The origin of the allometric scaling of lung's ventilation in mammals
Frédérique Noël, Cyril Karamaoun, Jerome A. Dempsey, Benjamin Mauroy
TThe origin of the allometric scaling of lung’s ventilation in mammals
Fr´ed´erique No¨el, Cyril Karamaoun, and Benjamin Mauroy Universit´e Cˆote d’Azur, CNRS, LJAD, Vader center, Nice, France (Dated: May 28, 2020)A model of optimal control of ventilation recently developed for humans has suggested that thelocalization of the transition between a convective and a diffusive transport of the respiratory gasdetermines how ventilation should be controlled to minimize its energetic cost at any metabolicregime. We generalized this model to any mammal, based on the core morphometric characteristicsshared by all mammals’ lungs and on their allometric scaling from the literature. Since the mainenergetic costs of ventilation are related to the convective transport, we prove that, for all mammals,the localization of the shift from an convective transport into a diffusive transport plays a criticalrole on keeping that cost low while fulfilling the lung’s function. Our model predicts for the firsttime where this transition zone should occur in order to minimize the energetic cost of ventilation,depending on the mammals’ mass and on the metabolic regime. From that optimal localization, weare able to derive predicted allometric scaling laws for both tidal volumes and breathing rates, atany metabolic regime. We ran our model for the three common metabolic rates – basal, field andmaximal – and showed that our predictions accurately reproduce the experimental data availablein the literature. Our analysis supports the hypothesis that the mammals’ allometric scaling lawsof tidal volumes and breathing rates are driven by a few core geometrical characteristics shared bythe mammals’ lungs, the physical processes of respiratory gas transport and the metabolic needs.
In animals, the main molecular source of energy, theadenosine triphosphate (ATP), is produced through along and intricate chain of biochemical reactions that,in fine, aims at recovering the chemical energy storedin primary energy sources, such as glucose. This energyconversion occurs mainly through oxidative processes col-lectively grouped under the term of cellular respiration.Glucose oxidation requires oxygen to be brought fromthe atmosphere through the tissues to each individualcell. Various evolutive strategies have emerged over time,from the gills of fish to the respiratory tracheae of arthro-pods. The mammal’s lung has been selected and shapedby evolutive processes to fulfill the body needs in oxygenand to eliminate the carbon dioxide, a major by-productof cellular respiration. Thus, the architecture of this or-gan has evolved following the needs for gas exchange.In mammals, the lung is composed of two major parts:the bronchial tree and the alveolar zone. The bronchialtree is structured as a nearly dichotomous tree structurewherein the airflow circulates to and from the alveolarzone, depending on the ventilation phase, i.e. inspirationor expiration. At inspiration, fresh air is brought intothe alveolar region where oxygen exchange takes placeby diffusion from the alveoli to the pulmonary blood cir-culation, through the alveolar–capillary membrane. Itallows erythrocytes to be reloaded in oxygen. In paral-lel, the carbon dioxide by-product is transferred from theblood stream to the alveoli. Then, at expiration, a highercarbon dioxide/lower oxygen air is expelled from the lunguntil fresh air comes in again at the next inspiration [41].The thin alveolar gas exchange membrane is combinedwith a large alveolar surface in connection to a com-pact bronchial tree. These characteristics have been se-lected by evolution to fulfill the gas exchange require-ments in mammals while satisfying the structural body needs, namely a compact and rib-covered thorax cav-ity [24].The transport of air in the lung, especially in thebronchial tree, requires a certain amount of energy due tophysical constraints. A hydrodynamic resistance to theair flow in the bronchi arises from friction effects, mainlydue to air viscosity [21]. In parallel, mechanical energyis needed to expand the thoracic cage and the lung tis-sues during inspiration. That energy is lost at expirationby the viscoelastic recoil of the tissues. Without care-ful regulation, the amount of energy induced by thesephysical constraints of ventilation might correspond to ahigh metabolic cost, even at rest [28]. However, naturalselection fovors configurations that require low amountsor minima of energy. Moreover, the process of optimiza-tion by evolution is performed under the constraint ofthe lung’s function: the gas exchange requirements haveto fit the metabolism activity.The typical functional constraint associated to that en-ergy cost was up to recently based on the total air flowrate entering the lung only [18, 25, 28], without account-ing for the gas transport in the organ. More recently,No¨el & Mauroy [27] optimized the energy spent for ven-tilation in humans with a more realistic functional con-straint, based on the oxygen flow in the