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Edited by: Umberto D'Ortona, UMR7340 Laboratoire de Mécanique, Modélisation et Procédés Propres (M2P2), France

Reviewed by: Mathias J. Krause, Karlsruhe Institute of Technology (KIT), Germany; Aurora Hernandez-Machado, University of Barcelona, Spain

This article was submitted to Biophysics, a section of the journal Frontiers in Physiology

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

Ventilation is at the origin of a spending of energy coming from air circulation in the bronchial tree and from the mechanical resistance of the tissue to motion. Both amplitude and frequency of ventilation are submitted to a trade-off related to this energy, but they are also submitted to a constraint linked to the function of the lung: to transport enough oxygen and carbon dioxide in order to respect metabolism needs. We propose a model for oxygen and carbon dioxide transport in the lung that accounts for the core physical phenomena: lung's tree-like geometry, transport of gas by convection and diffusion, exchanges with blood and a sinusoidal ventilation. Then we optimize the power dissipated by the ventilation of our model relatively to ventilation amplitude and period under gas flow constraints. Our model is able to predict physiological ventilation properties and brings interesting insights on the robustness of different regimes. Hence, at rest, the power dissipated depends very little on the period near the optimal value. Whereas, at strong exercise any shift from the optimal has dramatic effect on the power. These results are fully coherent with the physiological observation and raises the question: how the control of ventilation could select for the optimal configuration? Also, interesting insights about pathologies affecting ventilation could be derived, and our model might give insights on therapeutical responses, with specific breathing strategies or for better driving mechanical ventilation.

The respiratory system's function consists in supplying the body with oxygen and in removing carbon dioxide. Each part of the respiratory system is regulated according to the others. Hence, the control of ventilation is coupled to the control of heart rate, so that both lung ventilation and blood circulation are coordinated in order to cope with body needs. Regulation is performed through inputs from sensors, amongst which sensors to oxygen and carbon dioxide partial pressure in blood play a crucial role. As a consequence, partial pressure in oxygen and carbon dioxide in blood are strongly regulated in mammals (Weibel,

Ventilation decomposes into inspiration and expiration. During inspiration, an amount of ambient air is internalized in the lung and fresh oxygenated air with low carbon dioxide is brought into the bronchi, closer to the exchange surface. From there, diffusion occurs downward the bronchial tree for oxygen, as blood acts as an oxygen pump, and upward the bronchial tree for carbon dioxide, as blood acts as a carbon dioxide source. The internalized volume of air is then expelled during expiration and replaced by new fresh air during the next inspiration. The control of ventilation is based on the regulation of the volume of air that is internalized (ventilation amplitude) and the frequency at which this volume of air is renewed (ventilation frequency) with the aim to keep oxygen and carbon dioxide partial pressure constant in blood.

Ventilation amplitudes and frequencies are stereotypic in human and mammals (West et al.,

Trade-off between elastic energy stored in the tissue and viscous energy dissipated in the air circulation (exercise regime, computed from our model).

In this work, we propose a transversal approach for this problem, by developing a model based on the core physical phenomena involved in lung's ventilation and function (Weibel,

We show that the system built from these hypotheses can predict ventilation amplitudes and frequencies compatible with the physiology at different functional regimes and allows to understand finely and quantitatively the lung's inner dynamics of gas transport.

In the next section, section 2, we will describe our model for oxygen and carbon dioxide transport in the lung and the energy we propose to optimize. In section 3, we present and analyse the optimal ventilation in three emblematic cases: exercise, altitude and changes in lung's geometry. In section 4, we discuss the model's predictions and limitations, before concluding in section 5 on model's perspectives.

Our model is based on the set of following assumptions:

The lung is modeled as a tree with symmetric bifurcations.

The lung is divided in two parts: the convective tree (17 first generations) and the acini (6 last generations).

The size of the branches in the convective tree decreases at each bifurcation with a constant ratio.

All branches in the acini are of the same size

Oxygen and carbon dioxide are transported with air by convection and by diffusion. Exchange with blood is accounted for in the acini with a reaction term.

Air fluid mechanics is incompressible and reduces to mass conservation in the bifurcations.

