The presence of air-filled spaces in the vertebrate cranium has attracted the attention of anatomists since centuries, but only with the advent of noninvasive medical imaging methods such as computed tomography has it become possible to analyze these complex three-dimensional structures in detail. Three basic (and mutually nonexclusive) questions are typically asked during the study of cranial pneumatization: (1) Do patterns of presence/absence represent phyletic history? (2) How are air-filled spaces formed during ontogeny? (3) Is the morphology of sinuses and air cells associated with specific functions?
Because of a special interest in the evolutionary history of our own genus Homo, various studies are now available documenting patterns of pneumatization in higher primates (Koppe et al., 1999) and in fossil and modern hominoids (Wind, 1984; Seidler et al., 1997; Marquez et al., 2001; Sherwood et al., 2002; Prossinger et al., 2003; Rossie, 2005; Balzeau and Grimaud-Hervé, 2006). However, even within this restricted group of primates, cranial pneumatization exhibits astonishing diversity and variability. Patterns of presence/absence of pneumatization in specific cranial regions and the complex and highly variable morphology of sinuses and air cells defy comprehensive phyletic and/or functional interpretation.
Within higher primates, for example, the presence/absence of the maxillary and frontal sinuses is linked to the taxonomic status of subgroups (Rae, 1999). At the same time, however, it has been shown that the evolutionary loss and subsequent reacquisition of sinuses seem to be a common process, such that complex patterns of homoplasy are the rule rather than the exception (Rae, 1999; Witmer, 1999). This makes the phyletic interpretation of pneumatization in fossil hominins, where only few specimens are available per taxon, especially difficult.
A recent review of the literature on sinus morphology and physiology showed that functional interpretations remain inconclusive. Although, for example, maxillary sinuses have peculiar physiological properties such as elevated levels of nitric oxide production (Stierna, 2001), it is difficult to correlate such properties with specific functions. Likewise, structure–function relationships remain elusive. For example, maxillary sinus size shows clinal variation in macaques, and also in humans, but the correlation of sinus size with cold stress is at odds with the expectations of previous functional models (Rae et al., 2006). While straightforward functional interpretation of cranial pneumatization is thus prone to lead to Panglossian fallacy, i.e., association of a specific structure with an ad-hoc adaptationist scenario (Gould and Lewontin, 1979), interpreting sinuses as multifunctional structures tends to converge with interpreting them as entities without specific function.
Given our lack of understanding of possible sinus functions, and applying Occam's razor (“entities should not be multiplied beyond necessity”), it is thus reasonable to consider air-filled spaces as Gouldian spandrels (Gould and Lewontin, 1979), i.e., nonfunctional phenotypic features which owe their existence and specific form to developmental and functional constraints imposed by surrounding structures. The assumption that pneumatic elements of the skull are spandrels is thus a viable null hypothesis during the study of cranial pneumatization, against which more elaborate phyletic, developmental, and functional hypotheses can be tested.
The ontogeny of sinuses and air cells has been studied intensively from a clinical perspective, in humans as well as model animals (Ruf and Pancherz, 1996; Weiglein, 1999; Shah et al., 2003; Tan et al., 2003; Mey et al., 2006), and various studies are now available that document sinus growth in nonhuman primates such as Pongo and Macaca (Koppe et al., 1995; Koppe and Nagai, 1997). However, beyond issues concerning growth and allometry of sinus volumes (i.e., size), relatively little is known about how their often complex and highly variable shape is formed during development, and only few studies attempt to quantify sinus morphology beyond volume measurements and description of anatomical variants (Krennmair et al., 1999; Farke, 2007).
Understanding the morphogenesis of air-filled spaces in the cranium is a key to testing the “spandrel” null hypothesis. In operational terms, this means that, before we start drawing functional and phyletic implications, we must try to understand the presence and specific structure of air-filled regions in the cranium as the result of a generalized morphogenetic process (i.e., pneumatization), which itself is governed by the ontogenetic constraints of the cranium.
The aim of this article is to propose a generalized morphogenetic model, which implements what we name here the “invasive tissue hypothesis” of cranial pneumatization. We combine empirical data about epithelial tissue growth with physical theories of surface growth and develop a minimum model, i.e., we define growth equations with a minimum number of system parameters to model the invasion of mucous tissue and air-filled spaces into the cancellous compartment of cranial bones. Computer simulations are used to explore the parameter space of the model system and to experiment with various spatial boundary conditions that constrain the process of pneumatization. This permits generation of a wide diversity of virtual morphologies, which can then be compared with real morphologies of air-filled spaces, notably the paranasal sinuses and the temporal air cells.
