For a long time, biology was a qualitative (descriptive) science. The investigations failed to fully explicate the functional designs of whole organisms and their constituent parts. About half a century ago, at an interdisciplinary meeting which was held in Feldberg (Germany), the International Society of Stereology (ISS) was formed. Mathematicians, statisticians and physical and biological scientists combined their skills to create a new scientific discipline of stereology that allowed for reliable and reproducible quantitation of structural entities of composite physical and biological materials and extrapolation of measurements made on two‐dimensional profiles/images to their three‐dimensional forms. With time, novel bias‐free sampling and quantitation techniques have been developed and tested. Presently, there is no justification for totally descriptive biological studies. Numerous books, publications, computer programmes and applications and dedicated microscopes exist for cost‐effective analysis. Within the relatively short time, it has been in existence, the ISS has actively advanced stereology which is now applied by scientists all over the world in various biological disciplines. Only basic understanding of mathematics, geometry and statistics is needed to do good stereology. Here, analysis of the avian (bird) lung is given to show the versatility and robustness of stereological techniques in analysing biological structures.
- avian lung
In biological studies, by default or design, the primary objective is to understand the relationships between the structure and the function of an organism or animal. The nature and organization of the cellular and tissue constituent parts specify its form and performance. For a long time, in rigidly compartmentalized scientific disciplines, it was not possible to correlate these aspects mainly because disciplines such as physiology and biochemistry generated functional (i.e. quantitative) data, structural studies such as morphology and anatomy were fundamentally descriptive. During the last half century, however, methods that allow biological structures to be meaningfully analysed have been developed by essentially synthesizing statistical and mathematical methods. Biology has profoundly changed from a mere observational, descriptive and classificatory discipline into a fully fledged branch of science with complex theories and concepts of quantitation and formulation of comprehensive mathematical models that powerfully explicate the forms and functions of complex dynamic systems. Quantifying and comparing different biological systems that share common design principles powerfully explain evolutionary and adaptive changes .
Quantitative studies require careful planning and execution. Before performing such a study, thorough understanding of the structure of a biological entity and its formative parts is imperative. A pilot study should precede the study and proper research questions formulated from it. A well‐planned and properly executed quantitative study must adequately support or refute articulated research questions:
The tissues analysed must be handled and processed carefully. This is to avoid introduction of artefacts during stages like sampling, processing, fixing and staining. The primary goal should be that the tissue analysed is preserved in its initial (natural) state. Since only a small fraction of the organ or tissue can routinely be analysed (Figure 1), rigorous unbiased and reproducible sampling is critical to meaningful quantitative studies [2, 3].
Morphometry means measurement of form (
A tailor who takes measurements of a person in order to make him or her a suit performs morphometry and not stereology. In the use of the highly informative transmission microscopes, be it light or electron, thin sections of tissues have to be cut to allow light or electron beam to pass through. The inescapable action converts 3D structures to 2D profiles (Figure 2). The profiles that are generated after sectioning cells, tissues or organs depend on the plane of sectioning (Figure 3). For example, circular profiles can be generated by sectioning tubular or spherical structures.
As a scientific discipline, stereology did not formally start until ∼55 years ago at a meeting of researchers from as varied disciplines as biology, geology, engineering and materials sciences in 1961. A biologist, Professor Hans Elias (specifically a histologist), organized a meeting at Feldberg in Germany which was attended by scientists with one goal in mind:
The start of stereology in the 1960s occurred simultaneously with important technological developments that included availability of powerful and affordable microscopes and introduction of innovative tissue preparation and staining techniques such as immunocytochemistry. These developments increased interest in quantitative biology in general and in the emerging field of stereology in particular. Investigators began to prefer the more objective stereological approaches over subjective evaluations that were known to be greatly affected by inter‐observer errors. Mathematicians and statisticians in particular have greatly contributed their unique skills to the theoretical aspects of the interdisciplinary discipline of stereology . The major weaknesses in the previous techniques that applied Euclidean formulas based on classical geometric shapes were identified and improved or replaced with more robust ones. Stereology is a developing science where new innovations continue to make important improvements in the efficiency of the techniques [2, 3]. Utilizing random systematic sampling and different analytical methods, stereology provides unbiased quantitative data.
This account illustrates the versatility of stereological techniques that have contributed greatly to the better understanding of the functional design of the avian lung [6–11]. Many publications, books and programs and algorithms (softwares) are now available on the discipline. Among these are Weibel [2, 5], Stuart , Gundersen and Jensen , Gundersen and Østerby , Mouton , Howard and Reed  and West .