Gas concentration at the trachea inlet is equal to that in the air.

In blood, oxygen is mainly linked to hemoglobin and carbon dioxide is mainly linked to bicarbonate ions.

All these assumptions are detailed in the next sections.

The geometrical model mimicking the bronchial tree is based on a symmetric dichotomic bifurcating tree (Mauroy et al.,

To account for the core geometrical properties of the lung, we assume that the size of the branches in the conductive tree decreases from one generation to the next with a ratio 0 <

with _{i} the length and _{i} the radius of a generation _{i} of a branch in generation _{i} and the lumen area of the branch _{i}. As air compressibility effects in the lung are considered small during forced expiration (Elad et al., _{i} in bronchi,

The last six generations mimic the acini. We assume that the size of the branches in acini remains constant (_{A} and the length _{A} are respectively equal to the radius and the length of the last generation branches of the conductive tree. Likewise the lumen area of branches remains constant and is

where _{A,i} is the mean fluid velocity in a branch in generation

The transport of oxygen and carbon dioxide in the bronchial tree is driven by three main phenomena: convection, diffusion and exchange with the acini walls. In the proximal conductive branches, the transport of the two species is mostly made through convection as transport velocity is higher than diffusive velocity. In the distal conductive branches, gas conductive velocities have decreased enough, and diffusion is dominating the species' transport thanks to the exchange occurring on the acini walls. We will describe the fluid motion along the axis of the bronchi, using a one dimensional model in bronchi. As bronchi and fluid properties are the same in all branches from the same generation, equations of transport are the same for each branch in a same generation.

The transport dynamics of the partial pressure of oxygen and carbon dioxide in a single branch of the conductive tree is given by the following convection and diffusion equation,

_{i}(^{−1}), ^{2} · s^{−1}) and _{i} is the _{2} or _{2} partial pressure (mmHg).

Oxygen and carbon dioxide transport and exchange in the acinus consists in a convection and diffusion equation with a reaction term mimicking the exchanges with blood. In a branch belonging to an acinus, partial pressure of oxygen or carbon dioxide checks

where _{A,i} is the partial pressure of the gas (mmHg), _{blood} is the partial pressure of the gas in the blood (mmHg), and finally β is an exchange coefficient. We can express β as follow (Weibel,

^{−2} · s^{−1} · mmHg^{−1}) represents the permeability of the alveolar membrane. The coefficient _{gas,H2O} is the diffusion coefficient of the gas in water (m^{2} · s^{−1}), σ_{gas,H2O} is the solubility coefficient (mol · m^{−3} · mmHg^{−1}) of the gas in water and finally τ is the thickness (m) of the alveolar membrane.

Equations (1, 2) can be adimensionalized and bring forth three adimensional numbers:

α_{i} represents the relative amplitude of the transitory effects and of the diffusion; the Peclet number _{i} represents the relative amplitude of the convection and of the diffusion; and γ_{i} represents the relative amplitude of the gas capture by blood and of diffusion and is meaningful only in the acini. These numbers are plotted on

Adimensional numbers at rest

These numbers show interesting insights on the behavior of gas transport. At rest, diffusion becomes dominant near the acini inlet only. Convection is dominant in the whole conductive tree. Transitory effects are slightly smaller than convection effects in the convective tree. Interestingly, in the acini, transport by diffusion, absorption by blood, and transitory effects are all similar in amplitude. At exercise, convection is dominant on every other phenomena down to the middle of the acini, where diffusion, absorption by blood and transitory effects become dominant and with similar amplitude. Diffusion and transitory effects are of the same amplitude because of the geometrical properties of the acini. The fact that absorption by blood is also of the same amplitude is the consequence of the value of

This analysis shows that all transport phenomena have to be included in the equations in order to reach satisfactory predictions for different metabolic regimes. Depending on these regime, the physics of transport and exchange with blood is driven by the three adimensional numbers previously defined.

The bronchi are connected together with bifurcations. We define a new variable _{bif,i} which represents the mean gas partial pressure in a bifurcation of generation

with _{bif,i} (m^{3}) the volume of the bifurcation

We approximated the volume of a bifurcation as the volume of three tubular extension of the three branches involved in the bifurcation: one with radius _{i} and length _{i}/2 and two with radius _{i+1} and length _{i+1}/2.