Our overall approach is exploratory rather than explanatory. The computer model of pneumatization proposed here is not expected to explain real-life morphogenetic processes of pneumatization, but it offers a new perspective on how the diverse and complex morphologies of air-filled spaces could have resulted from evolutionary modification of one and the same basic growth mechanism.
MATERIALS AND METHODS
A Generalized Model of Cranial Pneumatization
Modeling the morphogenetic process of pneumatization requires both empirical and theoretical foundations. The empirical foundations are given by an important body of literature documenting the ontogeny of pneumatization from the cellular to the macroscopic level (Koppe and Nagai, 1997; Sherwood, 1999; Weiglein, 1999; Behar and Todd, 2000; Shah et al., 2003; Tan et al., 2003; Pohunek, 2004; Smith et al., 2005; Mey et al., 2006; Rae et al., 2006; Rossie, 2006; Chaiyasate et al., 2007; Guimaraes et al., 2007). It is generally acknowledged that sinuses and air cells are formed by invasive mucous epithelium from the nasopharynx, which expands into different cranial bones, primarily the ethmoid, maxillae, frontals, sphenoid, and, via the Eustachian tube, into the temporal bones. It was proposed that, during this process, the mucous tissue invades neighboring bones in an opportunistic manner (Witmer, 1999) until an equilibrium is reached between pneumatization and biomechanical stability (Sherwood, 1999). Direct support of this hypothesis comes from the clinical observation that failure to form epithelial tissue results in lack of pneumatization (Mey et al., 2006). From a macroscopic perspective, there is overall consensus that sinuses and air cells have highly irregular morphologies, and exhibit considerable deviation from bilateral symmetry. Any growth model of pneumatization thus needs to incorporate a stochastic component.
Theoretical foundations for a morphogenetic model of pneumatization come from both biology and physics. Following the functional matrix hypothesis proposed by Moss (1986), the vertebrate head can be seen as a complex structure comprising various functional components. Each component consists of a functional soft tissue matrix and a skeletal unit. One point that needs special attention is the fact that air cells and sinuses are structures within single bones. Adopting Moss' hypothesis, their development and morphology can thus be expected to be constrained by the development and morphology of the bone in which they are contained. An additional point worth of attention is the fact that almost all cranial bones are developmental units, but each of them is associated with more than one functional matrix. For example, the greater wing of the sphenoid is involved in three matrices: its internal surface contains the temporal lobes of the brain, its lateral wall delimits the temporal fossa, and its anterior wall forms part of the orbital cavity. The shape of the greater sphenoid wing is thus constrained by the relative size and orientation of the brain, the masticatory muscles, and the eyes. Accordingly, pneumatization of the resulting compartment within the greater wing is also constrained by these units. Similarly, the inner and outer tables of the cranial vault bones typically belong to different functional units and exhibit diverging patterns of growth. While the inner table closely follows brain growth, the outer table, and especially its superstructures, reflect growth trajectories mainly defined by the size and shape of the nuchal and masticatory muscles attached to the neurocranium. Divergence between internal and external modes of growth thus entails extensive pneumatization of the frontal and temporal bones.
Based on these considerations, we hypothesize that pneumatization, whether in the form of air cells or in the form of sinuses, occurs whenever a substantial spatial gap exists, or is being formed, between different surfaces of one and the same cranial bone. From this perspective, bone pneumatization appears as an effect of the diverging developmental and functional constraints/demands imposed onto different regions of one and the same bone, but belonging to different functional matrices. In other words, air-filled spaces are probably the most literal implementation of a biological spandrel, as proposed by Gould and Lewontin (1979).
Invasive growth of mucous epithelium into cancellous bony compartments as the principal mechanism of pneumatization (Sherwood, 1999) has an interesting parallel in applied physics, where surface growth at the interface between two media is an active area of research. An array of theoretical concepts and models are today available to describe how interfaces between materials with different properties develop in time and space. One model deserves special attention here: the “viscous fingering” or Saffman–Taylor model of surface propagation (Saffman and Taylor, 1958). Viscous fingering occurs when a low-viscosity medium intrudes into a medium with higher viscosity. The interface between the two media becomes unstable and develops an irregular shape with characteristic finger-like protrusions. Such effects can be observed when water is injected into oil, or air into the liquid-filled space between two glass plates. Depending of the viscosity of the media, the interface develops a wide variability of forms, ranging from dendritic to finger-like structures (Fig. 1).