2. The functional design of the avian lung
2.1. Formulation of research question(s)
It is less costly for an animal to fly a given distance per unit time than it is for it to run on the ground across the same distance in the same period of time. This is notwithstanding the fact that powered (active) flight is energetically a very costly form of locomotion [17–19]. The capacity of overcoming gravity and remaining stationary in air (hovering) requires high metabolic capacity and concomitant considerably high consumption of large amounts of O2 [20, 21]: a hovering hummingbird supports its body weight entirely by power generated by the flight muscles (Figure 5). Showing the highly selective nature of flight, powered flight has evolved in only a few animal taxa. It was achieved by insects ~350 million years ago (mya), by the now extinct pterodactyls ~220 mya, birds ~150 mya and bats ~50 mya, chronologically in that order. In terms of speed, endurance and high altitude travel, birds are excellent flyers. For example, a diving peregrine falcon,
2.2. Research question
From the above account, the research question posed was:
3. Stereological study of the avian lung
The health of a bird was ascertained before it was killed by sodium pentobarbitone (200 mg cm−3) at a dosage of 0 ± 5 ml kg−1 injected into the brachial vein. The body mass was measured for normalization of data to allow intra‐ and inter‐specific comparisons.
3.1. Fixation of the lung
A longitudinal incision was made along the neck and the trachea exteriorized. To avoid cutting any of the blood vessels associated with the neck and inadvertently introducing blood into the respiratory system, the trachea was cannulated through the larynx. The lungs were fixed by intratracheal instillation with 2.5% glutaraldehyde buffered with sodium cacodylate (pH 7.6 and osmolarity 350 mOsm) by gravity at a pressure head of 3000 Pa (1 cm H2O = 1 mbar = 102 Pa). The osmolarity of the fixative was made close to the physiological one of the blood plasma  to avoid tissue shrinkage. Osmolarity is a critical factor in morphometric studies because hyperosmotic reagents cause tissue shrinkage, while hypo osmotic ones may cause tissue swelling. Where necessary, shrinkage factors should be determined and introduced in the final calculations. The pressure that was used to fix the lung and the air sacs have been found to be sufficient to drive the fixative into the very narrow terminal respiratory units, the air capillaries, thereby causing optimal fixation . The fixation of the lung was performed by intratracheal instillation because the method preserves the erythrocytes that are important in the characterization and modelling of the lung: vascular perfusion of the lung washes away the erythrocytes. According to Crapo et al. , it also causes relatively greater degree of shrinkage compared to airway instillation. When the fixative stopped flowing down the trachea, the coelomic cavity was gently compressed to expel air trapped in the air sacs and that way increase the penetration of the fixative to different parts of the respiratory system. The trachea was then ligated and the fixative left
3.2. Determination of the lung volume
The extrapulmonary primary bronchus was trimmed at the hilum of the lung and the adhering fat and connective tissue removed before the volume of the lung was determined. This was done by weight displacement method  (Figure 6) which is based on Archimedes’ principle that states that a floating body displaces its own weight. Since the specific gravity of the water is unity, the increase in the weight caused by the volume of the fluid displaced by freely suspended lung equals its volume. As the specific gravity of the lung is less than one and hence the lung floats to the surface of the water, a metal wire of known volume was used to keep the lung submerged under water. At least three close measurements were made, and the average volume of the lung calculated.
Scherle’s  method is very accurate in determining volumes of small objects. There being no morphological and morphometric differences between the left and the right avian lungs , the left lung was used for light microscopic analyses and the right one for electron microscopy.
3.3. Light microscopic analysis
3.3.1. Tissue processing and sampling
Very small lungs were processed by the standard laboratory techniques, embedded in paraffin wax and serial transverse sections cut craniocaudally at a thickness of 10 µm. The sections were then stained with haematoxylin and eosin. Eight equidistantly spaced, i.e. stratified, sections were taken for analysis. The lungs were cut into slices along the costal sulci and the slices were then cut into halves just dorsal to the primary bronchus (Figure 7). Facing cranially, the half slices were processed and embedded in paraffin wax. The first technically adequate section from the cranial face of each half slice was stained with haematoxylin and eosin for analysis.
3.3.2. Determination of volume densities
The volume densities, i.e. the fractional or proportional volumes, of the exchange tissue, the lumina of the parabronchi and secondary bronchi, the blood vessels larger than blood capillaries and the primary bronchus were determined by point‐counting at a magnification of 100× using a 100‐point Zeiss integrating graticule (Figure 8). For example, for the exchange tissue, the volume density (
The absolute volumes of the structural components were calculated from the volume of the lung (
The volume densities of the components of the lung that comprise insignificant volume of the lung, e.g. the lymphatics and the inter‐parabronchial septa, were not determined. The parabronchi and the secondary bronchi were combined because most of the secondary bronchi have a gas exchange tissue mantle surrounding them, and the small secondary bronchi cannot be well differentiated from the parabronchi.
