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# Physical and geometric constraints shape the labyrinth-like nasal cavity

Edited by Leslie Greengard, New York University, New York, NY, and approved January 26, 2018 (received for review August 29, 2017)

## Significance

Although nasal cavities fulfill similar tasks across animals, their geometry varies widely. One such task is heating and humidifying the inhaled air, which works best if the nasal cavity is narrow. However, narrow geometries have a large resistance to flow. We show that these opposing geometrical requirements are critical for shaping the nasal cavity and strongly restrict the local gap width. In contrast, the overall shape has little influence on the resistance and air conditioning, so the observed labyrinth-like patterns could emerge from geometric constraints imposed by the head. Our theory predicts geometric parameters of nasal cavities quantitatively, and it suggests that the surprisingly small nasal cavities of humans force us to become oral breathers during heavy exercise.

## Abstract

The nasal cavity is a vital component of the respiratory system that heats and humidifies inhaled air in all vertebrates. Despite this common function, the shapes of nasal cavities vary widely across animals. To understand this variability, we here connect nasal geometry to its function by theoretically studying the airflow and the associated scalar exchange that describes heating and humidification. We find that optimal geometries, which have minimal resistance for a given exchange efficiency, have a constant gap width between their side walls, while their overall shape can adhere to the geometric constraints imposed by the head. Our theory explains the geometric variations of natural nasal cavities quantitatively, and we hypothesize that the trade-off between high exchange efficiency and low resistance to airflow is the main driving force shaping the nasal cavity. Our model further explains why humans, whose nasal cavities evolved to be smaller than expected for their size, become obligate oral breathers in aerobically challenging situations.

The nose not only allows us to smell but also humidifies, heats, and cleans inhaled air before it reaches the lungs. All these vital tasks depend critically on nasal airflow, which is driven by the pressure difference created by the lungs and depends on the complex geometry of the nasal cavity. Nasal geometries vary considerably among vertebrates in general (1) and among mammals in particular (2⇓–4), ranging from the complex labyrinth-like internal nasal cavity of dogs to the unique structure of humans that combines relatively simple geometry in a short internal nasal cavity with an additional external nasal vestibule; see Fig. 1. These qualitative differences in nasal geometry were likely selected as adaptations to different functional requirements, but how the geometry of the nose influences the airflow and thus the function of the nose is a long-standing unsolved problem.

The nasal cavity is a complex, air-filled space that connects the two nostrils with the throat; see Fig. 1*A*. All mammals have an internal nasal cavity, but humans are unique in having an additional external vestibule with inferiorly oriented nostrils (5). The two sides of the cavity are separated by the nasal septum and merge only behind the posterior nasal cavity (choanae) that separates the nasal cavity from the pharynx. Each side can be further divided into the main pathway (turbinates) and large side chambers (sinuses). The walls of the nasal cavity are covered by a highly vascularized bed of epithelial tissue overlain by a

We here study how the geometry of the nasal cavity influences the airflow and the associated processes of heating and humidifying the inhaled air. Generally, we expect that a narrower geometry improves the efficiency of heating and humidification at the expense of greater resistance to airflow. Since this trade-off likely plays an important role in shaping nasal cavities, we determine the shape that has the lowest resistance to airflow for a given conditioning of the inhaled air. Here, we have to take into account geometric constraints imposed by the shape of the head that determine the length of the nasal cavity, its cross-sectional area, and, generally, the shape of the space that it occupies. To tackle this complex problem, we first show that, without geometric constraints, optimal shapes have slender cross-sections. We then demonstrate that these shapes can be compacted into the typical labyrinth-like shapes without much loss in performance.

## Results

### The Flow in the Nasal Cavity Is Laminar.

It has been suggested that the flow in the nasal cavity is turbulent (7, 8), since the speeds are high and nasal geometry is complex. Indeed, turbulence can easily be induced in the surrounding air by exhaling heavily, as is apparent on a cold winter day. Inside the nose, turbulent flow would induce additional mixing that improves the heating and humidification of the inhaled air (9), but it also implies a larger resistance to flow. It is thus unclear whether turbulence would be beneficial.

