4.041 [38]. The relatively greater length-based D_Fs suggested that compared to diameters, branch lengths may contribute more to the fractality of the structures. The high D_Fs of the bronchial and the vascular systems of the structural components of the human lung show the high fractality, i.e., intense space-filling properties, of the three main structural parts of the organ (Figure 3).

2.2. Parabronchial (Avian) Lung

Among the air-breathing vertebrates, the avian respiratory system (the lung-air sac system) is structurally the most complicated and functionally the most efficient [43,100,101,102,103,104,105,106] (Figure 4a-f). The bronchial system of the parabronchial (avian) lung comprises a three-tiered hoop-like assemblage of bronchial passageways which comprise an intrapulmonary primary bronchus, four sets of secondary bronchi (SB) and many parabronchi (tertiary bronchi) which interconnect the SB [103,106] (Figure 4a-d). Like for the pig (Sus scrofa)- [98] (Figure 1a-d) and the human lungs [38] (Figure 1e-h), the bronchial- and the vascular systems of the avian lung closely track and pattern each other (Figure 4e). However, while the three main structural parts of the mammalian lung branch by regular dichotomy [85,86,89,94,107] [Figure 1; Figure 2 (ii, iii)], the avian ones develop by monopodial branching [108,109] (Figure 4g-j). For the avian lung, the different developmental process may increase the number of hexagonal-shaped parabronchi which contain gas exchange tissue in relatively smaller lungs [43,48,49,100,101,103,104,106,110,111]. The structure of the circulatory system of the avian lungs has been well-investigated [112,113,114,115,116,117,118]. The pulmonary artery and vein, which respectively deliver deoxygenated (venous) blood to the lung (from the heart) and return oxygenated (arterial) blood to it, closely follow each other (Figure 4e). Although the D_Fs of the bronchial-, the pulmonary artery- and the pulmonary veins of the avian lung have not yet been determined, the particularly complex morphological organizations of the structures (Figure 4a-f) suggest high fractalities.

2.3. Insectan Tracheal System

Among the gas exchangers that have evolved in multicellular animals, the tracheal system of insects is unique in that O₂ is delivered directly from air to the body tissues and cells at extremely high partial pressure of O₂ (PO₂) [119,120,121,122]. The respiratory mechanism profoundly varies from that which exists in ‘lunged’ animals, where a circulatory system separates the gas exchanger (respiratory organ) from the tissues and cells of the body, a point at which the PO₂ has decreased close to zero. Compared with the bronchial system of the vertebrate lungs, the insectan tracheal system delivers ten times more O₂ per gram body tissue [123]. The tracheal system, which saturates the insectan body, comprises a copious system of branched conduits (Figure 5a) which are strengthened by spiral coils of chitin that are called taenidia. Adaptively, the volume density of the tracheal system of an insect’s body per unit body mass correlates with attributes such as body size, metabolic capacity and the PO₂ in the air [120,121,122,123,124,125]. As the trachea branch, they narrow and become thin-walled [120,121,126,127]. At the point they enter tissues, the passageways are termed tracheoles [Figure 5a (ii)]. In the extremely highly metabolically active insects such as dragon flies (Order: Odonata), for tissues such as the flight muscles, to reduce the diffusional distances of O₂, the tracheoles depress the cell walls to end very close to the mitochondria [121,123,128]. Like for the bronchial system of the mammalian- [60,61,62] (Figure 1) and the avian lungs [85,86,129] (Figure 4a-d), the tracheal system of insects develops by the conserved process of branching morphogenesis which is controlled by well-regulated spatiotemporal expression of growth (molecular) factors [38,129,130,131]. For the insectan tissues and cells, the branching and hence the fractality of the tracheal system provides efficient delivery of O₂ and elimination of CO₂.

2.4. Labyrinthine Organs of Catfishes

Depending on environmental conditions, bimodal-breathing or transitional fish such as the catfishes (Order: Siluriformes) adaptively acquire O₂ from both water (using gills) and air by means of so-called accessory respiratory organs (AROs) [132,133,134,135,136,137,138,139]. The AROs comprise the suprabranchial chamber membranes and the labyrinthine organs (LOs) [132,133,134,135,136,137,138,139] (Figure 5(ii). The LOs largely originate from the second- and fourth gill arches and slot into the suprabranchial chambers, concavities that are located dorsal to the brachial (gill) arches [Figure 5a (i)]. Termed ‘dendritic or arborescent organs’, morphologically, the LOs are ‘tree-like’ dichotomously branched structures (Figure 5b). Their surfaces are well-vascularized [Figure 5(iii)]. The fractality of the LOs optimizes the RSA in the limited space which is bordered by the hard-walled suprabranchial chambers.

2.5. Chorioallantoic Membrane of the Avian Egg

Regarding the cleidoic (self-supporting = self-containing) avian eggs, gas exchange occurs by diffusion across the pores which perforate the calcaneous shell and during incubation across the chorioallantoic membrane (CAM) to support embryonic development (Figure 5c). The CAM, which is a simple extraembryonic membrane (Figure 5c), also plays other important functions such as transfer of calcium from the shell to the embryo for growth and development especially of bones, maintaining acid-base homeostasis, ion- and water reabsorption from and storage of excretory waste products in the allantois [140,141,142,143]: from its respiratory- and nourishment roles, the CAM is analogous to the placenta of the viviparous mammals. The CAM develops by fusion of the splanchnic mesoderm of the allantois and the somatic mesoderm of the chorion [144]. To promote acquisition of O₂ during embryonic development, vascularization of the CAM increases significantly [145,146,147,148]: the process occurs by the morphogenetic process of branching morphogenesis which is regulated by well-regulated spatiotemporal expression of molecular (growth) factors [78,81,82,145,149]. The fractality of the CAM vasculature (Figure 5c) increases the RSA and blood volume, features which increase respiratory efficiency.