3.3.3. Sample size sufficiency
The adequacy of the number of sections analysed was determined in a pilot study by plotting cumulative average graphs on analysis made on stratified sections, and the sufficiency of the number of points counted for a particular structural component from nomograms given in Weibel . Since the sections were analysed entirely, i.e. field by field, the number of points counted for the three main structural components, i.e. the exchange tissue, the lumina of the parabronchi and the secondary bronchi and the blood vessels larger than blood capillaries, surpassed those needed to give a standard error of the mean of 5% or less.
3.4. Electron microscopic analysis
3.4.1. Tissue processing and sampling
The right lung was cut into slices along the costovertebral sulci and the slices were then cut into halves dorsal to the primary bronchus. The half slices were diced, and the pieces (~1 mm3 in size) were processed for electron microscopy by the standard laboratory techniques (Figure 9). From each half slice, 4–10 blocks were prepared. One block was picked at random and trimmed to remove the rest of the structural components, leaving only the exchange tissue. Ultrathin sections were cut and mounted on 200‐square wire mesh copper grids. Five micrographs were taken from a predetermined corner of the grid squares (the top right corner) to avoid bias at a primary magnification of 3000×. The images were enlarged by a factor of 2.5 and a quadratic lattice grid superimposed on the image (Figure 10). On average, for each bird, a total of 40 electron micrographs were analysed at a final magnification of 7500×. In a pilot study, the magnification used for the analysis provided a large field of investigation while providing adequate resolution and permitting the counts and the measurements to be made accurately.
3.4.2. Determination of volume densities
The volume densities of the components of the exchange tissue, namely, the air capillaries, the blood capillaries, the supportive tissue, i.e. the tissue of the blood‐gas barrier and the parts of the exchange tissue that are not involved in gas exchange, and the erythrocytes were determined by point counting (Figure 10). The intersections of the vertical and the horizontal lines constituted the points used for the determination of the volume densities. For example, the volume density of the air capillaries (
The absolute volume of the air capillaries (
3.4.3. Determination of surface densities and surface areas
The surface densities of the blood‐gas (tissue) barrier, the blood capillary endothelium and the cell membrane of the erythrocytes were determined by intersection counting, i.e. by counting the crossings of the test system, i.e. the horizontal lines of a quadratic lattice grid, with particular tissue barrier (Figure 10). In a pilot study, it had been determined that the number of vertical and horizontal intersections with the barriers were not statistically different. This showed that the exchange tissue of the avian lung is anisotropic, i.e. homogeneous. For example, the surface density of the blood‐gas (tissue) barrier (BGB) (
The surface area of the blood‐gas (tissue) barrier (
3.4.4. Determination of harmonic mean thickness
The harmonic mean thicknesses of the blood‐gas (tissue) barrier (τht) and the plasma layer (τhp) were determined from the sum of the reciprocals of the respective intercept lengths (
4. Pulmonary modelling
4.1. General considerations
Following a hierarchy that can be examined on a scale from microscopic to gross, living things are highly organized and structured entities. In larger organisms, cells combine to comprise tissues, which are groups of similar cells performing similar or related functions. Organs are collections of tissues grouped together executing a common function to meaningfully explicate how and why animals work the way they do, biologists must adopt appropriate conceptual models of engineers. On their own, quantitative data do not quite explain the function of an organism or that of its constituent parts. Interestingly, the sum total of the functions performed by the different parts of an organism, e.g. organelles, cells, tissues, organs and organ systems, surpass the function expressed by the whole organism [38, 39]. Hammond et al.  stated that ‘natural selection operates on organismal‐level traits that are usually manifestations of the integrated functioning of a suite of organs and organ systems’. In the complex dynamic biological entities, measurement of physiological changes and processes by testing different structures and measuring separate functions therefore leads to wrong deductions.
Like other organs and organ systems, gas exchangers possess a complex cascading assemblage of structural components that span from cellular‐ to organ‐system level: the structural components are organized as discrete but functionally integrated units. For respiratory organs, the physiological process (gas exchange) that manifests at organismal level is a product of infinitely many small events that are generated by various structural entities at the different levels of organization. Gans [41, 42] pointed out that animals very rarely exhibit one‐function‐one‐structure designs: ‘each activity tends to involve multiple aspects of the phenotype and each aspect of the phenotype may be involved in multiple activities’.
Comparative respiratory biologists focus on the structure of gas exchangers and determine how they correlate with properties like function, phylogeny, environment, body size and lifestyle. An insightful study of a gas exchanger or for that matter any other organ should involve application of comprehensive physical models that mathematically integrate the structural and functional aspects of the whole organism/animal.
4.2. Biological models
Scheid  remarked that ‘models are not only helpful but often indispensable in quantitative biology’ and that a model is ‘an image of part of the physical or conceptual world apt to explain or predict observations’. Gutman and Bonik  termed a model ‘an abstraction of a real situation that describes only the essential aspects of the situation’. A mathematical model separates a complex biological structure into its functional parts, sets apart those that are most important in answering particular research questions and then integrates them. A biological model is a mathematical simplification of a complex structure that satisfactorily characterizes the system it describes. It should be simple to apply, easy to understand and theoretically and practically testable.
