sign in sign up
<<
Home , Exhalation, Human head, Human nose, Nasal cavity

Physical and geometric constraints shape the labyrinth-like

David Zwickera,b,1, Rodolfo Ostilla-Monico´ a,b, Daniel E. Liebermanc, and Michael P. Brennera,b

aJohn A. Paulson School of Engineering and Applied Sciences, Harvard University, Cambridge, MA 02138; bKavli Institute for Bionano Science and Technology, Harvard University, Cambridge, MA 02138; and cDepartment of Evolutionary Biology, Harvard University, Cambridge, MA 02138

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

The nasal cavity is a vital component of the take into account geometric constraints imposed by the shape that heats and humidifies inhaled air in all vertebrates. Despite of the that determine the length of the nasal cavity, its this common function, the shapes of nasal cavities vary widely cross-sectional area, and, generally, the shape of the space that it across animals. To understand this variability, we here connect occupies. To tackle this complex problem, we first show that, nasal geometry to its function by theoretically studying the air- without geometric constraints, optimal shapes have slender flow and the associated scalar exchange that describes heating cross-sections. We then demonstrate that these shapes can be and humidification. We find that optimal geometries, which have compacted into the typical labyrinth-like shapes without much minimal resistance for a given exchange efficiency, have a con- loss in performance. stant gap width between their side walls, while their overall shape can adhere to the geometric constraints imposed by the Results head. Our theory explains the geometric variations of natural The Flow in the Nasal Cavity Is Laminar. It has been suggested that nasal cavities quantitatively, and we hypothesize that the trade- the flow in the nasal cavity is turbulent (7, 8), since the speeds off between high exchange efficiency and low resistance to air- are high and nasal geometry is complex. Indeed, turbulence can flow is the main driving force shaping the nasal cavity. Our model easily be induced in the surrounding air by exhaling heavily, as further explains why , whose nasal cavities evolved to is apparent on a cold winter day. Inside the , turbulent flow be smaller than expected for their size, become obligate oral would induce additional mixing that improves the heating and breathers in aerobically challenging situations. humidification of the inhaled air (9), but it also implies a larger resistance to flow. It is thus unclear whether turbulence would be | fluid dynamics | scalar transport | evolution beneficial. To see whether turbulence occurs inside the nasal cavities of u¯ he nose not only allows us to smell but also humidifies, heats, animals of various sizes, we first estimate the mean speed using Tand cleans inhaled air before it reaches the . All these experimentally determined scaling relations of respiratory quan- vital tasks depend critically on nasal airflow, which is driven by tities with body mass; see Table 1. In particular, we combine the Q V the pressure difference created by the lungs and depends on the volumetric flux , the volume of the nasal cavity, and the length of the as a proxy for the length L of the cavity, to complex geometry of the nasal cavity. Nasal geometries vary con- 0.15±0.06 siderably among vertebrates in general (1) and among obtain u¯ = QL/V ≈ 0.3 m/s · (M /kg) . Since u¯ is much in particular (2–4), ranging from the complex labyrinth-like inter- smaller than the speed of sound, the flow is incompressible. For nal nasal cavity of dogs to the unique structure of humans that the flow to be turbulent, inertia has to dominate viscous dampen- combines relatively simple geometry in a short internal nasal cav- ing. The ratio of these two effects is quantified by the Reynolds ity with an additional external nasal vestibule; see Fig. 1. These qualitative differences in nasal geometry were likely selected as Significance adaptations to different functional requirements, but how the geometry of the nose influences the airflow and thus the func- Although nasal cavities fulfill similar tasks across animals, tion of the nose is a long-standing unsolved problem. their geometry varies widely. One such task is heating and The nasal cavity is a complex, air-filled space that connects humidifying the inhaled air, which works best if the nasal the two with the ; see Fig. 1A. All mammals cavity is narrow. However, narrow geometries have a large have an internal nasal cavity, but humans are unique in having resistance to flow. We show that these opposing geometri- an additional external vestibule with inferiorly oriented nostrils cal requirements are critical for shaping the nasal cavity and (5). The two sides of the cavity are separated by the nasal sep- strongly restrict the local gap width. In contrast, the over- tum and merge only behind the posterior nasal cavity (choanae) all shape has little influence on the resistance and air condi- that separates the nasal cavity from the . Each side can tioning, so the observed labyrinth-like patterns could emerge be further divided into the main pathway (turbinates) and large from geometric constraints imposed by the head. Our the- side chambers (sinuses). The walls of the nasal cavity are cov- ory predicts geometric parameters of nasal cavities quantita- ered by a highly vascularized bed of epithelial tissue overlain by tively, and it suggests that the surprisingly small nasal cavities a 10-µm-thick layer of , which is slowly propelled backward of humans force us to become oral breathers during heavy by cilia (6). The mucus consists mainly of water and thus humidi- exercise. fies inhaled air. Additionally, the nasal warms the air and absorbs airborne particles, like odorants. Author contributions: D.Z., R.O.-M., D.E.L., and M.P.B. designed research, performed We here study how the geometry of the nasal cavity influ- research, analyzed data, and wrote the paper. ences the airflow and the associated processes of heating and The authors declare no conflict of interest. humidifying the inhaled air. Generally, we expect that a narrower This article is a PNAS Direct Submission. geometry improves the efficiency of heating and humidification Published under the PNAS license. at the expense of greater resistance to airflow. Since this trade- 1 To whom correspondence should be addressed. Email: [email protected]. off likely plays an important role in shaping nasal cavities, we This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10. determine the shape that has the lowest resistance to airflow 1073/pnas.1714795115/-/DCSupplemental. for a given conditioning of the inhaled air. Here, we have to Published online March 5, 2018.