In order to close the system of equations, boundary conditions at both ends of the tree are needed. We assume _{0}(0, _{air} at the inlet of the root of the tree that models trachea. _{air} is the partial pressure of the gas considered in the air. And for the end of the last generation of the acini, we use a flux boundary condition, based on the exchange laws previously defined,

Finally, an initial condition is needed. We suppose that at time _{0}(0) = 0, _{A,i} constant in the acinus. We also suppose that _{bif,i} = _{i}(_{i}) = _{i+1}(0). With this hypotheses, we can compute the explicit solution of this system in the whole conductive tree :

We solve this system with an implicit scheme in space using the language Julia. Given a ventilation pattern, we compute sufficient ventilation cycles to reach a periodic pressure pattern in time.

In Equation (2), the term _{blood} is actually dependent on time, space and gas species. Oxygen can be found in blood dissolved in the plasma and linked to hemoglobin. To compute _{blood,O2} for oxygen, we use the same formulation as in Felici (_{s} as blood velocity (m · s^{−1}), we can relate _{blood,O2} with several physiological quantities,

The first term in the right hand side represents the link to hemoglobin. The factor 4 corresponds to the fact that a molecule of hemoglobin can link 4 molecules of oxygen. _{0} represents the concentration of hemoglobin in the blood (mol · m^{−3}). The function

The second term in the right hand side of the equation represents oxygen solubility in the plasma. σ is the solubility coefficient (mol · m^{−3} · mmHg^{−1}) of oxygen in blood and _{aO2} is the partial pressure of oxygen in pulmonary arterial blood.

Carbon dioxide can be dissolved in plasma, linked to hemoglobin and linked to bicarbonate ions. Similarly as oxygen, we can build an equilibrium law on carbon dioxide flow using (Sun et al.,

with _{s} the blood velocity (m · s^{−1}), σ the solubility coefficient (mol · m^{−3} · mmHg^{−1}) of carbon dioxide in blood, _{0} the hemoglobin concentration (mol · m^{−3}) and _{2} the oxygen-hemoglobin saturation (percents).

Practically, the partial pressure of oxygen _{aO2} and carbon dioxide _{aCO2} seen by the acini might be different to that of the pulmonary arterial circulation, as the history of the blood flowing in the acini wall is unknown. Blood could have already been in contact with acini air upstream. Consequently, we compute and use, instead of _{aO2} and _{aCO2}, efficace partial pressures in oxygen

To compute the exchanges between alveolar air and blood, we need to estimate the efficace gas partial pressure in pulmonary arterial and venous blood (Feher, _{aO2} = 40 mmHg and _{aCO2} = 47 mmHg. For oxygenated blood (pulmonary venous blood), we have _{vO2} = 100 mmHg and _{vCO2} = 40 mmHg.

Tidal volume (_{T}), mean air flow velocity in trachea (_{0}) and trachea cross-sections (_{0}) are related with:

The mean airflow velocity writes :

During rest ventilation, a human breathes around 12 times a minute, which corresponds to a period of ^{−3}m^{3} (Feher, ^{−1}. With all these parameters our transport model gives us the amount of oxygen captured by blood (_{2}) : 2.33 · 10^{−4}mol · s^{−1} and the amount of carbon dioxide expelled from blood (_{2}): 1.06 · 10^{−4}mol · s^{−1}. These values are in the range of physiology which is around 1−2 · 10^{−4}mol · s^{−1} (Jett et al.,

The respiratory exchange ratio (RER) is defined as follow:

This coefficient is supposed to be between 0.7 and 1. Using in our model typical arterial and venous partial pressures in blood, this coefficient is predicted to be about 0.45. The physiological value at rest is however about 0.8 (Feher,

Respiratory exchange ratio in function of the partial pressure of the respiratory arterial blood.

Number of molecules of oxygen and carbon dioxide per second during five respiratory cycle. The partial pressure in the arterial blood is set to _{art} = 88 mmHg.