The Saffman–Taylor model of surface propagation represents one specific case of a large class of surface growth phenomena known as Laplacian systems. Laplacian growth is well studied theoretically and has been used to demonstrate basic similarities of a wide spectrum of surface growth phenomena in physical and biological systems (Casademunt and Magdaleno, 2000; Bogoyavlenskiy, 2001). In all these systems, local surface growth velocities (v) depend on an external field (φ). This field may reflect diverse physical properties, such as pressure in the case of viscous fingering, an electrolyte concentration gradient in dendritic crystal growth, an electrostatic potential in dielectric breakdown during lightning, and a nutrient concentration gradient in growing bacterial colonies on a Petri dish (Matsushita et al., 1986; Fleury, 1997).
In Laplacian systems, growth velocity perpendicular to the surface (vn) is proportional to the local field gradient (▿φ):
In the present case, growth velocities describe the rate of cell divisions in the expanding epithelial tissue, and the field gradient describes pressure distribution inside a bone (Kurbel et al., 2004). Under far-from-equilibrium conditions, Laplacian systems obey the steady-state diffusion equation:
in which the Laplacian term ▿2φ = 0 characterizes a field with zero net flux.
A further generalization of Laplacian surface growth is Poisson growth. Poisson growth is of special interest here, as it describes the behavior of a viscous medium contained between two boundaries that are separated from each other over time (Laroche et al., 1991). Poisson growth occurs when air is entering the oil-filled space between two glass plates, or when a muddy shoe is being lift off the floor. The key point is that, in all these cases, the total mass of the included medium (oil, mud) remains constant. Whereas in Laplacian systems the distribution of φ values in the field results from solution of the steady-state diffusion Eq. (2), in Poisson growth the distribution of φ is a function of the separation of the two boundaries,
where h is the distance between the boundaries and dh/dt is the rate of separation.
Here, we use the Poisson equation to model the formation of air-filled spaces within a growing bone. This corresponds to a situation where the size of a bone increases through growth at its cortical surfaces, but without additional deposition of cancellous bone inside its volume. In this model, the expanding volume of the intra-osseous compartment is invaded by mucous epithelial tissue, which forms the active interface between air and cancellous bone. Laplace growth, on the other hand, is used to model invasion of mucous epithelial tissue into bones with constant dimensions, by replacement of cancellous bone space.
Generally, differential equations of the above form (2 and 3) can be solved by numerical integration. However, these methods are time-consuming and take more than 10 hr to perform appropriately sized two-dimensional simulations on current workstations (Fast and Shelley, 2006). On the other hand, it has been shown that the solutions of Laplacian and Poisson systems can be approximated by particle-based simulations (Mathiesen et al., 2006). Such a simulation takes place in a finite 2D or 3D grid consisting of N1 × N2 or N1 × N2 × N3 grid cells, respectively (Fig. 2). An initial aggregate is placed in the grid as a seed. By convention, cells occupied by the aggregate are defined as “filled”; however, in our simulation, they represent air-filled spaces. The grid contains particles that perform Brownian random motion through the “empty” space of the system: at each time step, each particles randomly moves from its cell to one of its next-neighbor cells. Particles are either released at the boundary of the grid (“infinity”; Laplacian growth) or at an “empty” random location within the grid (Poisson growth). Note that “empty” cells represent cancellous bone in our simulation, and that the mucous epithelial tissue is represented by the interface between “filled” and “empty” cells.
If a particle collides with a wall of the system, it is reflected. Whenever a particle collides with the aggregate, it is added to the aggregate, provided that viscosity conditions are met (Fig. 2). Viscosity at the interface between “filled” (aggregate) and “empty” cells is simulated as follows: Whenever a particle collides with the aggregate, the number of filled cells in the neighborhood of the particle is considered. The particle is only added to the aggregate if this number exceeds a certain threshold value (Fig. 2A). A low threshold represents low viscosity, whereas a high threshold represents high viscosity. Figure 1 shows the influence of viscosity on the resulting aggregate for a two-dimensional grid with 100 × 100 cells.