2936–2941 | PNAS | March 20, 2018 | vol. 115 | no. 12 www.pnas.org/cgi/doi/10.1073/pnas.1714795115 Downloaded by guest on September 28, 2021 Downloaded by guest on September 28, 2021 emtyo h aa aiywt h oetrssac oairflow to resistance lowest Width. the Gap with cavity Uniform nasal a the Has of geometry Cavity steady Nasal Optimal analyze The simply can associated we pulsatil- The the similarly, since here. animals . flow but, all larger rate, affects respiratory a ity the require set larger thus might a imply trade-off and would volume frequency lower tidal a while strongly, frequency increase the about if since by in flow, increases show steady We resistance animals. all the mass for that body similar the is pulsatility on of weakly find only we depends 1, which Table the from when frequency ings important the only and with is ciated humidification this However, the length air. characteristic limits induces the generally also of which and heating , resistance natural additional by caused of motion (15). development rats the and (14), prevents dogs cavity (13), humans nasal in the turbulence of about geometry than row heavier animals in only 10 occurs turbulence is that (11) cates plates lower parallel a between at flow turbulent example, for geometry; flow when turbulent are about flows in is Typically, mass. given body scalings with find increases the we Using 1, respectively. Table geometry, the of area geometries, complex In flow. the of and (10) air of viscosity wce tal. et Zwicker number 31 ref. and glider), greater and (potoroo in bars 3 (Scale ref. pig). (dog), (guinea 14 ref. from mission and (3), glider) (greater weight): body decreasing (14), by (dog) right to to left perpendicular from is (sorted airflow the The of (C region). plane. shape (gray the complex chambers the Coro- showing nasal (B) cavity air-filled . nasal two during human cavity a of nasal cross-section the nal though flows air how showing 1. Fig. C AB nte bqiosfaueo aa iflwi h oscillatory the is airflow nasal of feature ubiquitous Another 3 g ned ueia iuain aesonta h nar- the that shown have simulations numerical Indeed, kg.

