Shape and material characteristics of the trachea in the leatherback sea turtle promote progressive collapse and reinflation during dives

In terms of the respiratory system, all air-breathing vertebrate lineages have been constrained by the nature of structures evolved primarily for a terrestrial existence. These consist of lungs of varying degrees of complexity, plus the upper respiratory system: the larynx and the trachea, together with bronchi that connect the trachea to the lungs (in some groups via a branching bronchial tree). Lungs have little resistance to compression, while the tracheae and bronchi of terrestrial reptiles, birds and mammals have a common structure designed to maintain a patent airway, yet provide flexibility, especially in the neck region. They are all composed of a series of circular, semi-rigid cartilaginous rings (complete in some groups, incomplete in others), interspersed with sections of thin and stretchable connective tissue.

Diving air-breathing vertebrates encounter a number of problems that are exacerbated by increasing depth and duration of dives (Kooyman, 1989). First, they require adequate oxygen supplies for each dive, but all diving groups feature enhanced blood and tissue storage of oxygen, and generally have rather smaller lungs than do related groups of terrestrial vertebrates. Second, the air spaces of the respiratory system become compressed and this decreases the buoyancy of the animal concerned, increasing locomotory costs. Third, the possibility of highly compressed air being in close proximity to moving blood has the potential for causing N₂ accumulation and decompression sickness (DCS). Scholander developed a simple ‘balloon and pipe’ model that addressed the DCS problem (Scholander, 1940). He envisaged that the rib cage and lungs of a diving mammal would be compressed, and the latter finally collapse, at relatively shallow depths (30–50 m), displacing air into the rigid (and poorly vascularised) tubes of the upper respiratory system, where little gaseous exchange would be possible and the threat of DCS eliminated. In support of this concept, it was observed that the tracheae of diving mammals tended to have much wider tracheal rings, with far less connective tissue between them than terrestrial mammals, and that the rings were composed of more rigid materials, sometimes calcified.

In the subsequent 70 years this model has become extensively modified as understanding of the complexity of diving physiology and biochemistry has improved. Some very deep divers (e.g. Weddell seals, Leptonychotes weddellii, and northern elephant seals, Mirounga angustirostris) exhale before they dive to minimise lung volume and nitrogen accumulation (LeBoeuf et al., 1986; Ponganis et al., 1993; Sato et al., 2003), while king penguins (Aptenodytes patagonicus) and sperm whales (Physeter macrocephalus) inhale before diving (Sato et al., 2002; Miller et al., 2004), presumably to maintain buoyancy and delay alveolar collapse to greater depths. Evidence indicates that lung collapse and gaseous exchange can be delayed to depths of well over 100 m in some seals (Kooyman and Sinnett, 1982; Moore, M. J. et al., 2011). A recent comprehensive review (Hooker et al., 2012) argues that diving mammals ‘manage DCS’ rather than eliminate it. Whereas several shallow divers (e.g. marine otters and extinct mosasaurs) feature calcified tracheal rings that help to provide the rigid pipes envisaged by Scholander (Scholander, 1940), this appears not to be the case in deep divers such as Weddell seals and northern elephant seals (both of which exceed 1000 m depth on many dives). They show reduced tracheal volume when compressed in hyperbaric chambers (Kooyman et al., 1970), and it is now clear that this is caused by ‘slipping’ of overlapping cartilaginous rings (Moore, C. et al., 2011). Bostrom and colleagues demonstrated that a compliant trachea causes increased depth of alveolar collapse and a gradual rather than abrupt collapse (Bostrom et al., 2008).

Our study considers the structure of the leatherback sea turtle, Dermochelys coriacea (Vandelli 1761). Dermochelyidae diverged from other chelonians 100–150 million years ago (Wood et al., 1996), so the leatherback turtle can be regarded as the living deep-diving vertebrate with the longest evolutionary history. Adult leatherbacks are large animals, typically weighing in the range of 300–500 kg but are less than 916 kg (Eckert and Luginbuhl, 1988), and overlap in size with many marine pinniped and cetacean species.