Generally, simulations using a higher viscosity take a longer time to grow an aggregate of comparable size. To keep CPU time within reasonable limits (minutes to hours), we used an additional set of relaxation rules to simulate growth under high viscosity conditions: when a particle collides with the aggregate and meets the threshold conditions, it relaxes local surface tension by moving to the neighboring cell with the highest number of neighbors (Fig. 2B).
Our simulation software was implemented in C++, using OpenMPI [ref. www.openmpi.org], to parallelize the workload on a small cluster of AMD and Intel processor machines running Linux.
Simulations were performed with the aim to explore the parameter space of the morphogenetic system (viscosity: threshold and relaxation conditions), and to assess the impact of different initial conditions (seed aggregate) and boundary conditions (configuration of the walls of the grid) on the resulting form of the aggregate. Changes in viscosity parameters were used to simulate variability of growth parameters of the epithelial tissue (which is represented in our model system by the interface between filled and empty cells). Changes in the initial configuration of the seed aggregate were used to approximate the initial conditions of pneumatization in different bones: In the temporal bone, the mastoid and squamous regions are pneumatized from extrusions of the mastoid antrum in posterior and anterior directions, respectively (Sherwood, 1999). All paranasal sinuses develop, as their name says, from the nasal cavity, but initial geometries are quite distinct. The ethmoid sinuses develop along the left and right lateral nasal walls. The maxillary sinus develops from an opening in the lateral nasal wall just below the ethmoid infundibulum. Frontal sinuses typically grow as separate left and right entities from the middle nasal meatus; their expansion is confined laterally by the orbits, and anteroposteriorly by the inner and outer tables of the frontal bone.
We implemented model versions of these boundary conditions (Fig. 3 and Table 1) and visualized the growing aggregates at different points along the time axis (Figs. 4–6). Our simulations show that variation in system parameters leads to a wide variety of aggregate forms. Before focusing on details, some general trends can be seen in our simulations. Changes in viscosity parameters have a profound influence on the resulting aggregates. Low-viscosity growth leads to formation of a large quantity of ramifications, which appear as a dense aggregate of interconnected bubbles (Fig. 4). A high viscosity threshold suppresses particle aggregation at cells with only few nearest neighbors, i.e., at “tips,” and relaxation tends to fill crevices that form at the base between neighboring branches. Overall, thus, aggregates grown under high-viscosity and relaxation conditions assume a smooth, sparsely ramified morphology, forming relatively few fingers, lobes, or bubbles (Figs. 5 and 6).
Table 1. Geometric boundary conditions and parameter settings for simulations
As outlined above, Laplace and Poisson growth equations represent different spatiotemporal boundary conditions of growth. Under Laplace conditions, air-filled spaces lined by mucous epithelium invade a bone with fixed dimensions while, under Poisson conditions, bone dimensions increase. Because particles are released only from the walls of the grid Laplace growth favors aggregation of particles at the tips of already existing branches, which ultimately leads to a more ramified aggregate morphology than under Poisson growth conditions (Fig. 6).
In addition to these general trends, Figs. 4–6 show an array of specific results, which permit direct comparisons of simulated with real morphologies. Figure 3 shows invasion of a cube by air cells growing from a corner site. This geometry is intended to mimic pneumatization of the mastoid region of a temporal bone, where growth starts at the mastoid antrum (seed in one corner) and extends into the mastoid process (opposite corner). Simulation of Poisson growth at low to mid viscosity conditions leads to a complex ramified aggregate of bubbles (Fig. 4), which are similar in appearance to the system of air cells in the temporal bone of hominoids (Fig. 7A,B).
The same overall geometry was combined with high-viscosity Poisson growth conditions and a slightly different seed geometry (at the center of a face of the cube) to simulate growth of a maxillary sinus (Fig. 5). High viscosity leads to formation of relatively few protrusions, which ultimately fill the entire volume, while being separated from each other by crest-like structures protruding from the outer walls into the air-filled space. This condition approximates the formation of a lobed maxillary sinus, as it is characteristic for chimpanzees (Fig. 7H). In an additional experiment, while keeping all growth parameters constant, we introduced a canal-like inner surface into the cube geometry in order to mimic structures such as the infraorbital canal of the maxilla, which represent potential obstacles for pneumatization of the entire maxillary volume. Our simulation shows that air sacs grow around the obstacle, which acts as a source for the generation of additional lobes (Fig. 5C).