naris External R aitlcross-section Sagittal (A) cavities. nasal of cross-sections Schematic 10 Re h ooostridactylus Potorous 2 = ooa rs-etoso h ih aa hmeso mammals of chambers nasal right the of cross-sections Coronal ) Dog 3 =¯ ihtepeietasto eedn ekyo the on weakly depending transition precise the with , uR V Mucus-covered walls /S h /ν Re f where , R ai porcellus Cavia where , Re R =2LQ eemc ihr h eitnewould resistance the higher, much were h h B scmaal to comparable is hni ie 1) eades hsindi- this Regardless, (12). pipes in than R ssalrta h length the than smaller is and h ln-oe ooo)(3), potoroo) (long-nosed V stecaatrsi ierdimension linear characteristic the is pharynx Naso- f /(ν ν mm.) 5 C, R and Potoroo ≈ fteflw(6.Uigtescal- the Using (16). flow the of h to lungs (f S 1.5 ) gie i)(1.Rpitdwt per- with Reprinted (31). pig) (guinea /ν ≈ S · ) R 70 r h oueadsurface and volume the are 10 1/2 h −5 · sgvnb h hydraulic the by given is ≈ (M 0% 50 m 0.7 (f Greater glider /kg) 2 /ν stekinematic the is /s · ai uu familiaris lupus Canis (M M ) etcabrrightchamber chamber left 0.36±06 earie volans Petauroides 1/2 oprdwith compared f (ν/ oteeffect the so , /kg) ese the seek We oethat, Note . IAppendix SI ) 0.09±07 Guinea pig 1/2 so , asso- Re Re , etof dent with scales which Eq. Solving and ity, plane, cross-sectional the in Laplacian with Quantity efficiency exchange scalar the by quantified is cavity the of cross-sectionally the cavity, value and scalar the scalar temperature averaged through constant body flowing While at a walls. kept implies the is which boundary humidity, consider the maximal first we where simplicity, case For the tissue. epithelial the by imposed where scalar the the state, with stationary cav- exchanged a scalar In is the cavity. passive and through the diffuses, a of flowed flow, walls as it the described with after be transported air is can the quantities concentra- in Both and vapor temperature ity. in water change of the tion by quantified be can simply on by effect airflow the its on studying shape cross-sectional the of influence the see shape; cross-sectional the where ouercflux Volumetric Cross-section rate Respiratory volume Nasal area Surface length Skull kilograms, of quantities units respiratory in of given Scaling 1. Table problem λ closer the larger is which c that equation only field velocity component the a has incompressible, and stationary, laminar, is P cavity. varying nasal on focus the shape. first cross-sectional of airflow we the straight, geometry the typically are the of cavities on dependence nasal Since properties the physical calculate its to cross-sectional and need and thus length, We flux, volumetric area. the given under air a the of humidifying and constraint heating of efficiency given a for ¯ 0 = (0) n h o hog h aiyi rvnb rsuedifference pressure a by driven is cavity the through flow The h etn n uiicto rpriso h aa cavity nasal the of properties humidification and heating The eeae ytelns ic h o nasrih aa cavity nasal straight a in flow the Since lungs. the by generated r eghsae htflo rmtegnrlzdeigenvalue generalized the from follow that scales length are c ¯ u (z C c D 0 = Q a ) K η y a eepesdas expressed be can oapproach to eoe h ifsvt,adbudr odtosare conditions boundary and diffusivity, the denotes ≈ n a eepesdas expressed be can and sanniesoa aaee htdpnsol on only depends that parameter nondimensional a is L .Here, ). (SI walls the at S 1 V A 1.8 V t for Q f · c ¯ 10 u u (z P eoti h ouercflux volumetric the obtain we , PNAS nteaildrcin hc by h Poisson the obeys which direction, axial the in −5 h resistance The . ) C E Prefactor = 0 9cm 19 ∇ 4cm 14 iltu hnefo t min value ambient its from change thus will c K stednmcvsoiyo i (10). air of viscosity dynamic the is Pa.s 9s 0.89 cm 0.4 cm 7. cm 2.9 cm 7 ln = b | ⊥ 2 . c ¯ K h xeto hspoesa h end the at process this of extent The . D u (L) ac 0 2018 20, March y 2 3 (x ∇ =  0.78 /s −1 = 2 3 3 ecntu quantify thus can We Appendix. SI , C 2 esto gets c ¯ c ¯ a c ¯ y c a(M (L) (0) K (z = ) − · = ) /kg) ∂ − A η − − z L K 2 c −0.26 (uc c c c b η , b b P L Exponent b 0.32 0.63 1.04 0.96 0.74 eso in show We . b L = |  + ), stelnt ftecav- the of length the is y , P , o.115 vol. P ihbd mass body with ± ± ± ± ± ± ± /Q 1(33) (18) 0.04 (18) 0.01 0.03 0.03 1(33) 0.02 0.01 (18) 0.01 n a ∇ ste indepen- then is b n ⊥ 2 e Q | c −z hsobeys thus o 12 no. = eoe the denotes IAppendix SI /λ R n Q u A where , Source c = dx | = c M b 2V that V 2937 dy [1] [2] [3] [4] /L at t f ,

PHYSIOLOGY PHYSICS 2 −1 − D∇⊥cn (x, y) = λn u(x, y)cn (x, y), [5] A rectangle

and the coefficients an can be determined from the initial value c¯(0) at the inlet. Note that we here neglected the small axial dif- fusion term, since it is dominated by advection (uL¯  D). More- over, in the simple case of a long cavity, entrance effects can be ellipse neglected, and only the mode with the largest λn contributes to E. In this case, we have DL optimized shape E = C · , [6] E Q

where CE = Q/(λ1D) is a nondimensional factor associated with B the largest λn . We show in SI Appendix that CE depends only on the cross-sectional shape and captures how the shape affects the

scalar exchange. gap width To find the geometry that has minimal resistance K for a given distance along centerline exchange efficiency E, we determine the respective prefactors C C C Fig. 3. (A) Sensitivity to shape perturbations of three cross-sectional shapes. K and E for several simple shapes; see Fig. 2. K is low- The arrows indicate the magnitude of the decrease of the resistance pref- est for a circular shape, but the associated CE is also minimal. actor CK (arrow length) when the shape is locally perturbed in the normal As expected, both CK and CE increase when shapes become direction (arrow direction) at fixed area A and scalar exchange efficiency narrower for larger aspect ratios, which illustrates the trade-off CE ; see SI Appendix for numerical details. (B) Width perpendicular to the between low resistance and large exchange efficiency. In fact, for centerline as a function of position for the three shapes shown in A. This all shapes considered, the ratio CK /CE approaches a constant gap width is uniform in the midsection of optimal shapes and rectangles, for large aspect ratios, indicating that both quantities increase while it varies significantly in ellipses. proportionally. However, not all shapes perform similarly: The resistance of rectangular shapes is up to 40% lower than that of ellipses with the same exchange efficiency. and SI Appendix, Fig. S2 show that optimal cross-sections are To understand what geometric features lead to good perfor- dumbbell-shaped, with a slender midsection. We compare the mance, we numerically determine how shapes need to be altered midsections of different shapes by quantifying the width of the to lower the resistance at constant exchange efficiency; see Fig. 3 shape perpendicular to the centerline as a function of the dis- and SI Appendix for details. This sensitivity analysis suggests that tance along the centerline. Fig. 3 shows that this gap width is sharp corners and narrow regions are detrimental, as indicated uniform for the optimal and rectangular shapes, while it varies by the large arrows in Fig. 3. However, it does not show clearly significantly for the ellipse. Taken together, optimal shapes are why rectangular shapes outperform ellipses. To understand this thus rounded and posses a uniform gap width. aspect better, we use the sensitivity analysis to obtain optimal The optimal width of the gap can be estimated from the shapes by iterative adaptation as described in SI Appendix. Fig. 3 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 −2 scalar exchange in this geometry is CE = 7.54 A` , where ` is the plate separation. Using Eq. 6, we can then solve for the ` that results in a given scalar exchange efficiency E, r Dτ ` = 2.75 , [7] E