alveoli, includingthe physics of oxygen and carbon dioxide transport in asymmetric branched model of the lung. This approachwas not only able to predict physiological ventilation pa-rameters for a wide range of metabolic regimes, but italso gave highlights on the distribution and transport ofoxygen and carbon dioxide in the lung.Actually, the progression of air in the lung is a combi-nation of two mass transport processes: convection anddiffusion. In the upper and central part of the bronchialtree, the convective transport largely dominates the mass a r X i v : . [ q - b i o . T O ] M a y transport, driven by the pressure gradient imposed bythe airflow. However, as the cumulative surface of thebronchi section area increases at each bifurcation, the airvelocity decreases while progressing towards the distalpart of the tree. At some point, the characteristic ve-locity of convection equilibrates with, and even becomessmaller than the characteristic velocity of diffusion; themass transport becomes dominated by the diffusion pro-cess. The localization of the transition zone between con-vection and diffusion depends on the geometry of the lungand on the ventilation parameters. The previous work ofNo¨el & Mauroy showed that the control of ventilation inhumans localizes the transition zone as a trade-off be-tween the oxygen demand and the availability and acces-sibility of the exchange surface deeper in the lung [27, 31].The lungs of mammals share morphological and func-tional properties, raising the question on whether theprevious results for human can be extended or not toall mammals. These shared properties are known to de-pend on the mass M of the animal with non trivial powerlaws, called allometric scaling laws [10, 15, 17, 29, 40].The physics of ventilation, and hence its control, islinked to the geometry of the lung. Consequently,the morphological differences amongst mammals also af-fect the control of ventilation. This is confirmed bythe allometric scaling laws followed by the ventilationfrequency and tidal volume. Breathing rate at basalmetabolic rate (BMR) has been estimated to follow thelaw f BMR b (cid:39) . M − Hz [42] and tidal volume tofollow the law V BMR T (cid:39) .
14 10 − M L [13, 40]. Atother metabolic rates, less data is available in the litera-ture except for the breathing rate of mammals at maxi-mal metabolic rate (MMR), estimated to follow the law f MMR b (cid:39) . M − . Hz [2]. Whether or not theseallometric scaling laws are reflecting the optimizationthroughout the whole class of mammals of the energyspent for ventilation remains still to be uncovered. Amodel able to predict these laws for mammals would bea powerful tool to derive them at other regimes, such asat submaximal exercise, at maximal exercise or at thefield metabolic rate (FMR).In this work, we develop two mathematical models:one to estimate the amount of oxygen captured from am-biant air by idealised mammals’ lungs; and one to esti-mate the energetic cost of ventilation. These two modelsdepend on the mammals’ mass and are coupled togetherto form a mathematical model for the natural selectionof breathing rates and tidal volumes. Under our models’hypotheses, our analysis shows that the physiological al-lometric scaling laws reported in the literature for bothbreathing rates and tidal volumes are actually minimiz-ing the mechanical energy of breathing. Moreover, weshow that the selected configurations are mainly drivenby the geometries of the mammals’ lungs and by thephysical processes involved in oxygen transport in thelung.
MODELLINGVentilation pattern and energy cost of ventilation
Airflow velocity can be idealized by a sinusoidal pat-tern in time, i.e. under the form u ( t ) = U sin(2 πt/T ).The quantity U is the maximal velocity amplitude and T is the period of ventilation, inverse of the breath-ing frequency f b = 1 /T . Denoting S the surface areaof the tracheal cross-section, the tidal volume is then V T = (cid:82) T S u ( s ) ds = US Tπ . The parameterization us-ing ( U, T ) or ( V T , f b ) are equivalent. The flow rate isthen ˙ V E = V T f b .The biomechanics of lung’s ventilation involves two ac-tive physical phenomena that are sources of an energycost [18, 27]. First, the motion of the tissues out oftheir equilibrium position implies that the diaphragm hasto use, during inspiration, an amount of energy that isstored in the tissues as elastic energy. This energy is thenused during expiration for a passive tissues recoil. Thepower spent P e ( U, T ) is related to the elastic propertiesof the thoracic cage and of the lung. These propertiesdepends on the lung’s compliance C [1] which is definedas the ratio between the change in volume of the lungand the change in pleural pressure. Then, P e ( U, T ) = 1 T (cid:90) T C V ( t ) dVdt ( t ) = 1 C U S T π , where V ( t ) = (cid:82) t S u ( s ) ds is the volume of the lung as afunction of time. Second, the airflow inside the bronchiinduces an energy loss due to viscous effects that has tobe compensated by the motion of the diaphragm duringinspiration. The viscous power dissipated depends on thehydrodynamic resistance of the lung R , P