In order to validate the choice for efficace partial pressure, we did a perturbation analysis on the RER at rest and at exercise, see

Respiratory Exchange Ratio at rest with A the amplitude and T the period of the respiration.

0.9 | 0.77 | 0.80 | 0.80 |

1 | 0.80 | 0.80 | 0.80 |

1.1 | 0.81 | 0.80 | 0.79 |

We have satisfactory little variation on RER when rest ventilation amplitude and frequency are perturbed and we remain in the physiological range.

During intense exercise, human can exhibit up to 40 breaths a minute, which represents a respiratory period ^{−3}m^{3} (Feher, ^{−1}. The values of the RER during exercise predicted by our model are also shown on

Respiratory Exchange Ratio during exercise with A the amplitude and T the period of the respiration.

12 | 0.90 | 0.90 | 0.90 |

13 | 0.91 | 0.91 | 0.91 |

14 | 0.92 | 0.92 | 0.92 |

These last analyses show that our hypothesis to use an efficace partial pressure in oxygen for exchanges in our model leads to predictions fully compatible with expected physiological responses.

Our model is based on a set of parameters that needs to be quantified from physiology. The parameters' list and values are shown on _{0}) of the root of the branch, mimicking the trachea, and from the homothetic ratio _{0} = 6_{0}. Although this value is not fully accurate for the main bronchi, it is a good approximation for the other branches. Since, the global behavior is mainly driven by the most numerous bronchi, extending the length over diameter ratio to all the branch of the tree is a reasonable approximation.

Table of parameters for the environment.

Radius of the trachea | m | 10^{−2} (Mauroy, |

Homothetic ratio (h) | Dimensionless | 0.7937 (Mauroy, |

Thickness of the alveolar membrane | m | 1 · 10^{−6} (Felici, |

Velocity of blood flow | m · s^{−1} |
5 · 10^{−4} (Felici, |

Hemoglobin concentration in blood | mol · m^{−3} |
9.93 (Davis, |

pH | Dimensionless | 7.4 (Sun et al., |

pK | Dimensionless | 6.09072 (Sun et al., |

Once the geometry of the lung defined, we can define the parameters linked to oxygen and carbon dioxide behavior, see ^{o} Celsius).

Table of parameters for the oxygen and the carbon dioxide.

Diffusion coefficient in air | m^{2} · s^{−1} |
0.2 · 10^{−4} (Mauroy, |
0.14 · 10^{−4} (Mauroy, |

Diffusion coefficient in water | m^{2} · s^{−1} |
3.3 · 10^{−9} (Felici, |
2.505 · 10^{−9} (Lu et al., |

Solubility coefficient in the blood | mol · m^{−3} · mmHg^{−1} |
1.34 · 10^{−3} (Linder and Melby, |
3.07 · 10^{−2} (Higgins, |

Henry solubility | Dimensionless | 2.592 · 10^{−2} (Sander, |
0.594 (Sander, |

The power spent by the lung to bring air in contact with the exchange surface can be divided in two parts: a viscous part due to the resistance of the bronchial tree to air flow and a mechanical part due to the elasticity of the tissue. The viscous power is computed using the hydrodynamic resistance of the lung ^{5}Pa · m^{−3} ·

where ^{−7}m^{3} · Pa^{−1} (human). Compliance is commonly used to evaluate lung's elasticity, and relates lung's volumes with lung's pressures under static conditions. Compliance is a synthetic variable, mixing the effects of many biophysical phenomena occurring in the lung, such as tissue elasticity, surfactants' effects, etc. Compliance depends notably on lung's volume when deformation is high, as shown by the pressure-volume curves in Agostoni (

We assume elastic energy is brought during inspiration only, i.e., for

Consequently the total power dissipated is

The relative influence of each power source depends on the frequency

as π^{2}

We aim at minimizing _{O2}(_{O2}(

Practically, ventilation period

consequently, we search for zero of

Predicted link between ventilation amplitude and period when the oxygen flow is constrained (rest regime).