where τ = LA/Q is the time it takes air to cross the cavity. The gap width must thus be similar to the typical distance (Dτ)1/2 the scalar diffuses while passing through the cavity (17). The result of our theoretical considerations is twofold: First, we qualitatively predict that natural selection favors nasal cavi- ties where the separation between the walls is approximately con- stant everywhere. Second, we quantitatively predict the optimal Fig. 2. Comparison of the relative resistance K/E for different cross- gap width, either from the aspect ratio of realistic ducts that lead sectional shapes as a function of a prescribed exchange efficiency E. Only to a given E or by using Eq. 7 as an approximation. the prefactors CK and CE are shown, but the top and right axes display the respective values for human with parameters given in Table 2. The Theory Agrees with Experimental Measurements. Nasal cavities The considered shapes are a circular pipe (large red circle), a square duct described in the literature are typically narrow and exhibit little (green square), elliptical ducts (red line, ratio of half-axes indicated at sup- variation in gap width (2, 3), which agrees with our theory qual- port points), rectangular ducts (green line, ratio of sides indicated at sup- itatively. For a quantitative comparison, we consider geomet- port points), rectangular ducts with rounded corners (violet line), N circu- ric measurements of nasal cavities of canid and arctoid carnivo- lar pipes in parallel (blue dots), and numerically optimized shapes (black stars). The resistance of the proper cross-sections is bounded by the value for rans that have been reconstructed in silico from CT scans (18). parallel plates (orange dotted line). Solid lines are quadratic interpolations The associated scalings of the geometric parameters with body between the support points (dots) where values were calculated numerically mass are summarized in Table 1. The volumes V of the cavities and cross-checked with the literature (32). and the lengths of the scale isometrically, but the surface