The leatherback has an unusual feeding ecology, biogeography and physiology (for a review, see Davenport et al., 2009b). Adult Dermochelys are accomplished deep divers. Diving beyond 1000 m was first inferred (Eckert et al., 1986; Eckert et al., 1989) and subsequently confirmed by satellite tags (Houghton et al., 2008). Maximum recorded dive depths and dive durations have now been extended to 1280 m (Doyle et al., 2008) and 86.5 min (López-Mendilaharsu et al., 2009), respectively. However, it is has recently been determined from satellite telemetry that very deep dives (>300 m) are rare (Houghton et al., 2008), making up only 0.4% of all dives, and exceed calculated aerobic scope for the species. Leatherbacks are mostly (99.6%) limited to aerobic dives of <300 m, but they conduct thousands of dives per year, so there are appreciable total annual numbers of very deep dives.

Dermochelys is a most unusual chelonian anatomically, with a highly modified and reduced shell structure (no bony elements protect the belly) that, with the thick leathery skin, allows the animals to vary their shape considerably (Davenport et al., 2011). Spotila reported that the ventral surface of the leatherback becomes visibly concave when leatherbacks start dives, which is consistent with respiratory tract collapse (Spotila, 2004). The turtle therefore shows convergent features with marine mammals and differs considerably from the hard-shelled marine turtles, which cannot reach such great depths. The anatomy of the leatherback trachea has attracted recent anatomical study (Davenport et al., 2009b; Fraher et al., 2010). It may be seen that, as in other reptiles, the larynx and the tracheal structure begins at the anterior of the floor of the mouth [fig. 4 in Davenport et al. (Davenport et al., 2009a)]. The larynx has a rectangular cross-section and is thick walled. At the rear of the hyoid plate (Fraher et al., 2010) it joins the trachea, which is a continuous long tube, consisting of fused, irregularly shaped, tracheal rings, with minimal amounts of connective tissue between them (Davenport et al., 2009b). The cross-section of the trachea [fig. 2 in Davenport et al. (Davenport et al., 2009b)] is near elliptical in shape, but thinner walled dorsally than ventrally. The material of this nearly elliptical tube is predominantly uninterrupted, uncalcified cartilage, with a thin vascular lining. Davenport and colleagues (Davenport et al., 2009b) reported that the trachea could easily be compressed between the fingers, suggesting that it should collapse progressively given small pressure differences between lumen and exterior.

The first aim of our study was to investigate the mechanical properties of the leatherback trachea. Such study has rarely been performed on tracheae of marine vertebrates, though limited measurements have been made on sections of the tracheae of the striped dolphin (Stenella coeruleoalba) and the bottlenose dolphin (Tursiops truncatus) (Cozzi et al., 2005; Bagnoli et al., 2011). We investigated the forces required to cause tracheal deformation in the laboratory and used the data recorded to investigate material behaviour. We also studied the hysteresis properties of the tracheal material, as diving and surfacing take place thousands of times per year. The second aim of the study was to examine the likely changes in the tracheal volume (expressed as changes in the luminal area) during the diving process through a simplified application of Boyle's law and to model these changes in a finite element (FE) structural model. In doing so we were aware that most previous considerations of tracheal behaviour have treated the trachea as an isolated structure. However, from previously published figures [fig. 4 in Davenport et al. (Davenport et al., 2009a); fig. 2 in Davenport et al. (Davenport et al., 2009b)] it may be seen that the forepart of the trachea is surrounded by tissue (mostly muscle and blubber), to which it is connected. Only when it emerges posteriorly from the neck (which has low mobility in this species) into the general body cavity (sea turtles do not have separate thoracic and abdominal cavities) does it become less constrained, and even then it is bound to the dorsal wall of that cavity. We have therefore taken into account the surrounding tissues in our modelling. In addition, while it is possible to model the behaviour of the trachea assuming a fixed mass of air within it, we have also taken into account the dynamic situation of real dives, during which air from the collapsing lungs moves into the upper respiratory tract and so serves to keep the trachea inflated to greater depths (Bostrom et al., 2008). A recent paper (Fossette et al., 2010) has demonstrated that Dermochelys inspires before diving and proposes that this is primarily concerned with buoyancy regulation. However, this inspiration will also delay lung collapse to greater depths. Our calculations in the application of Boyle's law rely on the estimates of leatherback lung volume made by Lutcavage et al. (Lutcavage et al., 1992).