Previous studies in the literature have been looking for optimal ventilation using modeling approaches, but they did not account for oxygen and carbon dioxide flows in the lung, or only in a basic way, without detailing the transport and exchanges of gas within a tree geometry and with detailed physics (Otis et al.,

We ran the model for different amount of oxygen needs, mimicking physical activities with increasing intensity (Jett et al.,

Activities and the amount of oxygen consumed.

Walking for exercise (5 km/h) | |

Bicycling (15 km/h) | |

Jogging (9 km/h) | |

Basketball | |

Ice hockey |

Up: Energy in function of amplitude and period for different types of exercise. Ticks represent +5% relative energy variations. Down: Partial pressures distributions, results are plotted for peak flows. At exercise, “fresh” air is pushed deeper in the lung. _{2} and _{2} distribution in the tree at rest, and _{2} and _{2} distribution in the tree at exercise.

At low regimes, the model predicts relatively high optimal period ^{2}^{2}

Model response in term of oxygen flow to an increase of membrane thickness at rest to mimic pulmonary oedema. As shown in Sapoval and Filoche (

When the exercise intensity increases, the power profiles as a function of the period become steeper and steeper and focus the optimal value within a tighter region. So our model predicts a finely tuned ventilation period at high regimes, in a way much more critical than at low regimes. An opposite behavior is observed for ventilation amplitude, but it is quantitatively lower. As amplitude increases, oxygen source goes deeper within the lung, entering the acini and increasing the exchange efficiency, but by thus draining more quickly oxygen from air. Renewing of the internalized air becomes more crucial to keep sufficient oxygen flow. A similar behavior occurs for carbon dioxide, but in the opposite direction. The increase in RER is due to a stronger response of carbon dioxide to a reduced screening as its diffusion coefficients are smaller than those of oxygen.

Oxygen and carbon dioxide exchanges are tightly related in our model, but only oxygen exchanges are constrained. However, the model predicts a RER fully compatible with physiology: as expected it increases from 0.8 at rest up to 0.9 at high exercise, see

Respiratory Exchange Ratio as a function of oxygen flow.

Hypoxia is the consequence of an alteration of the flow between alveolar gas and blood. This is typically one of the response observed in high altitude (Peacock,

Partial pressures in oxygen as a function of altitude in percents of sea-level partial pressure.

0 | 100% |

1,000 | 89% |

2,000 | 79% |

3,000 | 69% |

4,000 | 60% |

Our model predicts an increased ventilation in order to compensate the lower oxygen partial pressure, with higher amplitude and lower periods as shown on

Period and amplitude in function of altitude.

Optimal ventilation depends on the relative values of elastic and viscous powers. The power profile (but not its scaling) is driven by the quantity

For increases of resistance that do not affect notably the amount of inhaled air volume, typically for local or proximal constrictions, we observed very low changes in the optimal configuration. The energy increases proportionally with the dissipative term. In this case, the relevant parameter for computing the optimal value is the product

For increases in resistance (see

Energy in function of amplitude and period for different resistance values. Ticks represent +5% relative energy variations.

The optimal configuration for the ventilation of our lung's model is based on physical phenomena known to play a crucial role in the lung's function: gas convection, diffusion and exchange with blood (Weibel,

Our model allows to analyse how the coupling of these phenomena could drive gas exchanges in a way that minimizes a total power, sum of the mechanical power (compliance) and of the dissipative power (hydrodynamic resistance) (Johnson,

The control of ventilation in the lung is a complex dynamical and biophysical system that relies on many sensors' signals of different types (mechanical, chemical), with delays and complex integration. Many models have been proposed to mimic how the brain can control ventilation in accordance with metabolic regimes, see for example the reviews (Raux et al.,

In the case of rest regime, our model predicts an optimal ventilation with an amplitude for air velocity in trachea of about 0.9 m/s, very close to the acknowledged physiological value of 1 m/s (Weibel,

Our model is however not able to answer why the selected normal ventilation at rest stands in the lower values of the low energy plateau. A shift to lower period's values favors a lower mechanical energy and consequently higher amplitudes and dissipative energy. The answer might be linked to robustness criteria that are known to play an important role in strategies selection (Mauroy et al.,

Amongst the scenarios tested, three are considered intense: jogging, basketball, and ice hockey. At these regimes, measured data for the physiological period (Blackie et al., ^{−1}. The predictions of our model for amplitude are in this range, but more tightly packed toward the lower values, with amplitudes ranging between 7 and 10 m · s^{−1}. A possible explanation for this packing toward the low values might be that our model does not account for the wide range of possible human physiology and body needs as our studies is based on one set of parameters only.