2938 | www.pnas.org/cgi/doi/10.1073/pnas.1714795115 Zwicker et al. Downloaded by guest on September 28, 2021 Downloaded by guest on September 28, 2021 S find about to raised is humidity about relative to heated with ically equilibrate about humidity to and walls the temperature that implying efficiency mass, exchange scalar relations the scaling sider reported for mass use body we with animals, be measured all could been not in parameters have they independently other since but, The dif- animal, each heating. slower for different its the than consider better just slightly we similar, are fusivity, values both temperature, Since room at heat of fusivity ie ytesaedaee ucin(4,frtesadrie aa aiy(3.(C (23). cavity nasal standardized the indicate for assumed bars we (24), cross; Here, function geometry). (orange diameter Eq. square in humans shape indicates given the in part plates by parallel dashed and of green; given ( dots) width (light 1. gap (blue geometry Table optimal rectangular (18) the in a from given carnivorans on follow arctoid lines) based (green solution and predictions numerical Our canid a 2). in Table in areas parameters surface deviation; standard turbinal respiratory measured experimentally 4. Fig. (with conse- isometrically efficiency scaled direct exchange width a gap is width the gap 22), if optimal that, (5, the which of animals allometry area, of negative surface the range of wide the quence a of cavities. in allometry nasal observed positive is real correctly the of theory particular, properties scaling In geometric simple important our predicts Consequently, data. sured 4A width gap optimal the calculate can we ` where plates, parallel vapor, water of parameters diffusivity nasal constrained the the of of geometry and function optimal a the as of cavity area surface the calculate geometric from expected than that larger allometry; is positive it scaling. animals, significant heavier exhibit in cavities is, the of areas wce tal. et Zwicker predicted the area cross-sectional with compared line) (blue (23) 4A. Fig. see shape rectangular case; the this the determine in numerically from we accurate derived this, not scalings correct have To are the will model and animals plate unity, smaller parallel to thus particular, closer ratio In ratios aspect the mass. aspect the that body and above isometrically with showed scale varies we not but does duct, width rectangular real- gap more a the For of of approximation shape conditions. good istic a boundary is this ratios, periodic aspect large unphysical contains which scalar the and airflow the nasal of the physics exchange. the of of parameters consequence a geometric is of cavity scaling allometric the that as scales it because ABC xlctyuigEq. using explicitly ≈ efis rdc h ufc area surface the predict first We we data, observed the explain can theory our whether test To ofr ecniee h daie emtyo aallplates, parallel of geometry idealized the considered we far, So hw htti rdce cln geswl ihtemea- the with well agrees scaling predicted this that shows 2V E ` Here, . ≈ ufc area Surface (A) data. experimental to predictions theoretical the of Comparison n hssae as scales thus and /`, 6 D mm = D D · h Inset (M mligta uiicto ftearwl be will air the of humidification that implying , stesaa ifsvt,wihi ihrtemass the either is which diffusivity, scalar the is A 60% ftesadrie aiya ucinof function a as cavity standardized the of /kg) h soitdvolumes associated The ) E A, ≈ E sn h clnsdsrbdaoe we above, described scalings the Using 7. ol rpsgicnl nlre animals, larger in significantly drop would 65% and L, nalaias ned nae i styp- is air inhaled Indeed, animals. all in 0.09±04 ` −2 D ae oehr u hoyshows theory our together, Taken . w fbd eprtr 2) n the and (20), temperature body of S 2 = Q ≈ h soitdsraeae is area surface associated The . e al .Fnly econ- we Finally, 1. Table see ; 9.8 .5 80% S · E 10 cm D ntesml emtyof geometry simple the in 1 = ` −5 h 2 ie yEq. by given 2)i humans. in (21) 2 = V · (M m needn fbody of independent ftenslcvte safnto of function a as cavities nasal the of 2 .2 1) rtedif- the or (10), /s /kg) · M 10 1/3 −5 0.87±07 7 D ,tescalar the ), genln) hc a acltdfor calculated was which line), (green m , z A, 2 . (19). /s Note `. L, Fig. . Q , bv.Ti ifrnei ieyacneuneo te factors, other shapes wide of the consequence since constraints, a determined geometric likely additional shape including is optimal difference theoretically This the above. which resemble cross-section, not labyrinth-like complex does a have cavities nasal Cross-Sections. Labyrinth-Like Imply Constraints Geometric cavity. nasal the constrains head nasal the in of human point shape the this the to this of how back consider and behavior come mammals, exceptional we an tested Before cavity. at other hint all thus for might well since surprising worked is theory This be theory. our should our predic- from width quantitative significantly gap the deviate exchange, tion the scalar cavity efficient that nasal an result human for constant qualitative the of our geometry with the in agrees whereas summarized Thus, parameters 2. respiratory Table typical the with together width gap optimal dicted cavity, nasal 4C, measured (Fig. standardized the significantly However, the varies remarkably area of cross-sectional is fraction the width while large gap a the over that constant indicate radius quantifications hydraulic Both area the cross-sectional and the of 4B), point from (Fig. every at (24) walls surface in nearby of the width function, distance diameter gap average shape the local the gives which the using cavity compute nasal we standardized human the closely, of more geometry the cavities respiratory examine To nasal standard- typical 2. with Table a together in given (23), from parameters humans reconstructed calculated 30 averaging by of was obtained geometries was humans the which cavity, and for nasal 4A, suggests, ized (Fig. point prediction scaling the data the of 10% what The about of only even half is about volume is cavity nasal 4A Fig. in prediction cal 3). cross- (2, simpler observed have been animals indeed, small has, of which sug- cavities sections, thus geometries nasal small the in that efficiency gests high This 4A). Fig. in line exceeds efficiency exchange for the that deviations 1 area find surface we the large plates fact, to In contribute are parallel significantly. walls side the there the where where Conversely, animals, small ratio accurate. have aspect is they correction high that model this with confirms that which cavities Note animals, nasal law. large cannot scaling now for which simple insignificant mass, is a body as of expressed function a be as area surface ated area cross-sectional efficiency given a with o ml nml vni h aeo qaegoer (dashed geometry square a of case the in even animals small for tiigy n on htdvae togyfo h theoreti- the from strongly deviates that point one Strikingly, yrui radius Hydraulic ) M oa a width, gap Local (B) 1. Table in given scaling the with together E S 1 = fnslcvte safnto fbd mass body of function a as cavities nasal of h ih-re iei i.4A Fig. in line light-green The . E PNAS = R h n h aaeesgvni al .(Inset 2. Table in given parameters the and 1 ln h anaxis main the along | sfrhmn.Tesraeae ftheir of area surface The humans. for is R ` h ac 0 2018 20, March ≈ ssgicnl mle hntepre- the than smaller significantly is 3.7 A n h perimeter the and m hc olw rmEq. from follows which mm, A htlast h exchange the to leads that | z E ftesadrie cavity standardized the of o.115 vol. = n sdtescalings the used and 1 enext we Discussion, hw h associ- the shows 7 | dr re)and green) (dark o 12 no. M Γ R hw are Shown . Fg 4C). (Fig. h 2 = Natural Inset). Inset). | A/Γ The ) 2939 7