To see whether turbulence occurs inside the nasal cavities of animals of various sizes, we first estimate the mean speed

Another ubiquitous feature of nasal airflow is the oscillatory motion caused by natural breathing, which generally induces additional resistance and also limits the humidification and heating of the air. However, this is only important when the characteristic length *SI Appendix* that the resistance increases by about

### The Optimal Nasal Cavity Has a Uniform Gap Width.

We seek the geometry of the nasal cavity with the lowest resistance to airflow for a given efficiency of heating and humidifying the air under the constraint of a given volumetric flux, length, and cross-sectional area. We thus need to calculate the dependence of the airflow and its physical properties on the geometry of the nasal cavity. Since nasal cavities are typically straight, we first focus on varying the cross-sectional shape.

The flow through the cavity is driven by a pressure difference P generated by the lungs. Since the flow in a straight nasal cavity is laminar, stationary, and incompressible, the velocity field only has a component u in the axial direction, which obeys the Poisson equation*SI Appendix*). Here, **1** for u, we obtain the volumetric flux *SI Appendix*. We can thus quantify the influence of the cross-sectional shape on the airflow by simply studying its effect on

The heating and humidification properties of the nasal cavity can be quantified by the change in temperature and concentration of water vapor in the air after it flowed through the cavity. Both quantities can be described as a passive scalar c that is transported with the flow, diffuses, and is exchanged with the walls of the cavity. In a stationary state, the scalar c thus obeys*SI Appendix* that *SI Appendix* that

To find the geometry that has minimal resistance K for a given exchange efficiency E, we determine the respective prefactors

To understand what geometric features lead to good performance, we numerically determine how shapes need to be altered to lower the resistance at constant exchange efficiency; see Fig. 3 and *SI Appendix* for details. This sensitivity analysis suggests that sharp corners and narrow regions are detrimental, as indicated by the large arrows in Fig. 3. However, it does not show clearly why rectangular shapes outperform ellipses. To understand this aspect better, we use the sensitivity analysis to obtain optimal shapes by iterative adaptation as described in *SI Appendix*. Fig. 3 and *SI Appendix*, Fig. S2 show that optimal cross-sections are dumbbell-shaped, with a slender midsection. We compare the midsections of different shapes by quantifying the width of the shape perpendicular to the centerline as a function of the distance along the centerline. Fig. 3 shows that this gap width is uniform for the optimal and rectangular shapes, while it varies significantly for the ellipse. Taken together, optimal shapes are thus rounded and posses a uniform gap width.

The optimal width of the gap can be estimated from the asymptotic geometry of two parallel plates, which provides a lower bound for the achievable resistance; see Fig. 2. This geometry corresponds to a rectangular duct with the two small sides replaced by unphysical periodic boundary conditions, so the cross-sectional area A is still well defined. The prefactor for the scalar exchange in this geometry is **6**, we can then solve for the ℓ that results in a given scalar exchange efficiency E,

The result of our theoretical considerations is twofold: First, we qualitatively predict that natural selection favors nasal cavities where the separation between the walls is approximately constant everywhere. Second, we quantitatively predict the optimal gap width, either from the aspect ratio of realistic ducts that lead to a given E or by using Eq. **7** as an approximation.

### The Theory Agrees with Experimental Measurements.

Nasal cavities described in the literature are typically narrow and exhibit little variation in gap width (2, 3), which agrees with our theory qualitatively. For a quantitative comparison, we consider geometric measurements of nasal cavities of canid and arctoid carnivorans that have been reconstructed in silico from CT scans (18). The associated scalings of the geometric parameters with body mass are summarized in Table 1. The volumes V of the cavities and the lengths of the skulls scale isometrically, but the surface areas of the cavities exhibit significant positive allometry; that is, in heavier animals, it is larger than expected from geometric scaling.