Simulation of frontal sinus growth involved construction of more complex boundary conditions (Fig. 3D). The frontal squama has a quasi-two-dimensional geometry, which was modeled as a flat cuboid. Frontal sinus growth is further constrained by the orbits, which were modeled as discs delimiting the interorbital space. Two small areas at the bottom of the “interorbital space” served as initial seeds. Laplace growth was used to simulate sinus expansion into a frontal squama of fixed size (i.e., after it had reached its final thickness), and Poisson growth was used to simulate sinus expansion into the growing frontal squama. The resulting morphologies differ in the amount of ramification of the sinus. These fan-like structures exhibit considerable bilateral asymmetry (Fig. 6A,B), as can also be observed in many hominoids (Fig. 7D).
According to the elongated geometry of the nasal meatus, ethmoid pneumatization was simulated in an elongated cuboid, with two epithelial lines as the initial seeds (Fig. 3E). Poisson growth under high-viscosity conditions promotes formation of two rows of similar-sized, spheroid air cells (Fig. 6C). This corresponds to the observation that ethmoid air cells form arrays of similar-sized structures.
The morphogenetic model of pneumatization proposed here is based on the invasive tissue hypothesis, which states that air-filled spaces in the cranium are formed by invasion of nasal or otic mucous epithelium into neighboring bones. One of the main results of this study is that a growth model with relatively few parameters can generate a wide variety of three-dimensional structures, which exhibit many of the features observed in real-world pneumatization of cranial bones. Using simple model systems to explain complex patterns has a long history in biology that can be traced back to A. Turing's reaction-diffusion (RD) models (Turing, 1952). RD models are widely used to model patterns of mammalian fur or butterfly wing pigmentation (Nijhout, 1990; Murray, 1993; Sick et al., 2006) and ammonoid shell ornamentation (Hammer and Bucher, 1999). Like viscous fingering models, which represent a subclass of diffusion-limited aggregation (DLA), RD models describe far-from-equilibrium (steady-state) morphogenetic processes, in which variation in a small set of parameters generates a wide range of morphologies (Pearson, 1993).
In a biological context, two-dimensional DLA models have been used previously to simulate growth of bacterial colonies on nutrient-depleted media (Matsushita et al., 1995) and to study the formation of ammonoid sutures (Garcia-Ruiz et al., 1990). To our knowledge, the models presented here are the first three-dimensional implementations (Goold et al., 2005) of Laplacian and Poisson growth functions to simulate biological growth processes.
An important result of our studies consists in the fact that the model system is capable of replicating an important array of properties of real-world air-filled spaces, as they can be found in hominoid and other vertebrate crania. One salient feature of sinuses and air cells is their irregularity, both between individuals of the same taxon and between left and right sides of one and the same individual (Fig. 7). These properties naturally emerge from the specific stochastic properties of Laplacian and Poisson growth, which tend to reinforce random fluctuations at the growing interface, and which favor formation of several large fingers at the expense of many small structures.
The model system also produces a wide diversity of distinct morphologies of pneumatized spaces, such as “bubbles” (Fig. 4), bulbous or spherical sinuses (Fig. 5), and fan-like protrusions (Fig. 6), which mimic air cells in the temporal bone, maxillary, ethmoid, and frontal sinuses, respectively. In the model system, formation of these distinct morphologies can be traced back to small changes in growth parameters (viscosity), and in initial and boundary conditions. According to these findings, it is reasonable to assume that the diverse forms of real cranial air-filled spaces have a common morphogenetic origin.
In real crania, paranasal sinuses are typically described as being composed of chambers, protrusions, lobes, etc., which are separated from each other by septa, lamellae, and crests. The latter structures have been proposed to act as struts and pillars to increase the biomechanical stability of air-filled spaces [biomechanical fastening of the otherwise thin-walled outer tables of these bones (Bookstein et al., 1999; Prossinger et al., 2000)], but these arguments were received with skepticism (Ravosa et al., 2000). In our model system, crest-like and septum-like structures emerge as a well-known property of Laplace and Poisson aggregation phenomena. Growth probabilities at the extremities of the ramifying aggregate are consistently higher than at the base and between ramifications, such that deep valleys tend to be conserved between growing bubbles, lobes, or fingers. This offers a new perspective on real-world septa and crests between air-filled spaces: these structures most probably reflect properties of the invasive mucous tissue, and hypotheses about their potential biomechanical function must be tested against the invasive tissue hypothesis.