PHYSIOLOGY PHYSICS Table 2. Typical physiological parameters for humans To study the trade-off between heating the inhaled air and c Quantity Value Source recapturing heat during exhalation, we vary the scalar value w that is prescribed at the walls of the nasal cavity. Given that Length of nasal cavity L (6.5 ± 0.7) cm (23) cw is the same for inhalation and exhalation, we can calculate Volume of nasal cavity V (21 ± 5) cm3 (23) how the scalar transported with the air changes during these two Surface area S (200 ± 25) cm2 (23) processes. In particular, we can define a scalar exchange effi- 2 Cross-sectional area A (3 ± 1) cm (23) ciency Eex for exhalation analogously to Ein for inhalation, given Tidal volume Vt (0.5 ± 0.1) L (34, 35) by Eq. 4. In our calculations above, we considered cw = cb and −1 f (15 ± 4) min (34, 35) Ein = 1, which implies Eex = 0. Heat can be recaptured when cw Body mass M (70 ± 10) kg (34, 36) is lowered to cw = 0.5(ca + cb), where ca is the ambient value. Volumetric flux Q (15 ± 5) L/min Q = 2Vtf A simple analytical model presented in SI Appendix shows that, in this case, Ein = Eex ≈ 0.4. These values can be improved to Ein = Eex ≈ 0.5 when a linear gradient from ca at the tip of the that we predict would simply not fit into the head. However, the nose to cb at the nasopharynx is used. Numerical simulations fact that our theory agrees well with experimental data suggests presented in SI Appendix confirm this picture and show that the that natural nasal cavities function close to optimally. This would scalar exchange efficiency is actually higher than predicted by suggest that the bending and branching of the nasal cavity that Eq. 6, because the scalar profile is typically not fully developed is necessary to obtain labyrinth-like geometries does not signifi- and entrance effects matter. Taken together, we conclude that cantly affect the physical principles that led to the optimal gap a gradient boundary condition, as observed in nature (28), can width given in Eq. 7. To test this hypothesis, we examine the improve the recapturing of heat and humidity, at the expense bending and branching of the cross-section and calculate how it of a lowered exchange efficiency during inhalation. Note that affects the resistance and exchange efficiency. Fig. 5 shows that this lower efficiency is approximately compensated by entrance a U-shaped cross-section has virtually identical properties to a effects that improve the exchange efficiency, so we expect Eq. 7 rectangle of the same aspect ratio. Consequently, bending the to work for nasal cavities with gradient boundary conditions and optimal cross-sectional shape in-plane affects neither the resis- realistic lengths. tance nor the exchange efficiency significantly. To examine the consequence of branching, we consider a T-shaped junction with Discussion three rectangular branches of equal length. Numerical simula- A critical issue for the shape of the nasal cavity is the oppos- tions indicate that both K and E are affected more strongly ing geometrical requirement for low nasal resistance and high than in the case of bending, but still only change by a few per- exchange efficiency. Whereas resistance decreases with increas- cent compared with an equivalent rectangular shape; see Fig. ing gap width, the exchange efficiency is higher when the gap 5. While the T-shaped junctions can be directly compared with is thinner. The central result of this paper is the demonstra- rectangles, they also introduce additional corners, which increase tion that the optimal geometry that balances these requirements the resistance. We show in SI Appendix that optimal junctions has uniform gap width `, which we predict in Eq. 7. Strictly are more rounded and impact the resistance less than shown in speaking, the optimal design of two parallel plates will not fit Fig. 5. Taken together, neither bending nor branching affects the inside the head, but our calculations show that the bending and function of the nasal cavity strongly, implying that natural shapes branching of the thin duct has only a modest effect on nasal are close to optimal. efficiency. This suggests that the diverse morphology and Another geometric constraint on the nasal cavity comes from labyrinth-like patterns of nasal cavities provide the lowest resis- the fact that it must connect the pharynx (and thus the lungs) to tance for sufficient air conditioning under the geometric con- the outside world. In humans, this forces a curved flow, which straints imposed by the head. These physical and geometric con- could influence the functions of the nasal cavity (5); see Fig. 1A. straints thus explain the large-scale morphology of nasal cavities, In general, curved flow increases the resistance and the exchange while the details likely exhibit additional constraints, like suffi- efficiency significantly (25, 26), but, in the case of the human cient mechanical integrity and blood supply, which need to be nose, the bends are localized to the connecting regions, while the studied in the future. main nasal cavity is rather straight. We show in SI Appendix that Our theory predicts that the surface area of the nasal cav- the overall function of the nose is only slightly affected by the ity scales as S ≈ (VQE)1/2, which implies the observed posi- bent geometry, consistent with numerical simulations (27). This tive allometry (18, 22). This scaling explains why short-snouted 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. Con- sequently, it is impossible to both heat the air efficiently and Fig. 5. Bending and branching does not affect the resistance K and scalar recapture most of the heat during exhalation. Such a conflict- exchange efficiency E significantly. Shown are the prefactors CK (orange) ing requirement also holds for humidification, and we show in SI and CE (blue) of (Left) bent and (Right) branched shapes normalized to the respective values for the corresponding rectangular shape with sides and Appendix 1% a b that small animals would lose about of their body as a function of its aspect ratio a/b. C follows from a numerical solution of weight due to exhaled water each day. To understand how to K Eq. 1, while CE has been calculated from Eq. 4 using the first 16 eigenmodes prevent this loss while still heating and humidifying inhaled air, of Eq. 5 as described in SI Appendix. Parameters are given in Table 2, and the scalar exchange needs to be studied for both inhalation and the dashed blue line in Right shows CE for channels that are elongated by a exhalation. factor of 100, emphasizing the influence of the junction in longer channels.