Tracheal cartilage samples for material testing were obtained from a freshly stranded adult female leatherback turtle at Ballycotten, East Cork, Ireland (for a review, see Davenport et al., 2009b). Non-destructive compressive testing was performed on a single tracheal portion in order to investigate the response of the material to external loadings. The portion, some 20 mm front to back, was taken from the upper region of the trachea, about 50 mm caudal to the posterior end of the larynx and well before the point of bifurcation. It was estimated that the trachea had a length of about 450 mm, based on the assumption that the material and geometric properties of the remaining tracheal portion in the disposed carcass were similar to those in the preserved material. It had a near-elliptical cross-section with the following approximate measurements; external diameter 42×29 mm; internal diameter 33×23 mm; maximum dorsal thickness 2.1 mm, maximum ventral thickness 3.6 mm (Davenport et al., 2009b). The mean width of the sample tested was measured to be 8.6±0.3 mm. The sample was placed on the lower flat plate of a Tinius Olsen Benchtop Tester H5K-S UTM (Tinius Olsen, Salfords, Surrey, UK) and a compressive point displacement was applied to the dorsal side (Fig. 1). Testing of the sample adhered to all biomechanical standards and procedures. The sample was preconditioned five times (Fung, 1993) using a compressive velocity of 5 mm min^–1 at room temperature and a 50 N load cell (0.001 N and 0.001 mm precision), with the applied displacements and corresponding loads being recorded during all testing cycles.

The lung–trachea–larynx system is clearly a complex structural engineering problem that would be very difficult to model three-dimensionally. However, under due consideration of both the biology and physics of the system, a much simpler two-dimensional (2-D) model can capture the important responses. 2-D plane strain models assume that the structure is infinite in length so that no strain is possible in this direction. As the leatherback trachea is an almost solid tube of cartilage (Davenport et al., 2009b), and as the majority of the trachea is embedded in the musculature and fat in the neck, a 2-D plane strain slice through the neck section (capturing the tracheal luminal area) will simulate the response of the majority of the length of the trachea. It certainly will quantify, to within reasonable accuracy, the deformation of the trachea as the whole system is subject to increasing pressure. The geometry used in the 2-D structural model is taken from computed tomography (CT) transverse sectional images of the head and neck of the female turtle [fig. 4 in Davenport et al. (Davenport et al., 2009a)] and was imported into the FE analysis software system Strand7 (Sydney, NSW, Australia). Included in the model were the tracheal cartilage, the surrounding soft tissues and the vertebral column (Fig. 2). The stiff vertebral column was modelled as rigid and fixed in space to provide adequate fixity for overall stability. Both the tracheal cartilage (initial luminal area 571 mm²) and the soft tissues were modelled using separate hyper-elastic isotropic constitutive material models (described in detail in ‘Material behaviour’ below). There were 9733 degrees of freedom in the model, analysed using a mixed FE mesh of eight-node rectangular and six-node triangular isoparametric elements. The simulation of the structural response of the neck section during a dive was achieved using a non-linear static solver. Loading conditions were based on the application of Boyle's law (see ‘Loading conditions’ below). The reported results of this analysis consisted of the deformed shapes and the tracheal luminal area in 5 m increments from surface level to a maximum depth of 1500 m.

Cartilage samples responded almost elastically when compressed both by human touch (Davenport et al., 2009b) and between flat plates (this study). Cartilage behaviour resembled that of a rubber material in that it returned to its initial shape even after it was subjected to large deformations. Compressive testing of the material revealed that it exhibited both non-linear (although almost linear during loading) and viscoelastic behaviour (see Results). In order to model both the non-linear and apparent rubber-like behaviour, the cartilage has been treated as a non-linear hyper-elastic material and the Mooney–Rivlin rubber material formulation was applied to model its behaviour in the FE simulations. The two material constants required for this constitutive model were obtained using an integrated mathematical optimisation, FE simulation and experimental testing methodology (Murphy et al., 2010). Poisson's ratio, a measure of incompressibility, was assigned a value of 0.49, which correlates well with values cited in other studies (Teng et al., 2008; Teng et al., 2009a; Teng et al., 2009b; Trabelsi et al., 2010).