Interestingly, crests can be understood as remains of septa in our model system. When simulations are run until a volume is filled entirely with the aggregate (Fig. 5), gaps between lobes are successively filled with aggregate cells. This process leaves minor invaginations at the surface of the structure, which can be perceived as analogs to bony crests. Similarly, the development of bony “pillars” formed around nerve/vessel canals (e.g. around the infraorbital canal in the maxillary sinus) has its analog in model simulations of sinus growth in the vicinity of an “obstacle” (Fig. 5C). The obstacle elicits branching in the developing sinus, but is eventually wrapped up by the expanding lobes.
The stochastic nature of Laplacian and Poisson growth also has implications for the acquisition of morphometric data from sinuses and air cells. Classic measurements such as volume and maximum width, depth, and height of a pneumatized compartment provide hints about its general size and shape in relation to cranium size and shape (Zollikofer et al., 2008), but these do not permit in-depth comparisons of different sinus morphologies. Another possible approach is fractal geometry. Aggregates generated under Laplacian and Poisson laws of growth have fractal properties, but it was not clear until recently whether they represent simple fractals or multifractals (Ball et al., 2002; Mathiesen et al., 2006) (simple fractals can be characterized by one constant fractal dimension Df; in multifractals, Df itself has fractal properties). As shown by Ball et al. (2002), these aggregates indeed have normal fractal properties, so it is, at least in principle, possible to characterize different sinus and air cell morphologies by their fractal dimension Df. For three-dimensional Laplacian fractals, values of Df = 2.5 have been reported (Bowler and Ball, 2005) (this study used noise-reduction techniques, which are comparable to the concept of viscosity used here). Using the box count method (Mandelbrot, 1982), it is possible to evaluate fractal dimensions of virtual and real pneumatized regions: In our low-viscosity Poisson growth simulations (Fig. 4A,B), Df ranges between 2.6 and 2.7, and the air cell system of the temporal bone shown in Fig. 7A,B has an average fractal dimension of Df = 2.3. However, under high-viscosity conditions, as they apparently prevail during the formation of paranasal sinuses, relatively few large-scale ramifications are formed, such that empirical values for Df are uninformative.
While comparative morphometric analysis of air-filled spaces has principal limitations, it might be useful to consider another approach to quantify sinus morphologies. As shown by our simulations, distinct morphologies can be generated by specific parameter settings (viscosity, initial and boundary conditions). Each parameter setting is thus equivalent to a specific morphology. Moreover, each parameter setting can be thought of as a point in parameter space. Accordingly, different real-space morphologies can be imagined as different points in parameter space. It is now possible to calculate morphometric distances between points in parameter space, as an alternative to compare the corresponding real-space morphologies. One interesting aspect of this approach is that perceived real-space morphometric differences between two aggregates might be large, but the distance between corresponding points in parameter space might be small. In other words, minor changes in the parameter setting might cause major shifts in the apparent morphology of air-filled spaces. This is best illustrated in Figs. 3 and 4, which show quite diverse morphologies, whose underlying generating functions differ in only one parameter value (viscosity; see Table 1). In sum, the concept of quantifying morphologies not by their spatial characteristics but by their generating functions might provide important insights into evolutionary and developmental mechanisms underlying morphological diversity.
The model systems presented here are able to replicate the diversity of real-world patterns of pneumatization to an appreciable degree. The approach initiated here can be extended in various directions, including alternative growth models, and imposing more complex spatial, temporal, and biomechanical constraints on invasive tissue growth, all with the aim to achieve morphologies which are as close as possible to the real-world phenomena.
However, it is necessary to be aware of the special relationship between model systems and their natural counterparts. While the mechanisms of growth implemented in our model system of pneumatization are able to mimic real-world morphologies, they do not necessarily replicate real-world morphogenetic processes. Hence, convergence of model and empirical patterns does not imply equivalence of generating processes. Nevertheless, morphogenetic models offer the opportunity to propose most-parsimonious mechanisms of growth, which serve as a basis to understand large classes of diverse and complex natural growth mechanisms that are otherwise difficult to generalize (Akam, 1989; Hammer, 1998). Adopting this approach, we showed here that a quantitative general growth model offers a unifying perspective on the wide diversity of pneumatization patterns seen in nature.
The authors thank Marcia Ponce de León for critical revision of the manuscript and one anonymous reviewer. The authors also thank Sam Márquez for his invitation to contribute to this volume as well as for proof-reading the manuscript.