2940 | www.pnas.org/cgi/doi/10.1073/pnas.1714795115 Zwicker et al. Downloaded by guest on September 28, 2021 Downloaded by guest on September 28, 2021 3 h ,Kentee ,ZagZ(06 aia iflwadnnpril rvprdepo- vapor or nanoparticle and airflow Laminar pipe (2006) Z in Zhang C, transition Kleinstreuer Turbulence H, (2007) Shi J 13. Westerweel B, Hof TM, Schneider B, flow. Eckhardt channel in turbulence 12. to transition universal A (2016) K Tamai M, Sano 11. (2012) DM Gates 10. 5 hoK atnP agG,ShrrP 20)Nmrclmdln ftruetand turbulent of modeling Numerical (2006) PW Scherer GC, Yang verifica- P, Dalton and K, Development Zhao (2009) 15. MJ Lawson GS, Settles EG, Paterson BA, Craven 14. 7 qie M esne J aai R(08 aigi tc:Cneto,rato and reaction Convection, stick: it Making vis- (2008) SR and Manalis RJ, flow Messinger TM, of Squires rate velocity, 17. of calculation the for Method (1955) JR Womersley 16. wce tal. et Zwicker scalar passive as pollutants described of be might clearance both It the can transport. air. and which olfaction exhaled air air, study inhaled with inhaled from to heat of used expelling conditioning be of the also efficiency affect the pre- the changes swelling theory and by such Our mucus. narrowed how additional be dicts secreting could or width tissue from gap epithelial where, climates, adapta- the plasticity, different instance, evolutionary phenotypic in for by separate living caused species to adjustments animal. related short-term interesting the by of be depend e.g., state details tions, will physiological the exact it the the along and Here, but conditions environment optimal, the boundary generally on the is are cavity in the processes nasal gradient recapturing both When and a exhalation. air considered, during inhaled moisture and the heat conditioning between conditioning. off air the of part over relatively their take because could 22), (5, longer weight compared same cavities the nasal of smaller mammals have with (smaller birds why important explain less might when ter smaller is be humidifying can cavity and nasal heating the that result intuitive the tifies (smaller animals .BrmnT,IcoeaF (2011) FP Incropera TL, Bergman the 9. of Assessment Objective on Committee Standardisation F; Gordts PAR, and review. variation Clement Morphological A (2004) 8. mucus: MT Black nasal JN, of Georgi LL, rheology Shackelford The SE, Churchill (1998) J 7. Mason N, Jones M, Quraishi special 6. primates—with of capsule (2011) nasal D the Lieberman of Morphology 5. (2014) I Ruf W, adult Maier of skeleton 4. nasal internal the of morphology Comparative (2012) TE Macrini 3. .NgsV 15)Itouto otecmaaieaaoyo h oeadparanasal and nose the of comparative and the to (birds Introduction (1954) archosaurs VE Negus extant in 2. structures facial of Homology (1995) LM Witmer 1. h clrecag efficiency exchange scalar The aa iwy R,adES(05 osnu eoto cutcrioer n rhi- and rhinometry acoustic on report Consensus (2005) nomanometry. ERS and IRS, Airway, Nasal nose. human the in dynamics airflow Sci Allied Otolaryngol MA). bridge, Homo. and Daubentonia to reference tomography. computed x-ray 1–91. on based marsupials iini ua aa aiymodel. cavity nasal human a in sition flow. 12:249–253. sinuses. conchae. nasal and pneumaticity paranasal to Morphol J reference special with crocodilians), aia iflwadooattasotdrn nfn ntehmnadrtnose. rat and human the in sniffing during transport Chem odorant and airflow airflow.laminar nasal canine of model dynamics Eng fluid Biomech computational J high-fidelity a of tion osda natre hntepesr rdeti known. biosensors. is surface-based in diffusion gradient pressure the when 563. in drag cous nuRvFudMech Fluid Rev Annu n olSr Engl Surg Coll R Ann 225:269–327. 31:107–118. 131:091002. Rhinology ipyia Ecology Biophysical V h vlto fteHmnHead Human the of Evolution The 23:403–413. aesalrsraeae 2) n tquan- it and (29), area surface smaller have ) 43:169–179. 39:447–468. 15:141–171. nrdcint etTransfer Heat to Introduction Cuir ot hlsod MA). Chelmsford, North (Courier, mJHmBiol Hum J Am imc Eng Biomech J a Biotechnol Nat ntRec Anat E lodpnso h trade- the on depends also 297:1985–2006. 128:697–706. 16:625–638. 26:417–26. ulA u a Hist Nat Mus Am Bull HradUi rs,Cam- Press, Univ (Harvard Wly e York). New (Wiley, Physiol J E .Telat- The ). 