Knowledge of the mechanical properties of the adjacent soft tissues (striated muscle and connective tissue) is limited. In previous studies tracheal smooth muscle has been treated as a non-linear material using hyper-elastic constitutive material models (Begis et al., 1988; Costantino et al., 2004; Teng et al., 2009a; Teng et al., 2009b; Trabelsi et al., 2010; Bagnoli et al., 2011) with Poisson's ratio ranging from being completely incompressible (Begis et al., 1988; Teng et al., 2009a; Teng et al., 2009b) to nearly incompressible (0.475) (Costantino et al., 2004). A slightly lower value was also applied in another study (0.45) (Bagnoli et al., 2011). In this study all the surrounding soft tissues have been assumed to be homogeneous and are modelled as an incompressible hyper-elastic material using the generalised Mooney–Rivlin hyper-elastic material model. The constants for this constitutive model were obtained by applying the stress–strain data for smooth muscle published previously (Bagnoli et al., 2011).

The complete load–displacement history obtained from the compressive biomechanical testing of the single tracheal tubular sample is shown in Fig. 3. Like most other biological soft tissues, tracheal cartilage displays viscoelastic properties and this may be seen through the contrasting non-linear loading and unloading histories. Hysteresis was also evident between successive cycles. In order to obtain a meaningful, and thus accurate, representation of the behaviour of the material, preconditioning was required to reduce the hysteresis effects (Fung, 1993). Fig. 3 shows that the hysteresis becomes less noticeable with an increasing number of loading cycles. However, complete curve occlusion was not attained after applying six consecutive cycles, suggesting that some viscoelastic effects remained. For the purposes of determining the two material constants in the Mooney–Rivlin material formulation, only the loading history from the sixth and final cycle was correlated with the loading history obtained from the corresponding FE model, as this best reflects the behaviour of the material recorded in the laboratory. These material constants were determined to be 6.1 and 1.0 Mpa, respectively, and lead to a bulk modulus (a measure of resistance to uniform volumetric compression) of 690.4 MPa. The characteristic stress–strain curve was evaluated using the constants and indicates that the material is stiffer (slope of curve) in compression (negative quadrant) than in tension (Fig. 4).

For the simplified formulation, in which it is assumed that there is no volume change in either the larynx or the trachea, the depth at which the lungs collapse completely is 871 m (Eqn 2). Most of the collapse occurs in the first 50 m of the dive with subsequent increases in depth causing the volume of the system to decrease asymptotically (Fig. 5). In the similarly simplified state envisaged for tracheal collapse, the luminal area reduces gradually with increasing depth (Eqn 3). Complete luminal collapse would theoretically occur at 3891 m.

For the first 871 m in which the lungs are collapsing, there is very little change in the cross-sectional area (and hence volume) of the trachea. The tracheal luminal air pressure, representing air mass flow from the collapsing lungs, provides resistance to the external hydrostatic pressures and prevents premature collapse. Indeed, the percentage decrease of the luminal area, between surface level and 871 m, was measured as 1.14% and was deemed to be insignificant when considering the overall luminal area change of the trachea. The assumptions made with the simplified system were therefore considered to have been modelled sufficiently. Tracheal collapse, beginning at 871 m, is progressive with both the dorsal and ventral aspects yielding as the surrounding soft tissues compress inwards (Fig. 5). The trachea subsequently levels ventrally whilst bulges are formed laterally. Additional increases in depth compress the luminal area further, making the bulges more pronounced. The integral luminal area was measured for each depth and compared with the corresponding areas predicted theoretically (Fig. 5). Good agreement was achieved at all depths (Table 1) confirming that the model succeeded in reducing the luminal area in line with the areas predicted using the simplified application of Boyle's law. It is worth noting that a larger reduction in area is expected during early depth increments and this is captured well in the FE model results (Fig. 5).

The response of the upper airway, in particular the trachea, is an important factor when studying the physiological response of the vertebrate respiratory system in extreme pressure environments. Its deformation under external hydrostatic loadings determines the form of collapse and, thus, the depth that may be attained during diving apnoea. In this paper, a simplified representation of tracheal collapse based on Boyle's law was presented and a computational FE model of the behaviour of the trachea and the surrounding soft tissues based on this simplification was developed. The overall objectives of this study were to investigate how the trachea of the leatherback turtle responds during deep diving, particularly the luminal area, and to examine the importance of its structure and material properties in resisting collapse.