v ( U, T ) = 1 T (cid:90) T Ru ( t ) S ds = R U S , The total power spent is the sum of the power spentfor the displacement of the tissues and for compensatingviscous effects due to air motion in the bronchi, P ( U, T ) = P e ( U, T ) + P v ( U, T )= P e ( U, T ) (cid:18) π T RC (cid:19) = U S T π C (cid:18) π T RC (cid:19) . (1)The power can also be expressed using the equiv-alent ventilation parameters ( V T , f b ), ˜ P ( V T , f b ) = V T f b C (cid:0) π f b RC/ (cid:1) .The total power depends on the compliance of the lung C ∝ M [34] and on the hydrodynamic resistance of thelung R ∝ M − [40]. These two allometric scaling lawsallow to search for a minimum of the total power for eachmammals’ species, and to analyze the dependence of theminima on the mass.The function ( U, T ) → P ( U, T ) is to be minimized rel-atively to the ventilation amplitude U and the period T with the constraint f O ( U, T ) = ˙ V O , where f O ( U, T ) isthe oxygen flow resulting from a ventilation with char-acteristics (
U, T ) and ˙ V O the desired flow. The desiredoxygen flow ˙ V O depends on the metabolic regime. Allo-metric scaling laws have been derived for basal metabolicrate, ˙ V BMR O ∝ M [20, 29], field metabolic rate ˙ V FMR O ∝ M . [16] and maximal metabolic rate ˙ V max O ∝ M [39].Our analysis is based on these scalings that allow to setthe desired oxygen flow ˙ V O depending on the animal’smass. In the following, we develop a model of the lungof a mammal with mass M in order to estimate the oxy-gen flow f O ( U, T ) that results from the ventilation of itslung with a ventilation amplitude U and a period T . Core properties of the geometry of the mammals’lung
The derivation of the allometric properties of ventila-tion is based on the previous model developed by No¨el& Mauroy in [27], which is adapted to all mammals over5 orders of magnitude in mass. Mammal’s lung sharesinvariant characteristics [38]. First the lung has a tree-like structure with bifurcating branches. It decomposesinto two parts: the bronchial tree that transports, mainlyby convection, the (de)oxygenated air up and down thelung, and the acini where gas exchanges with blood occurthrough the alveolar–capillary membrane. The bronchialtree can be considered as auto-similar, as the size of thebranches is decreasing at each bifurcation with a ratioin the whole tree close to h = (cid:0) (cid:1) [19, 24, 38]. In theacini, the size of the branches are considered invariant atbifurcations [35, 38].Thus, the bronchial tree can be modelled as a tree withsymmetric bifurcations [24], as outlined in Figure 1. Ageneration of the tree corresponds to the set of brancheswith the same number of bifurcations up to the trachea.The bronchial tree is modelled by the G first generationsand the acini, where the exchanges with blood are takingplace, are modelled with the H last generations [38]. Thetotal number of generations of the tree is N = G + H .Hence, if the radius and length of the trachea are r and l , the radius r i and length l i of a bronchus in generation i is r i = r h i ( i = 0 ...G ) , r i = r G ( i = G + 1 ...N ) and l i = l h i ( i = 0 ...G ) , l i = l G ( i = G + 1 ...N ) . The quantity S i = πr i is the cross-section surface area ofa branch in generation i . In the bronchial tree ( i = 0 ...G ), S i = h i S , while in the acini ( i = G + 1 ...N ), S i = S G . ... ... ... ... lengths, diameters × hlengths, diameters × h .... .... .... .... .... .... .... .... ... ... ... ... ... lengths, diameters × ... ... ... ... ... ... ... Conductive zoneRespiratory zone r l FIG. 1. Outlines of the lung’s model used in this work. Ourmodel is based on the assembly of autosimilar trees with sym-metric bifurcations that mimic the two functional zones. Theconductive zone (beige) mimics the bronchial tree, where oxy-gen is only transported along the branches. The respiratoryzone (blue) mimics the acini, where oxygen is transportedalong the branches and also captured in the alveoli that coverthe walls of the branches.
As air is assumed incompressible in the lung in normalventilation conditions [7], flow conservation leads to (cid:40) u i = (cid:0) h (cid:1) i u ( i = 0 ...G ) ,u i = (cid:0) (cid:1) i − G u G ( i = G + 1 ...N ) . The derivation of a lung model that depends only onthe mass requires to relate explicitly the morphologicalparameters involved in our model with the mass of theanimal. We based our hypotheses on the datasets avail-able in [40], which brought a large set of theoretical al-lometric scaling laws for the cardiorespiratory system,compatible with the ecological observations. The mor-phological parameters used by our model are the trachearadius r , the reduced trachea length l and the gener-ations number G for the bronchial tree and H for theacini.From [40], r = aM . The bronchi diameters, and con-sequently the dead volume, are affected by the ventilationregime [5]. Since all bronchi diameters in our model arecomputed from the tracheal diameter, the value of theprefactor a depends on the regime and it is determinedbased on human data and dead volumes: a = 1 .
83 10 − m at BMR, a = 1 .
93 10 − m at FMR and a = 2 .