127:553– a Phys Nat 365: Clin 6 ilrW 18)Dsrbto fbd egtadhih:Cmaio festimates of Comparison height: and weight body frequency, of volume, Distribution tidal (1986) in WJ Changes Millar (1972) B 36. Wa J, Brodsky J, Auchincloss R, Gilbert 35. (2016) IP mammals. Herman in 34. variables respiratory of Scaling (1967) WR flow Stahl and transfer 33. heat convection forced flow Laminar (1971) AL London RK, Shah 32. 0 rml M ibra E(04 nuac unn n h vlto fhomo. of (2012) evolution Rodgers Jeri the Ms. and running 31. Endurance (2004) and DE feliform Lieberman in turbinals DM, olfactory Bramble and 30. Respiratory (2014) mam- al. small et some B, Valkenburgh of van passage flat- nasal 29. in the cavity in gradients nasal Temperature the (1976) within WD conditioning Schmid air 28. Impaired (2016) al. pipes et curved T, in transfer Nishimura heat convective 27. forced on Study (1967) W Nakayama Y, Mori skeletoni- 26. and partitioning It mesh Consistent 25. (2008) D Cohen-Or standardized A, a Shamir of L, Creation Shapira (2009) J Marsan 24. S, Kherani EA, Matida MR, Johnson Y, birds. Liu and mammals 23. lizards, monitor in innovations Respiratory (2004) T Owerkowicz 22. warm to nose (1982) human P the of Cole ability the 21. measure to technique A (1999) al. et tempera- P, at Rouadi air arctoid humid 20. and of properties canid transport in and Thermophysical size (2008) turbinal P Tsilingiris olfactory and 19. Respiratory (2012) al. et PA, Green 18. iosFudto.DZ a lofne yteGra cec Founda- Science the German by the as by 222/1-1. well funded ZW also as through tion was DMR-1420570, Science D.Z. Grant Research Foundation. (MRSEC) Materials Simons Centers and Engineering DMS-1715477 and Grant through Foundation more in ACKNOWLEDGMENTS. humans inter- be in thus breathing future. would oral the It and in cavity. detail nasal nasal hotter compare the the in to humidified as and esting with heated extent be dumped same cannot the be air to inhaled can the but heat air, wider exhaled Consequently, much efficiency exchange is lower. scalar cavity and is oral resistance its the cavity, Because nasal the (30). than become also heat which dump humans activity, to Instead, physical helps heavy energy. during stronger, breathers more made oral be require obligate could would lungs this the this, resistance but larger overcome a To implies airflow. width gap to a smaller accommodate the condition- to (5), evolved air likely smaller adequate cavity nasal ensure smaller to a While gap ing. narrower a requires then h xettosbsdo oyms.I at h oueo the of volume almost the with is fact, compared In cavity area mass. nasal surface body human on and based width expectations gap the reduced a with ities eas hwta uashv upiigysalnslcav- nasal small surprisingly have humans that show also We mals. homo. faced report). (3rd function. diameter shape the using sation cavity. nasal human the of geometry MA). Cambridge, Univ, (Harvard thesis PhD 163–189. pp Amsterdam), Biomed, (Elsevier air. humidify and c. 100 and 0 between range ture carnivorans. ae nsl-eotdadosre measures. observed and 40:319–323. self-reported on based measurement. their by induced ventilation and 460. Res, Naval (Off 75. solutions Report analytical Technical of VA), summary Arlington, ducts–A curved and straight in friction digimorph.org/specimens/Cavia Nature length. snout of influence The carnivorans: caniform 16)Lmnrflwi uvdpipes. curved in flow Laminar (1969) H o ¯ opBohmPyilA Physiol Biochem Comp 432:345–352. Anat J n etMs Transf Mass Heat J Int LSCmu Biol Comput PLoS h oe pe iwyPyilg n h topei Environment Atmospheric the and Physiology Airway Upper Nose: The plPhysiol Appl J hsc fteHmnBody Human the of Physics 221:609–621. PNAS unaPg iia opooy vial at Available Morphology. Digital Pig, Guinea porcellus, Cavia hsrsac a uddb h ainlScience National the by funded was research This | 54:305–308. 12:e1004807. oclu/ cesdAgs 6 2017. 16, August Accessed porcellus/. 87:400–406. ac 0 2018 20, March nryCnesManage Convers Energy 10:681–695. plPhys Appl J 90% plMt Mech Math Appl J Comput Vis mle hnepce,which expected, than smaller Srne,NwYork). New (Springer, plPhys Appl J 106:784–795. | pdmo omnt Health Community Epidemiol J ntRec Anat o.115 vol. 24:249–259. 49:1098–1110. 33:252–254. 49:653–663. plPhysiol Appl J 297:2065–2079. | o 12 no. | 22:453– 2941

PHYSIOLOGY PHYSICS