Boyle's law predicts that the lung volume will reduce progressively with increasing depth until complete collapse. The air mass in the lungs is, therefore, compelled to enter the trachea and the larynx, and will maintain a positive pressure on the interior wall of the tracheal lumen equal to the outside hydrostatic pressure, linearly dependent on depth. These figures rely on estimates of lung volume predicted by Lutcavage et al. (Lutcavage et al., 1992) and do not take into account the expected reductions in lung volume arising from the increasing partial pressures of all gases during descent. It is also assumed that air in the lungs is at capacity when diving commences (Fossette et al., 2010), and that continuity is maintained between lungs and upper airway throughout dives. For the adult turtle considered in this study, weighing 450 kg (Davenport et al., 2009b), the initial volume was calculated to be 28 l. It is likely that leatherback body size will influence the depth at which the lungs collapse completely and changes in tracheal volume commence. However, no data for tracheal and lung sizes are currently available for a size range of Dermochelys that would permit further calculations. As it stands, total lung collapse (and consequent cessation of air mass flow to the trachea) is expected to occur at approximately 871 m. Tracheal collapse commences subsequently, with net air mass transfer to the larynx, satisfying conservation of mass, as the larynx is considerably stiffer than the trachea. Collapse is gradual with increases in depth causing a corresponding decrease in luminal area (and hence volume, assuming constant length). The rate of change of the reduction in luminal area is greater in early depth increases (871–950 m) than at later depths, and is evident through varying slope values in the graph of depth versus luminal area (Fig. 5). At the deepest depth of 1280 m so far recorded (Doyle et al., 2008) the luminal area is estimated to be approximately 337 mm², which represents a 41% decrease. The percentage volume change in the trachea from its initial compression at 871 m to the maximum depth analysed in this study at 1500 m is very small representing just under 0.5% of the overall volume change in the system. The impact of gas compression at depth is therefore unlikely to contribute significantly to the leatherback's buoyancy state, in contrast to hard-shelled marine turtles (Hays et al., 2004).

In order to gain a visual representation of the extent of deformation of the trachea, a 2-D FE structural model of the trachea and adjacent soft tissues in the neck was created. The accuracy of the FE model in reproducing the volumetric changes predicted was measured through the calculation of the tracheal luminal area. For each increment of depth the luminal areas were evaluated and compared with the theoretical areas (Table 1). There is excellent agreement between these comparisons and between the two depth–luminal area relationships plotted in Fig. 5, confirming that the FE model succeeded in reproducing those areas computed through Boyle's law. For the depth of 1280 m the area computed by the FE model was 340 mm², which correlates well with the theoretical value. Overall, the results from the FE model reveal the ability of the trachea to withstand the colossal hydrostatic pressures at depth and its extraordinary capability to collapse accordingly. Progressive re-inflation during ascent also facilitates the return of air mass to the lungs. Complete V-shaped dive profiles of two adult leatherbacks to 1186 m [fig. 1 in López-Mendilaharsu et al. (López-Mendilaharsu et al., 2009)] and 1250 m [fig. 4C in Houghton et al. (Houghton et al., 2008)] reveal that changes occur in the rates of both descent and ascent. In the case of the dive to 1250 m there is a noticeable decrease in the rate of descent at approximately 750 m. This depth is similar to the depth predicted in this study (allowing for variations in turtle size and the assumptions made in this study) at which the lungs are expected to collapse completely and may be representative of descent rate alterations due to the cessation of air mass flow from the lungs. By applying the same collapse depth of 871 m predicted in this study (for a turtle with a curved carapace length of 168 cm) (Davenport et al., 2009b) the lungs would be expected to be completely collapsed for 19 min in the dive to 1250 m and 22 min in the dive to 1186 m. Essentially, it appears that the tracheal structure is perfectly suited for repeated collapse and reinflation.