To test whether our theory can explain the observed data, we calculate the surface area of the optimal geometry of the nasal cavity as a function of the constrained parameters D, A, L, Q, and E. Here, D is the scalar diffusivity, which is either the mass diffusivity of water vapor,

We first predict the surface area S in the simple geometry of parallel plates, where we can calculate the optimal gap width ℓ explicitly using Eq. **7**. Using the scalings described above, we find *A* shows that this predicted scaling agrees well with the measured data. Consequently, our simple scaling theory correctly predicts important geometric properties of real nasal cavities. In particular, the positive allometry of the surface area, which is observed in a wide range of animals (5, 22), is a direct consequence of the negative allometry of the optimal gap width ℓ. Note that, if the gap width scaled isometrically (with

So far, we considered the idealized geometry of parallel plates, which contains unphysical periodic boundary conditions. For large aspect ratios, this is a good approximation of the more realistic shape of a rectangular duct, but we showed above that the gap width does not scale isometrically and the aspect ratio thus varies with body mass. In particular, smaller animals will have aspect ratios closer to unity, and the scalings derived from the parallel plate model are not accurate in this case; see Fig. 4*A*. To correct this, we numerically determine the rectangular shape with a given cross-sectional area A that leads to the exchange efficiency *A* shows the associated surface area as a function of body mass, which now cannot be expressed as a simple scaling law. Note that this correction is insignificant for large animals, which confirms that they have nasal cavities with high aspect ratio where the parallel plates model is accurate. Conversely, there are large deviations for small animals, where the side walls contribute to the surface area significantly. In fact, we find that the exchange efficiency exceeds 1 for small animals even in the case of a square geometry (dashed line in Fig. 4*A*). This high efficiency in small geometries thus suggests that the nasal cavities of small animals have simpler cross-sections, which has, indeed, been observed (2, 3).

Strikingly, one point that deviates strongly from the theoretical prediction in Fig. 4*A* is for humans. The surface area of their nasal cavity is about half of what the scaling suggests, and the volume is even only about 10% of the prediction (Fig. 4*A*, *Inset*). The data point for humans was calculated from a standardized nasal cavity, which was obtained by averaging reconstructed geometries of 30 humans (23), together with typical respiratory parameters given in Table 2. To examine the geometry of human nasal cavities more closely, we compute the local gap width in the standardized nasal cavity using the shape diameter function, which gives the average distance of nearby walls at every point of the surface (24) (Fig. 4*B*), and the hydraulic radius *C*). Both quantifications indicate that the gap width is remarkably constant over a large fraction of the standardized nasal cavity, while the cross-sectional area varies significantly (Fig. 4*C*, *Inset*). However, the measured **7** together with the typical respiratory parameters summarized in Table 2. Thus, whereas the geometry of the human nasal cavity agrees with our qualitative result that the gap width should be constant for an efficient scalar exchange, the quantitative prediction deviate significantly from our theory. This is surprising since our theory worked well for all other tested mammals, and this might thus hint at an exceptional behavior of the human nasal cavity. Before we come back to this point in *Discussion*, we next consider how the shape of the head constrains the nasal cavity.

### Geometric Constraints Imply Labyrinth-Like Cross-Sections.

Natural nasal cavities have a complex labyrinth-like cross-section, which does not resemble the theoretically optimal shape determined above. This difference is likely a consequence of other factors, including additional geometric constraints, since the wide shapes that we predict would simply not fit into the head. However, the fact that our theory agrees well with experimental data suggests that natural nasal cavities function close to optimally. This would suggest that the bending and branching of the nasal cavity that is necessary to obtain labyrinth-like geometries does not significantly affect the physical principles that led to the optimal gap width given in Eq. **7**. To test this hypothesis, we examine the bending and branching of the cross-section and calculate how it affects the resistance and exchange efficiency. Fig. 5 shows that a U-shaped cross-section has virtually identical properties to a rectangle of the same aspect ratio. Consequently, bending the optimal cross-sectional shape in-plane affects neither the resistance nor the exchange efficiency significantly. To examine the consequence of branching, we consider a T-shaped junction with three rectangular branches of equal length. Numerical simulations indicate that both K and E are affected more strongly than in the case of bending, but still only change by a few percent compared with an equivalent rectangular shape; see Fig. 5. While the T-shaped junctions can be directly compared with rectangles, they also introduce additional corners, which increase the resistance. We show in *SI Appendix* that optimal junctions are more rounded and impact the resistance less than shown in Fig. 5. Taken together, neither bending nor branching affects the function of the nasal cavity strongly, implying that natural shapes are close to optimal.