2.6. Pores of the Ostrich Egg-Shell

Among the extant species of birds, weighing as much as 2 kg, the ostrich (Struthio camelus) egg [Figure 5c(i); d] is the largest and the mean thickness of the shell (Figure 5e) is as sizable as 2 mm [147,150]. Together with physically protecting the developing embryo and regulating water loss, the egg-shell performs an important respiratory function. Oxygen and CO₂ diffuse across the pores (Figure 5d, e) which are more plentiful on the blunt end on the shell, i.e., the air-cell part [147,150]. In various parts of the shell, the shapes and sizes of the pores remarkably vary in size and morphology [150] (Figure 5f). Generally, they display complex tree-like, i.e., branched, (fractal) morphology (Figure 5f) should promotes transfer of respiratory gases across a thick shell.

2.7. External Gills

Gills are evaginated respiratory outgrowths which are permeable to respiratory gases, i.e., O₂ and CO₂ (Figure 5g; h): well-perfused with blood and ventilated with water, they evolved for water-breathing. From their location and morphology, gills are categorized as either external- or internal types [151]: the former, which are structurally less elaborate and functionally less efficient, dangle freely into surrounding water while the later, which are more derived and structurally- and functionally more efficient are either covered by a mesenchymal tissue mass (e.g., for elasmobranchs, i.e., sharks) or a movable operculum (e.g. in teleosts) [Figure 5h(i)]. For the teleosts, the gills commonly comprise four pairs of gill arches that are contained in the branchial cavity [Figure 5h(i)].

Teleost gills comprise hundreds of gill filaments that branch into thousands of thin flap-like semicircular-shaped secondary lamellae [151,152,153,154,155,156] [Figure 5g, h]. Gas exchange occurs across the very thin water-blood barriers of the secondary lamellae [Figure 5h(ii)]. The hierarchical, i.e., stratified or branched, morphology of the teleost gills presents a fractal design which generates large respiratory surface area in a limited space, i.e., the brachial cavity. Compared to more sluggish species, more active fish species such as the streamlined tuna have relatively greater total gill respiratory surface areas which are achieved from greater numbers and closer spacing of secondary lamellae [151].

3. Conclusions

During the evolution of life, a process that has spanned a period of ~4 billion years [157,158], nature has tinkered with, crafted and optimized a collection of functional contrivances (traits) to overcome existing adversities. By exploiting such ‘solutions’, biomimetic studies have expedited engineering of solutions to challenges which humans face [159,160,161,162], [https://www.linkedin.com/pulse/biomimicry-engineering-design-learning-uohae, accessed on 18-11-2925]. Compared to the conventional ‘trial-and-error’ practice in engineering, mimicking nature (biomimicry) very so often presents more efficient and cost-effective resolutions to technology-centered problems, especially those regarding resource and energy utilization efficiency and resilience [163,164]. Fractal geometry explains how at various scales of organization, self-similar, i.e., iterated, properties generate profoundly robust and efficient biological assemblages. West [165] observed that ‘the fractal concept provides a mechanism for the morphogenesis of complex structures which are more stable than those generated by classical scaling, i.e., they are more error tolerant’. In biology, incontrovertibly, fractality is an adaptive novelty [7,10,27,32] which confers optimal utilization of spaces, which in animal bodies are always at premium [68]. For gas exchangers, branched or well-ordered morphology increases the RSA, a structural feature which increases passive diffusion of O₂ and CO₂ between respiratory fluid media. For the brain, copious neuronal networks specify efficient communication. Many other structural components of the animal body, e.g., the airway and the vascular systems, display complex fractal properties which cannot be meaningfully quantified by the traditional Euclidean geometry. In medicine, the non-integer fractal dimension (D_F) or scaling exponent is a highly instructive diagnostic tool [166,167,168,169,170]. Alteration from normal structure to abnormal, pathological- and disease states consequences in measurable change in fractality [25,166,167,168,169,170]: healthy organs/structures appear to have D_Fs which fall within a narrow range while disease and pathological conditions consequence in alteration of the normal fractality. Determination of D_FS from medical images such as those acquired by magnetic resonance imaging (MRI), microComputer tomography (μCT), X-ray and histological tissue preparations or time-series signals such as electroencephalogram (EEG), magnetoencephalography (MEG) and electrocardiogram (ECG) now provide physicians with resources on which after application of, e.g., fractal analysis various diseases and conditions can be diagnosed often much earlier than it is possible by traditional clinical methods [169,171,172,173,174,175,176,177]. Recently, application of FG was recognized as one of the important medical advances of non-invasively diagnosing diabetic retinopathy [170]. For the bronchial tree of the human lung, decrease of the D_F, which presents as decrease in branching complexity, ensues from respiratory diseases such as chronic obstructive pulmonary disease (COPD) [168] and for the brain, decrease of the D_F of the neural network is associated with Alzheimer’s disease [23,178,179,180]. Furthermore, application of FG provides a new method of evaluating physiological states such as cardiac risk and assessment of the aging process [20,173,177]. Although in the recent past noteworthy progress has been made in application of FG towards understanding the functional designs of biological states and processes, only very few organs have been thoroughly and comprehensively analyzed to the degree of determining the D_Fs. Differences of the fractalities of the structural components of the various organs and tissues of different animal species may explain the variations of the predispositions of tissues and organs to pathogens, diseases and pathological afflictions. Moreover, the data may be instructive in the formulation of effective human interventions.