34 10 − m at MMR. The effect of exercise on the hydrodynamicresistance of the lung, R , is difficult to evaluate since ex-ercise increases both inertia and turbulence, which tendsto increase energy dissipation due to air motion in thebronchi [23]. In the absence of a relevant evaluation ofthe balance between the changes in geometry and the Variables Exponent PrefactorPredicted [40] ObservedMorphometry V L : Lung volume 1 1.06 [34] 53.5 mL [34] r : Tracheal radius 3/8 0.39 [36] 1.83 mm ∗ l : Tracheal length 1/4 0.27 [36] 1.87 cm ∗ r A : Radius of alveolar ducts 1/12 0.13 [37] 0.16 mm ∗ l A : Length of alveolar ducts -1/24 N.D. 1.6 mm ∗ n A : Number of alveoli 3/4 N.D. 12 400 000 ∗ v A : Volume of alveolus 1/4 N.D. N.D.Physics f b : Respiratory frequency (rest) -1/4 -0.26 [34] 53.5 min − [34] V T : Tidal volume (rest) 1 1.041 [40] 7.69 mL [34] P : O affinity of blood -1/12 -0.089 [6] 37.05 mmHg ∗ R : Total resistance -3/4 -0.70 [34] 24.4 cmH O s L − [34] C : Total compliance 1 1.04 [34] 1.56 mL cmH O − [34] P pl : Interpleural pressure 0 0.004 [11] N.D.Variables Exponent at BMR Exponent at FMR Exponent at MMRMetabolism ˙ V O : O consumption rate 3/4 [20, 29] 0.64 [16] 7/8 [39] t c : Transit time of blood in pulmonary capillaries 1/4 [13, 40] 1/4 (hypothesized) 0.165 [4, 13]TABLE I. Predicted and observed/computed values of allometric exponents for variables of the mammalian respiratory system. ∗ : Prefactor computed using human values (M = 70 kg) at rest. BMR: Basal Metabolic Rate, FMR: Field Metabolic Rate,MMR: Maximal Metabolic Rate. N.D.: No data found. more complex fluid dynamics, we kept the same value for R whatever the regime. Nevertheless, we studied how thepredictions of our model where affected by an increase ora decrease of R .The allometric scaling law for the tracheal length canbe derived from [40]. Indeed the dead volume ( V dead ∝ M ) is assumed to be proportional to the tracheal volume[36]. The relationship V dead ∝ πr l ∝ M leads to l ∝ M .The computation of G and H requires to assume thatthe radius of alveolar ducts are similar to the radius r A of the alveoli [38], for which an allometric scaling law isknown, r A ∝ M [40]. The number of generations G ofthe bronchial tree is then obtained from r A = r G = r h G ,and the number of terminal bronchioles follows 2 G ∝ M . This last allometric scaling law can be rewritten inthe form G = (cid:104) log( r A /r )log( h ) (cid:105) = (cid:104)
78 log( M )log(2) + cst (cid:105) .The total lung’s gas exchange surface S A ∝ M [40]is distributed over the walls of the alveolar ducts. Asingle alveolar duct has a lateral surface s ad = 2 πr A l A with l A = l h G ∝ M − , hence s ad ∝ M . The to-tal surface of alveolar ducts in the idealised lung is then S ad = 2 G +1 (cid:80) H − k =0 k s ad = 2 G +1 (2 H − s ad ∝ M .Hence, the amount of exchange surface per unit of alve-olar duct surface, ρ s = S A /S ad is such that the prod-uct ρ s (2 H − ∝ M is independent on the mass ofthe animal. It is assumed that the number of genera-tions of alveolar ducts in the acinus is independent on the mass [12, 30]. Consequently ρ s is also independenton the mass in our model. Under these conditions, theamount of exchange surface in our model fits the allo-metric law S A ∝ M . Oxygen and carbon dioxide transport and capture
The transport of oxygen and carbon dioxide in the lungis driven by three phenomena: convection by the airflow,diffusion and exchange with blood through the alveoliwalls.The mean partial pressure of oxygen over the sectionof a bronchus is transported along the longitudinal axis x of the bronchus. In the alveolar ducts in the acini, theadditional phenomena of exchange with blood occurs. Itcan be represented with a reactive term based on a reac-tive constant β that accounts for the capture of oxygenby the duct wall.Hence, in each bronchus of the lung belonging to gener-ation i , the partial pressure of the respiratory gas followsthe transport equation ∂P i ∂t − D ∂ P i ∂x (cid:124) (cid:123)(cid:122) (cid:125) diffusion + u i ( t ) ∂P i ∂x (cid:124) (cid:123)(cid:122) (cid:125) convection + β i ( P i − P blood ) (cid:124) (cid:123)(cid:122) (cid:125) exchange with blood = 0 , for x ∈ [0 , l i ] , (2)where P i is the mean partial pressure of the gas (mmHg) A BFIG. 2. A: Predicted tidal volume as a function of the mammals’ mass (log-log). Solid line: BMR, V BMR T (cid:39) . M . ml;dashed line: FMR, V FMR T (cid:39) . M . ml, dash-dotted line: MMR, V MMR T (cid:39) . M . ml. B: Predicted breathingfrequency as a function of the mammals’ mass (log-log). Solid line: BMR, f BMR b (cid:39) . M − . Hz; dashed line: FMR, f FMR b (cid:39) . M − . Hz, dash-dotted line: MMR, f MMR b (cid:39) . M − . Hz. The