The structural configuration of the leatherback trachea differs from the tracheae of other air-breathing vertebrates studied so far. In terrestrial mammals the trachea is generally composed of a series of horseshoe C-shaped cartilaginous rings with flexible connective tissue between successive rings and smooth muscle tissue linking the tips of the rings (Wheater et al., 1979). This structural arrangement is different from that of deep-diving air-breathing oceanic mammals and reptiles such as dolphins, seals, whales and turtles. In these forms tracheae differ noticeably through the irregular shape of the cartilaginous rings, which are sometimes complete (O-shaped); they also have reduced amounts of connective tissue and lack muscle tissue (Tarasoff and Kooyman, 1973; Davenport et al., 2009b; Bagnoli et al., 2011). Studies examining reductions in human tracheal volume have shown that decreases are achieved through inward movement of the smooth muscle connecting the two tracheal tips (Griscom and Wohl, 1983; Lindholm and Nyrén, 2005; Fitz-Clarke, 2007). In a similar manner, increases in volume are managed through outward protruding of the smooth muscle. A recent study on the behaviour of the trachea–bronchial tree of the bottlenose dolphin (Tursiops truncatus) showed that the extent of its collapse was much less when compared with that of a goat (Capra aegagrus; whose geometrical configuration is similar to that of humans) under the same loading conditions and at the same depths (Bagnoli et al., 2011). The difference was credited to the geometric arrangements of the dolphin trachea in which there was a much greater stiffness. It appears that the structural compositions of oceanic tracheae are superior structures for enduring collapse by comparison with their terrestrial relatives. The leatherback trachea confers both homogeneity and strength and, combined with the stabilising effect of the surrounding soft tissues, appears perfectly designed for collapse. Most terrestrial tracheae are poor designs for diving because of premature collapse of the cartilaginous rings and inward movement of the connecting smooth muscle tissue towards the ventral aspect. The geometry of the incomplete C-shaped rings also implies that the stiffness distribution will not be uniform with a higher stiffness in the thicker central region than at the two tips. This configuration does not correlate well with withstanding the colossal pressures endured during repeated deep diving. In contrast to this, the particular geometry of the cartilage of Dermochelys, based on the O-shaped tracheal rings of reptiles, but with minimal connective tissue, appears to be advantageous for diving. The near-elliptical shape (thicker ventrally than dorsally) actually promotes collapse and ensures a more uniform stress distribution in extreme bending states. Although deep dives are rare, the leatherback undertakes a great number of dives each year. There is therefore an appreciable number of dives to very deep depths with extreme pressure environments. The fact that the turtle is able to perform these dives regularly suggests that the collapse mechanism of the trachea may be optimised in order to successfully undertake repeated dives. In the simplified application of Boyle's law presented in this paper, air mass flow from the collapsing lungs and alveoli in the early stages of the dive promotes progressive collapse of the lungs and maintains the tracheal geometry. Subsequent tracheal collapse is also progressive, with the entrapped air acting as a counterbalance to the hydrostatic pressures and resisting premature deformation. The near-elastic rubber-like behaviour of the cartilage facilitates progressive dilation during ascent, allowing the air mass to flow gradually back into the lungs and alveoli at shallower depths. The findings of this study also demonstrate that the bulk of aerobic dives to depths shallower than 300–400 m will involve minimal distortion of the trachea despite the rubbery nature of its uncalcified walls. Our calculations indicate that it will remain inflated by air displaced from the lungs; it is not necessary to invoke a rigid trachea if the diver inhales before diving.

The noted rubber-like behaviour of the tracheal cartilage was modelled successfully by applying the Mooney–Rivlin formulation. Using the two calculated material constants the characteristic stress–strain relationship was established (Fig. 4). The relationship suggests that the material displays higher stiffness in compression than in tension, as evidenced by the greater slope in compression. As the Mooney–Rivlin formulation is a non-linear hyper-elastic material model, stiffness increases with increments in strain. It is therefore not possible to make direct comparisons with many stiffness values for tracheal cartilage cited in the literature as a large proportion of these are based on linear elastic material behaviour. However, it may be seen from Fig. 4 that in tension the stress–strain relationship obtained in this study is approximately linear, yielding a stiffness value of 22.2 MPa. This is slightly higher than published values and suggests that leatherback tracheal cartilage is a stiffer material. For example, a tensile Young's modulus value of 13.389±4.230 MPa, was recorded in experimental research on striped dolphins (Stenella coeruleoalba) (Cozzi et al., 2005). Other estimates of Young's modulus from biomechanical tensile testing of human tracheal rings range from 13.6±1.5 MPa in the ablumenal superficial zone to 4.6±1.7 MPa in the central area (Roberts et al., 1997). Further studies on human tracheal cartilage used or obtained stiffness values of 3.33 MPa (Trabelsi et al., 2010), 2.5–7.7 MPa (Lambert et al., 1991), 16.67 MPa (Begis et al., 1988) and 10 MPa (Holzhäuser and Lambert, 2001). Maturity and location have also been determined to be factors affecting stiffness, with the age of samples (values between 1.8±2.1 and 15±5.1 MPa) (Rains et al., 1992), and depth (values between 1 and 20 MPa) (Roberts et al., 1997) giving rise to large variations. Studies on terrestrial vertebrate tracheae have recorded values of 5.364±1.591 and 3.735±1.433 MPa in goats and pigs, respectively (Cozzi et al., 2005). It appears that the characteristic material properties may be species, age and location dependent. In this paper the material properties were obtained by fitting a rubber material model to experimental data obtained from the compressive testing of a leatherback tracheal portion. This simulates the real non-linear behaviour of the material. The indication of higher stiffness in compression than in tension agrees with findings in other studies (Teng et al., 2008; Teng et al., 2009a; Teng et al., 2009b). More appropriate material models for cartilage are a current area of research. Another area of potential study lies in the tracheae of those diving vertebrates that apparently have stiffer tracheae by virtue of calcification of the tracheal rings (e.g. sea otters).