The optimal way of ventilating our model in a wide range of regimes are very similar to the physiological responses of real lung's ventilation. This is more particularly true when the metabolic regimes increases: the energy profile becomes steeper and steeper near the optimal limiting the margin for adjusting both ventilation amplitude and period. This suggests that control of ventilation might have built-in energetic optimizer, either encoded in neural control, learnt or computed. Our model shows that their efficiency is crucial at high regimes, where the energetic costs are the larger. At high regimes, a shift from the optimal configuration is predicted by our model to be very costly and maintaining the regime might prove difficult shifted from the optimal. This behavior is fully compatible with the fact that control of ventilation is stronger at exercise, preventing even talking. This raises a fundamental question by which mechanisms the control of ventilation can adjust the ventilation so that it is nearly optimal ni term of power dissipation.

In the case of an increase of lung's resistance due to a tighter geometry, the optimal strategy is to keep the same ventilation amplitude and to reduce the ventilation period. This leads to a ventilation process that is less energetic than a wider geometry. Although this sounds counter intuitive as higher hydrodynamic resistance is often correlated to tiredness and higher energy dissipation in the lung, our result shows that in the optimal strategy found, the resistance increase due to tighter bronchi is compensated by a volume of internalized air that is smaller (same amplitude, but for a lower time). This volume acts as a smaller oxygen source and ventilation period has to decrease for quicker renewal of the internalized air. In a whole, the mechanical energy gained with a smaller volume to move allows a decrease of the total energy. This behavior is however not the one expected from physiology, mainly because the effect of an increase of hydrodynamic resistance is first hypoxia, that induces ventilation control to use both a higher amplitude and a smaller period, as predicted by our model. The net result of hypoxia effects is an increase of the power spent for ventilation. So our analysis show that response to hypoxia might not be the best response for hydrodynamic resistance increase. Hypoxia is known to also be a typical maladaptive response to high altitude relocalization by activating processes that might be deleter for some individuals. Our results suggest that this might also be the case when mechanical characteristics of the lung are degrading.

We propose here a model of the gas transport in the human lung based on the core physical phenomena identified in the literature: tree structure of the lung, convective and diffusive transports of oxygen and carbon dioxide, exchange surface properties (size, thickness, etc.). We study the power dissipated (resistance to flow in the airways and mechanical energy stored in the tissue) for the ventilation of our model and how it depends on ventilation amplitude and period. We search for ventilation characteristics that minimize this power assuming the oxygen flow has to fit the need of the metabolism. The predictions of our model are very close to the physiology. At rest, the power depends weakly on the period and makes the system very robust to period adjustments; when the regime increases, the dependence in the period becomes steeper and steeper and any shift from the optimal value is critical in term of power dissipation. These results are fully compatible with the physiology, indicating that the phenomena included in our model might drive the main responses of control of ventilation in human. Since the power optimized in this study is based on physiological parameters routinely measured by physicians and physiologists, confirmation of the predictions of our model should be possible by experimental reconstruction of the power profiles.

Our study raises however two questions not answered by our model. First: why is the period of rest ventilation in human so stereotyped although power dependence is low? Second: What mechanisms make control of ventilation able to select at exercise for ventilation regimes that minimize power dissipation?

Analyses were performed by FN. Research topic was proposed by BM. Both authors contributed to modeling and writing of the paper.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

We thank the structuriong program IDEX UCA JEDI C@UCA. The authors would also like to thank Alona Ben-Tal for her insights about the modeling of control of ventilation.

The power dissipated by viscous effects of air circulation is based on the hydrodynamic resistance of the lung

where _{0} is the resistance of the branch of the first generation,

with μ = 1.8 · 10^{−5}Pa ·

Estimation of human lung's resistance have been extensively measured in the literature but its precise relation with the parameter