crossing of FMR and MMR curves predictedby our model is due to the increase of bronchi diameter at higher regimes extrapolated from humans [13] to all mammals. along the axis x of the bronchus, D is the diffusion coef-ficient in air of the gas considered and u i ( t ) is the air ve-locity in the bronchus of generation i . The reactive term β i that mimics the exchanges with the blood throughthe bronchi wall is equal to zero in the convective tree( i = 0 ...G ) and positive in the acini ( i = G + 1 ...N ). Avalue of β i different from zero means that gas exchangesoccur through the wall of the duct, with the gas crossingthe alveolar–capillary membrane whose thickness can beconsidered as mass independent, τ (cid:39) µ m [31]. Assum-ing that the diffusivity of the gas in the tissues can beapproximated with the diffusivity in water [31], the flowrate of the gas partial pressure per unit length of thebronchus is β i ( P i − P blood ) = ρ s πr i πr i kσ gas , H O D gas , H O τ ( P i − P blood )= ρ s kr i α ( P i − P blood ) , where k is the ratio relating partial pressure of the gasto its concentration in water, σ gas , H O is the solubil-ity coefficient of the gas in water and D gas , H O is thediffusion coefficient of the gas in water. The perme-ability of the alveolar membrane α is defined as follow, α = σ gas , H O D gas , H2O τ .Partial pressures continuity and mass conservation areassumed in the bifurcations. To determine the oxygenpartial pressure in the blood plasma that drives the ex-change, we assume that the flow of oxygen leaving thealveolar duct through its wall is equal to the flow of oxy-gen that is captured by the blood, accounting for theoxygen dissolved in the plasma and for the oxygen cap-tured by the haemoglobin [8, 27]. Finally, we evaluate the flow of oxygen exchanged withthe blood using f O ( U, T ) = N (cid:88) i = G +1 i (cid:90) t C + Tt C (cid:90) l i β i ( P i ( x ) − P blood ( x )) dx with t C a time at which the system has reached a periodicregime. RESULTSAllometric scaling laws of breathing rates and tidalvolumes
In 1950, Otis et al. showed that by constraining thealveolar ventilation ˙ V A = ( V T − V D ) f b within P ( V T , f b )with V D the dead volume, an optimal breathing fre-quency could be computed analytically by canceling thederivative of the power relatively to f b [18, 28], f b, pred = 2 ˙ V A /V D (cid:113) π RC ˙ V A /V D . At BMR, the allometric scaling laws of all the physio-logical quantities involved in this expression for f b areavailable in the literature: ˙ V A ∝ M [10], V D ∝ M [34], R ∝ M − [34, 40] and C ∝ M [34]. Hence we areable to derive an allometric scaling law for breathing rateat BMR, f BMR b , based on ventilation data in healthyyoung humans [13], f BMR b, pred ∝ . M − Hz. From thebreathing frequency and still based on ventilation datafrom [13], we can deduce the allometric scaling law forthe tidal volume at BMR, V BMR T = ˙ V A /f BMR b + V D .Since ˙ V A /f BMR b ∝ M /M − and V D ∝ M , we have V BMR T, pred = 7 . M ml. However, this approach is notable to predict the allometric laws at other regimes thanBMR. As the localization of the convection–diffusiontransition zone guides the blood-gas exchange, which hasto be adapted to the metabolic needs, the localizationand the refresh rate of the transition zone have to betuned depending on the regime. These properties dependon the morphology of the lung that has to be accountedfor in order to be able to reach predictions whatever theregime.Our analysis explores a set of masses ranging fromthe mouse (20 grams) to the elephant (5 tons) un-der three regimes: basal metabolic rate (BMR), fieldmetabolic rate (FMR) and maximal metabolic rate(MMR). The power P ( U, T ) is optimized with the con-straint f O ( U, T ) = ˙ V O . Both predicted breathing ratesand tidal volumes under the three regimes follow allo-metric scaling laws, as shown in Figure 2.At ˙ V BMR O : f BMR b (cid:39) . M − . Hz, V BMR T (cid:39) . M . mlat ˙ V FMR O : f FMR b (cid:39) . M − . Hz, V FMR T (cid:39) . M . mlat ˙ V max O : f MMR b (cid:39) . M − . Hz, V MMR T (cid:39) . M . mlThese results are only slightly sensitive to the allometricscaling law of the blood residence time in the pulmonarycapillaries. The hydrodynamic resistance R is positivelycorrelated to the exponent of the breathing rate f b . Ahydrodynamic resistance independent on the ventilationregime leads to good predictions for the breathing rate atboth BMR and MMR. If we neglect the increased inertiaand turbulence in the bronchi at MMR, the change indead volume at this regime leads the hydrodynamic re-sistance to be decreased by a factor larger than 3. In thiscase, the corresponding exponent for breathing rate dropsto − .