Some limitations could not be avoided when undertaking this study. As mentioned earlier certain assumptions were made in the application of Boyle's law regarding the response of the respiratory system. In particular, changes in length in both the trachea and larynx, as well as reductions in lung volume due to the increasing partial pressures of all air gases during descent were neglected. Assumptions were also made with regard to the constitutive material model used to model tracheal cartilage material behaviour. Collagen type II, which is one of the primary constituents of tracheal cartilage, has been shown to be a highly non-linear material (Sun et al., 2004). It is, therefore, rational to expect that tracheal cartilage should also exhibit non-linear behaviour, and this has been suggested in studies using mathematical models based on simple beam bending theory (Teng et al., 2008; Teng et al., 2009a; Teng et al., 2009b). In the past tracheal cartilage has also been treated as a linear elastic material in studies using three dimensional FE analyses (Begis et al., 1988; Costantino et al., 2004) and mathematical models based on thin curved beam theory (Lambert et al., 1991) and on the physics of a shell (Holzhäuser and Lambert, 2001). Additionally, linear elastic properties have been assumed in experimental material studies (Rains et al., 1992; Cozzi et al., 2005) as has hyper-elasticity (Teng et al., 2008; Teng et al., 2009a; Teng et al., 2009b; Trabelsi et al., 2010). The results from our material testing show that the load–displacement history is approximately linear during loading (Fig. 3). If the sample being tested also had uniform thickness, the stress–strain relationship would also be expected to be linear. This may explain why the material has been treated as linear elastic in previous studies. In order to investigate the rubber-like behaviour of the tracheal samples studied here the tracheal cartilage was analysed using the Mooney–Rivlin hyperelastic material model, the material constants of which were obtained from compressive experimental testing.

This study aspired to investigate the response of the trachea of the leatherback turtle in the extreme pressure environments it is subjected to during deep diving using a simplified application of Boyle's law and a corresponding FE model. The results produced both mathematically and from the FE model itself reveal that the trachea collapses and dilates progressively with variations in depth. It appears to be a superior structure for managing collapse when compared with the tracheae of terrestrial vertebrates and the tracheae of shallow divers (including humans). Complete (O-shaped) rings and the lack of connective tissue provide an almost solid elliptical tube structure that, when combined with air mass flow from the lungs, facilitates progressive deformation. Indeed, the structure appears perfectly designed to promote recoverable collapse. The results from this study should benefit further research on the response of biological soft tissues in extreme pressure environments.

 
• DCS

  decompression sickness

 
• FE

  finite element

 
• k

  Boyle's constant

 
• LG

  lung

 
• LX

  larynx

 
• P

  pressure acting on system

 
• P₀

  initial pressure at surface level

 
• P_ATM

  atmospheric pressure

 
• P_x

  pressure at depth x

 
• T

  trachea

 
• V

  volume of system

 
• V_LG,0

  volume of lungs at surface level

 
• V_LG,x

  volume of lungs at depth x

 
• V_LX,0

  volume of larynx at surface level

 
• V_LX,x

  volume of larynx at depth x

 
• V_T,0

  volume of trachea at surface level

 
• V_T,x

  volume of trachea at depth x

 
• x

  depth

 
• ρ

  density

We are grateful to Dr Tom Doyle and Mr Luke Harman for collection of the leatherback turtle material. We also thank Ms Tracy Cuffe for technical assistance and Dr Paddy McLaughlin for help with preparation of CT scans.