Another geometric constraint on the nasal cavity comes from the fact that it must connect the pharynx (and thus the lungs) to the outside world. In humans, this forces a curved flow, which could influence the functions of the nasal cavity (5); see Fig. 1*A*. In general, curved flow increases the resistance and the exchange efficiency significantly (25, 26), but, in the case of the human nose, the bends are localized to the connecting regions, while the main nasal cavity is rather straight. We show in *SI Appendix* that the overall function of the nose is only slightly affected by the bent geometry, consistent with numerical simulations (27). This is because the connecting regions are much wider than the main nasal cavity. Note that this effect is even weaker in animals that have a straighter airflow than humans.

### Gradients in the Scalar Exchange Limit Heat and Humidity Loss.

Up until now, we have derived the optimal geometry of the nasal cavity by focusing on the efficiency of heating and humidifying the inhaled air. However, improving this efficiency can come at the expense of heat and water loss during exhalation. This is because heating implies that the walls of the nasal cavity are warmer than the inhaled air, while the recapture of heat can only occur when the walls are colder than the exhaled air. Consequently, it is impossible to both heat the air efficiently and recapture most of the heat during exhalation. Such a conflicting requirement also holds for humidification, and we show in *SI Appendix* that small animals would lose about

To study the trade-off between heating the inhaled air and recapturing heat during exhalation, we vary the scalar value **4**. In our calculations above, we considered *SI Appendix* shows that, in this case, *SI Appendix* confirm this picture and show that the scalar exchange efficiency is actually higher than predicted by Eq. **6**, because the scalar profile is typically not fully developed and entrance effects matter. Taken together, we conclude that a gradient boundary condition, as observed in nature (28), can improve the recapturing of heat and humidity, at the expense of a lowered exchange efficiency during inhalation. Note that this lower efficiency is approximately compensated by entrance effects that improve the exchange efficiency, so we expect Eq. **7** to work for nasal cavities with gradient boundary conditions and realistic lengths.

## Discussion

A critical issue for the shape of the nasal cavity is the opposing geometrical requirement for low nasal resistance and high exchange efficiency. Whereas resistance decreases with increasing gap width, the exchange efficiency is higher when the gap is thinner. The central result of this paper is the demonstration that the optimal geometry that balances these requirements has uniform gap width ℓ, which we predict in Eq. **7**. Strictly speaking, the optimal design of two parallel plates will not fit inside the head, but our calculations show that the bending and branching of the thin duct has only a modest effect on nasal efficiency. This suggests that the diverse morphology and labyrinth-like patterns of nasal cavities provide the lowest resistance for sufficient air conditioning under the geometric constraints imposed by the head. These physical and geometric constraints thus explain the large-scale morphology of nasal cavities, while the details likely exhibit additional constraints, like sufficient mechanical integrity and blood supply, which need to be studied in the future.

Our theory predicts that the surface area of the nasal cavity scales as

The scalar exchange efficiency E also depends on the trade-off between conditioning the inhaled air and recapturing the heat and moisture during exhalation. When both processes are considered, a gradient in the boundary conditions along the nasal cavity is generally optimal, but the exact details depend on the environment and the physiological state of the animal. Here, it will be interesting to separate evolutionary adaptations, e.g., by related species living in different climates, from short-term adjustments caused by phenotypic plasticity, where, for instance, the gap width could be narrowed by swelling the epithelial tissue or secreting additional mucus. Our theory predicts how such changes affect the conditioning of inhaled air and the efficiency of expelling heat with exhaled air. It might also be used to study olfaction and the clearance of pollutants from inhaled air, which can both be described as passive scalar transport.

We also show that humans have surprisingly small nasal cavities with a reduced gap width and surface area compared with the expectations based on body mass. In fact, the volume of the human nasal cavity is almost

## Acknowledgments

This research was funded by the National Science Foundation through Grant DMS-1715477 and Materials Research Science and Engineering Centers (MRSEC) Grant DMR-1420570, as well as by the Simons Foundation. D.Z. was also funded by the German Science Foundation through ZW 222/1-1.

## Footnotes

- ↵
^{1}To whom correspondence should be addressed. Email: david.zwicker{at}ds.mpg.de.

Author contributions: D.Z., R.O.-M., D.E.L., and M.P.B. designed research, performed research, analyzed data, and wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1714795115/-/DCSupplemental.

Published under the PNAS license.

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