10. Consequently inertia and turbulence mightplay an important role on the control of breathing rate,but, interestingly, their influence seems to be balancedby the increase of the dead volume.
Transition between convection and diffusion
The localization of the transition between convectiveand diffusive transports can be estimated by the analy-sis of the variations with the mass of the P´eclet number,through the generations. This number arises by writ-ing the transport equations in their dimensionless form. From Eq. 2,2 l i DT ∂P i ∂s − ∂ P i ∂ξ + l i u i ( sT / D (cid:124) (cid:123)(cid:122) (cid:125) Pe i ( s ) ∂P i ∂ξ + β i l i D ( P i − P blood ) = 0 , for ξ ∈ [0 , . (3)The dimensionless time is s = 2 t/T with T / ξ = x/l i . The mean P´eclet number over a half breath isthen Pe i = V T f b l πr D (cid:0) h (cid:1) i . The localization of the transi-tion zone is reached when Pe i becomes smaller than oneover the ventilation cycle. This transition occurs at thegeneration k , with k such that2 k = (cid:18) V T l f b πr D (cid:19) = (cid:18) V E l πr D (cid:19) ∝ ˙ V E × M − At rest, the resulting allometric scaling law on the local-ization of the transition zone between convection and dif-fusion is 2 k r ∝ M . . At exercise, the scaling predictedby our model is different with 2 k e ∝ M . . Hence, thetransition generation k can be localized relatively to thegeneration of the terminal bronchioles G at both regimes: k r = G + 2 . − . M ) / log(2); k e = G + 6 . − .
275 log( M ) / log(2) . At exercise, the transition occurs deeper in the lung ofmammals than at rest. Animals with lower mass have atransition which occurs relatively deeper in their lung, asshown in Figure 3. In the acini, the oxygen is simulta-neously motioned along the alveolar ducts and capturedby the blood flowing in the alveoli walls. Consequently,the first alveolar ducts get higher oxygen concentrationthan those deeper. This phenomenon is known as thescreening effect [31] and results in an exchange surfacethat can be only partly active, depending on the local-ization in the lung of the transition between convectionand diffusion. Our model predicts that small mammalsare using almost all the volume of their lungs at rest,with low screening effect. To the contrary, large mam-mals present a clear difference in term of volume usagebetween rest and exercise, with a transition localized nearthe end of the bronchial tree at rest, with strong screen-ing effect, and a transition localized deeper in the aciniat exercise, with a lower screening effect.
Exhaled oxygen fraction
The oxygen flow captured by the lung is a propor-tion of the air flow inhaled, ˙ V O = ˙ V E ( f I − f E ) with˙ V E = V T f b the air flow rate, f I the oxygen fraction inambiant air and f E the mean exhaled oxygen fraction. FIG. 3. Localization of the transition between a convectiveand diffusive transport of the oxygen in the lung as a functionof the animal’s mass (logarithmic scale). The lines correspondto the localizations of that transition at BMR (rest, blue line)and MMR ( ˙ V max O , orange line). The vertical green line corre-sponds to human’s mass (70 kg). The lower beige region cor-responds to the convective zone of the lung, while the upperblue region corresponds to the exchange surface (acini). Smallmammals tend to transport oxygen mainly by convection andto use efficiently their exchange surface at rest, with the draw-back to have few reserve for increasing their metabolic rate.To the contrary, due to the screening effect [31], large mam-mals use only a small portion of their exchange surface at rest.Hence, large mammals have large reserve of exchange surfaceavailable for higher metabolic rates. The allometric laws predicted by our model for tidal vol-umes and breathing rates allow to derive similar laws forthe drop in oxygen fraction between ambiant and exhaledair, ∆ f = f I − f E : ∆ f BMR = 4 . M . %, ∆ f FMR =5 . M − . % and ∆ f MMR = 5 . M . %. The dropin oxygen fraction depends only slightly on the mass andis in the range 3 to 5%, whatever the ventilation regime.With an inhaled oxygen fraction in air of about 21%,the oxygen fraction in the exhaled air is ranging from16 to 18%, in full accordance with the physiology [38].The quantity η = ∆ f /f I can be considered as a measurefor the efficiency of oxygen extraction by the lung. Ourmodel suggests that the system extraction is optimal forvalues of η of about 20%. Differences in η exists betweensmall and large mammals because of the non zero ex-ponents in the allometric scaling laws of ∆ f . However,the values of these exponents are small and cannot beinterpreted as such. They might be the results of thesimplifications made in the model and/or of the numeri-cal approximations. DISCUSSION