A simple mechanistic model of a gas exchanger consists of a structure in which the external and the internal respiratory media are separated by a physical (tissue) barrier across which partial pressure gradient of oxygen (PO2) exists (Figure 12). Powell and Scheid  pointed out that ‘in an attempt at deriving a functional model for gas exchange from morphologic evidence, the physiologist has to identify the simplest functional subunit in the gas exchange organ’. Although biological models are mathematical abstractions of complex systems, since the functions of organs and organisms are regulated by many variables, models should be highly instructive in comparative studies where similar functions are performed by various structures in different ways. To simplify biological models, many structural details have to be omitted and certain assumptions made. The exclusions and the conjectures determine the predictive power of a model. Tweaking a model provides information on the relationship between the physical and functional parameters that drive a biological system. Mathematically adjusting one or more of the parameters and the prevailing conditions under which a system works while holding others constant allows for the identification of constraining, potentiating and redundant factors. The underlying multifaceted control mechanisms that drive the performance of biological components, systems and whole organisms can best be identified by careful modelling.
4.3. Morphological basis of pulmonary modelling
In accordance with the Fick’s law, the conductance or the volume of a gas (e.g. oxygen) that is transferred by diffusion across a tissue barrier per unit time (
It is the physiologist’s equivalent of Ohm’s law of electricity which is expressed as follows:
The morphometric diffusing capacities of the components of the lung can be estimated from the respiratory surfaces area, the thickness of the air‐haemoglobin pathway, the volume of blood in the blood capillaries, the Krogh’s O2 permeation coefficients and the O2 uptake coefficient of the whole blood [46, 47].
To various extents, the respiratory organs (gas exchangers) have been morphometrically analysed and functionally modelled. These include the fish gills  and the lungs of the lungfish (Dipnoi) , reptiles [50–53], birds [8, 54–57] and mammals [58–61]. From the concepts and findings of Roughton , Roughton and Forster , Staub et al.  and Sackner et al. , a morphometric model was developed by Weibel  and later revised in Weibel et al.  (Figure 13). The relationship between the total pulmonary diffusing capacity of the lung for O2 (
Although in all gas exchangers O2 diffuses across the so‐called air‐haemoglobin pathway that essentially comprises the blood‐gas (tissue) barrier, the plasma layer and to a certain extent, the cytoplasm of the erythrocyte before the molecule is biochemically bound to the haemoglobin, certain modifications of the basic model (Figures 12 and 13) have been necessary to satisfy the variations in the morphologies of the different respiratory organs and structures. For example, in birds where the erythrocytes are nucleated, the volume of the pulmonary capillary blood has to be adjusted by ‘subtracting’ the volume occupied by the nuclei in the erythrocytes [7, 8, 56]. The diffusing capacities of the blood‐gas (tissue) barrier (
The quantity of blood in the blood capillaries of a respiratory organ determines the amount of O2 bound by the haemoglobin [63, 68]. The diffusing capacity of the erythrocytes (
The barriers that form the air‐haemoglobin pathway are arranged in series, i.e. an O2molecule has to pass through these barriers in succession before it binds to the haemoglobin. Like for electricity when the resistances are arranged in series, in the lung, the total resistance that the molecule encounters can be mathematically expressed as follows:
According to the ‘older’ model of Weibel , the membrane‐diffusing capacity (
In the revised model of Weibel et al. , the thickness of the plasma layer is combined with that of the blood‐gas (tissue) barrier so that
The total morphometric pulmonary diffusing capacity (
4.4. Significance of mathematical modelling in biology
Inexperience and/or unawareness on the technique and lack of physical constants of O2 permeability through tissues have particularly hindered investigators from mathematically modelling respiratory organs. Comparisons of the functional designs of gas exchangers have largely been based on relating single structural parameters such as lung volumes, respiratory surface areas and thicknesses of the blood‐gas barriers. In some cases [50–52], the so‐called ‘anatomical diffusion factor’ (ADF) which is the ratio of respiratory area to the thickness of the blood‐gas (tissue) barrier has been used to assess functional efficiencies. Using individual morphometric parameters can lead to wrong conclusions. For example, among birds on which pulmonary morphometric data exist, the Humboldt penguin,
Application of integrative mathematical models on data acquired from biological structures provides robust answers to research questions.
Quantitative analyses of lungs of different species of birds have shown that pulmonary structural refinements correspond with factors such as body mass, lifestyle and habitat occupied [7, 8, 56] (Figure 14).
The writing of this chapter was supported by the National Research Foundation of South Africa (NRF). The views and opinions expressed here are, however, those of the author and are not necessarily of the NRF.