From a set of core morphometric parameters that rep-resent the lung’s geometry, our model allows to predict,at any metabolic regime, a set of dynamical parame-ters that represent the lung’s ventilation and that min-imize an estimation of the mechanical cost of ventila-tion. This approach is able to predict with good accu-racy the allometric scaling laws of mammals’ tidal vol-umes and breathing frequencies available in the literature(tidal volume at rest, breathing frequencies at rest and˙ V max O [2, 34, 40, 42, 43]). The validation of our modelat both minimal and maximal metabolic regime suggeststhat its predictions are able to bring insights in the phys-iology of ventilation whatever the regime, in the limit ofthe availability of the input parameters. This suggeststhat the mechanical energy spent for ventilation mighthave driven the selection of ventilation patterns by evo-lution. The optimization process was functionally con-strained, because the lung has to fulfil the function oftransporting the needed respiratory gas to and from theblood. However, the function is dependent on the phys-ical processes on which these transports rely. The mostcrucial physical phenomena is the screening effect [31].Screening effect affects how the exchange surface is ef-fectively used and drives at which depth in the lung theconvection has to bring oxygen so that diffusion couldtake over the transport. The lung’s response to changein the metabolic regime is to adjust the amount of ex-change surface effectively used. Hence, only an analysisincluding a reliable representation of the mammal’s lungand of the gas transport is able to reach predictions com-patible with the physiology, whatever the regime.The idealized representation of the bronchial tree andof the exchange surface used in this study accounts forfive core characteristics common to all the mammals’lungs, as identified in the literature [24, 27, 28, 38, 40]:a bifurcating tree structure; an homogeneous decrease ofthe size of the bronchi at the bifurcations; the size ofthe trachea; the size of the alveoli; and the surface areaof the exchange surface. These characteristics are themain determinants of the tuning of ventilation in orderto minimize the energetic cost of ventilation. This indi-cates that once the metabolic regime is fixed, the mor-phology of the lung is probably the primary driver of thephysiological control of ventilation. We tested this hy-pothesis by altering in our analysis the allometric scalinglaws related to the geometry of the lung. We observedcorresponding alteration of the laws predicted for tidalvolumes and breathing frequencies. Since morphology it-self has probably been selected by evolution in order tominimize the hydrodynamic resistance in a constrainedvolume [24], morphology and ventilation patterns are in-tertwined together in order for the lung to function witha low global energetic cost, i.e. a low hydrodynamic re-sistance R and a low ventilation cost ˜ P ( V T , f b ) that alsodepends on R . Interestingly, our representation of thelung does not account for interspecific differences knownto exist between the lungs of mammals, such as differentdegrees of branching asymmetry, monopodial or bipodiallungs, etc. [9, 22, 26, 35]. Nevertheless, the predictions ofour model for the localization of the convection–diffusiontransition (Pe = 1) in idealized lungs lead to good estima-tions of the allometric scaling laws for tidal volumes andbreathing frequencies, indicating that the morphologicalparameters included in our model might primarily drivethe control of ventilation.The generation index of the convection–diffusion tran-sition, shown in Figure 3, depends linearly on the log-arithm of the mass. Since the structure of the treeis also governed by allometric scaling laws, the genera-tion index at which the transition between the bronchialtree and the acini occurs also depends linearly on thelogarithm of the mass of the animal. However, theslopes are different and the convection–diffusion tran-sition is located in the acini for small mammals andin the lower bronchial tree for large mammals. Thereason is that larger mammals actually need less oxy-gen relatively to their mass than small mammals, as˙ V O /M ∝ M − / at rest and ˙ V max O /M ∝ M − / at˙ V max O . Hence, at rest, small mammals use a large por-tion of their exchange surface, while large mammals tendsto use a small portion of that surface. This suggeststhat large mammals have more reserve to increase theirmetabolic rates, in the limit of other physiological con-straints, such as their ability to dissipate the excess ofheat [32, 33]. Interestingly, for masses near that of a hu-man, the convection–diffusion transition occur near thebeginning of the acini [19, 27, 31].The ability to increase the metabolic rates plays a cru-cial role in animal life, for example for foraging or forresponding to environmental threat. Our model suggeststhat the proportion of oxygen extracted from the am-biant air by the lung, found to be about 20%, dependsonly slightly on the metabolic regime. More oxygen is ex-tracted at higher metabolic regimes because the volumesof air inhaled are larger. A larger volume of air inhaledallows to use a larger portion of the exchange surface,hence reducing de facto the screening effect and acceler-ating the exchange speed. As a consequence, air has to berenewed at a quicker pace and breathing rate is increased.This effect is however counterbalanced by the increaseof the dead volume during exercise [5]. A larger deadvolume allows to optimize the mechanical energy with abreathing rate that is lower than it would be for a restdead volume. In particular, our model suggests that thiseffect is predominant on the control of breathing rates insmall mammals. Since small mammals already use mostof their exchange surface at BMR, they have almost nomargin for increasing the available surface at exercise.However, the hypothesized larger dead volume allows to bring a larger oxygen reserve at the convection–diffusiontransition point. The balance between the increased vol-ume and the air renewal rate needed to maintain an ef-ficient diffusion gradient leads our model to predict forsmall mammals a breathing rate at MMR smaller thanat FMR, as shown in Figure 2. A more detailed analysisof the trade-off is however needed to confirm or infirmthis trend. CONCLUSION
Our results highlight how the transport of the res-piratory gas influences the control of ventilation, andmore generally, the behavior of the lung and respiratorysystem. Our results contribute to improve our under-standing of the allometric scaling of ventilation in mam-mals. They represent a new theoretical framework ex-plaining how the evolution might have driven the designof the respiratory system and its links with the organ-ism’s metabolism. Our work suggests that the dynam-ical characteristics related to the control of ventilationis highly dependent on the morphological characteristicsof the lung. This dependence comes from the physicalprocesses involved in oxygen transport. Moreover, it hasbeen suggested that several core morphological param-eters related to the bronchial tree minimize the hydro-dynamic resistance of the lung in a limited volume, sothat the exchange surface can fill most of the thoracicspace [24]. Consequently, the control of ventilation is,at least partially, a direct consequence of the repartitionof lung’s space between the bronchial tree and the acini.More generally, this highlights the importance of the geo-metrical constraints in the selection of organs’ character-istics, not only in terms of morphology, but also in termsof dynamics.
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Boundary conditions
As the bronchi are connected through symmetrical di-chotomous bifurcations, mass conservation over a bifur-cation leads to S i (cid:18) u i ( t ) P i ( l i , t ) − D ∂P i ( l i , t ) ∂x (cid:19) − S i +1 (cid:18) u i +1 ( t ) P i +1 (0 , t ) − D ∂P i +1 (0 , t ) ∂x (cid:19) = 0Partial pressure continuity at bifurcations writes P i ( l i , t ) = P i +1 (0 , t ). The mass conversation can berewritten using continuity, − DS i ∂P i ( l i , t ) ∂x = − DS i +1 ∂P i +1 (0 , t ) ∂x (4)Boundary conditions and initial condition are ex-pressed in order for the problem to be well-posed. We as-sume that P (0 , t ) = P air at the trachea entrance, where P air is the partial pressure of the respiratory gas in theair. At the bottom end of the tree, the gas exchangewith blood is neglected ( elaborer cette hypothese ), whichwrites − D ∂P N ∂x ( l A , t ) = 0. Blood partial pressures
The blood partial pressure P blood of oxygen dependsnon linearly on the local value of P i , as a result of a bal-ance between the amount of gas exchanged through thealveolo-capillary membrane and the amount of gas storedor freed during the passage of blood in the capillary [27].As oxygen is stored within haemoglobin and dissolvedin plasma, this balance writes α ( P O − P blood , O ) = 4 Z ( f ( P blood , O ) − f ( P aO ))+ σ O v s ( P blood , O − P aO ) , with Z the haemoglobin concentration, each of thesemolecules containing four sites of binding with oxygenmolecules. The function f ( x ) = x . / ( x . + 26 . ) isthe Hill’s equation [14] that reproduces the saturation ofhaemoglobin depending on the partial pressure of oxygen in blood. The quantity v s corresponds to blood velocityin the capillaries and σ O corresponds to the solubilitycoefficient of oxygen in blood. The pressure P aO is thepartial pressure of oxygen in the arterial lung’s circula-tion (low oxygenated blood).Blood mean velocity v s depends on the mass and onthe metabolic regime studied. It can be computed asthe ratio of the length of the capillary l c over the tran-sit time in the capillary t c . As in [40], we assume thatthe terminal units of the blood network are invariant insize. Hence, the capillary length is fixed constant in ourmodel and equal to 1 mm. The transit time in capillariesdepends both on the mass and on the metabolic regime, t c (cid:39) . M at basal metabolic rate [13, 40] t c (cid:39) . M . at maximal metabolic rate [4, 13]No data is available in the literature for the fieldmetabolic rate, so we derive the human value t c = 838 susing data from Haverkamp et al. [13]. Since the allo-metric laws for the field metabolic rate are very similarto the ones of the basal metabolic rate we take the sameexponent. The allometric law for the transit time is de-duced, t c (cid:39) . M at field metabolic rate . Initial conditions