Edited by: Rajat Mittal, Johns Hopkins University, USA
Reviewed by: Zhiliang Xu, University of Notre Dame, USA; William Andrew Pruett, University of Mississippi Medical Center, USA
*Correspondence: Cyril Karamaoun
This article was submitted to Computational Physiology and Medicine, a section of the journal Frontiers in Physiology
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In the human lungs, nitric oxide (NO) acts as a bronchodilatator, by relaxing the bronchial smooth muscles and is closely linked to the inflammatory status of the lungs, owing to its antimicrobial activity. Furthermore, the molar fraction of NO in the exhaled air has been shown to be higher for asthmatic patients than for healthy patients. Multiple models have been developed in order to characterize the NO dynamics in the lungs, owing to their complex structure. Indeed, direct measurements in the lungs are difficult and, therefore, these models are valuable tools to interpret experimental data. In this work, a new model of the NO transport in the human lungs is proposed. It belongs to the family of the morphological models and is based on the morphometric model of Weibel (
For more than 20 years now, nitric oxide (NO) has been shown to be of a striking importance in various physiological processes. This ubiquitous molecule contributes, among other roles, to vasodilatation (Palmer et al.,
Human lungs are a complex organ, organized around a dichotomous tubular “tree” structure (see Figure
an upper or bronchial part, composed of socalled airways, where convection is the dominant mechanism of gaseous mass transport. An airway can be seen as a hollow cylinder in which gas flows. The inner space of an airway is called the lumen. The lateral wall of an airway is composed of two tissue layers (see Figure
a lower or alveolar part. This part of the lungs is also composed of socalled airways, but with their lateral wall garnished by expansible bags called alveoli. These alveoli are optimized for mass exchange between the air and the blood. In this part of the lung, the inner space of an airway is also called the lumen (by definition, it does not include the inner space of the alveoli). Diffusion is the dominant mechanism of gaseous mass transport in this part of the lungs. NO is produced in the epithelial cells present in the wall of the alveoli. This alveolar part begins approximately at generation 17 (Weibel et al.,
Multiple models have been developed in order to characterize the NO transport in the lungs, owing to their complex structure. Indeed, direct measurements in the lungs are difficult and therefore these models are valuable tools to interpret experimental data such as the F
In the first developed models, the lungs are divided in two perfectly mixed compartments, a rigid bronchial compartment and an expansible alveolar compartment. These models are able to reproduce some observed features of the pulmonary NO transport (Hyde et al.,
As a step beyond these twocompartments models, a variety of socalled morphological models have been developed. In a morphological model, at the opposite of a twocompartments model, each generation of the lungs is represented and characterized by its number of airways, by the total volume of the lumen of its airways and, in the alveolar zone, by the total inner volume of its alveoli. It is assumed that each division in the tree structure of the lungs generates two identical airways (Paiva and Engel,
In this work, a new morphological model of the NO transport in the human lungs is developed and used. When compared to models published previously, several new features are introduced.
First, for the generations 0 to 18, the model includes a detailed description of the lateral wall of the airways. Diffusionreaction equations for the NO are written in the layers composing an airway wall (including a possible mucus layer). For a given airway, solving these equations allows expressing the NO exchange rate between the epithelium and the lumen of this airway as a function of fundamental parameters, such as the NO volumetric production rate in the epithelium, the airway and layers geometry. Thus, in our model and for generations 0–18, this exchange rate is not obtained by partitioning an experimentally determined total exchange rate. In the generations 19–23, the lateral wall of an airway is almost totally composed of alveoli (see Figure
Second, in our model, we also include the possibility to modulate the cross section of the airways in one or several generations of the upper part of the lungs and, therefore, to simulate bronchoconstriction (BC, the constriction of the airways in the lungs due to the tightening of muscles) which is a feature of multiple pathologies, as for instance asthma. In our model, when BC is imposed in a generation, the thicknesses of the epithelial and muscles layers of the corresponding airways are reevaluated, based on volume conservation. The thickness of a possible mucus layer is also reevaluated. Thus, by combining the possible modulation of the cross section of the airways in one or several generations and the layered description of the lateral wall of these airways, our model is able to take into account the influence of a BC on the NO exchange rate between the epithelium and the lumen of an airway in these generations, at the opposite of previous models.
Third, our model is written in a dimensionless form. This simplifies, accelerates and stabilizes the numerical calculations, when compared to previous models. It also brings out typical dimensionless numbers used in fluid mechanics, such as the Péclet number, and in chemical engineering, such as the Hatta number. Evaluation of these dimensionless numbers allows understanding the relative influence of different phenomena on the NO concentration profile in the lungs.
These new features allow our model to produce new insights into the NO transport in the human lungs, especially regarding the influence of BC on the F
In this paper, the developed model is first presented. Then, the model is used to discuss some features of the NO transport in healthy and unhealthy lungs. In particular, simulation results are compared with experimental information available in the literature. Moreover, using the model, the expected influence of the localization of BC and of its extent on the F
The notations introduced in this subsection are applicable to any lungs, whether they are healthy or not.
As for the morphological models developed previously, it is assumed that each division in the tree structure of the lungs generates two identical airways, even when mucus is present and after a possible BC (Paiva and Engel,
As mentioned previously, for the generations 0–18, the model includes a detailed description of the lateral wall of the airways. Such a wall is composed of two tissue layers: the epithelial layer and the muscles layer. In unhealthy lungs, an epithelial layer can also be coated by a mucus layer
The airways in generations 0–18 are considered to be perfectly axisymmetric (Weibel et al.,
As mentioned previously, in the generations 19–23, the lateral wall of an airway is almost totally composed of alveoli. Hence, the threelayers airway lateral wall representation is not relevant for these generations. Moreover, the alveoli are not surrounded by muscles and their inner surface cannot be coated with mucus, even in unhealthy lungs. Therefore, even if epithelial cells are found in the walls of each alveolus, the use of the terms “epithelial layer,” “muscles layer,” and “mucus layer” (and the use of the associated notations) is only restricted to the airways in generations 0–18.
For generations 0–18, if no mucus is present in generation i,
In our model, any lungs at rest are seen as healthy lungs at rest in which some alterations may have occurred.
For healthy lungs at rest, Ω_{i}(
0  12.00  1.53  1.83  1.83  0.892  15083 
1  4.76  1.03  1.68  1.68  0.869  6522 
2  1.90  0.67  1.54  1.54  0.841  2848 
3  0.76  0.48  1.44  1.44  0.809  1213 
4  1.26  0.38  1.79  1.79  0.785  1622 
5  1.06  0.30  2.24  2.24  0.760  1088 
6  0.90  0.24  2.86  2.86  0.733  726 
7  0.76  0.19  3.68  3.68  0.704  476 
8  0.64  0.158  5.02  5.02  0.676  294 
9  0.54  0.131  6.90  6.90  0.647  180 
10  0.46  0.110  9.67  9.67  0.618  110 
11  0.38  0.094  14.15  14.15  0.591  61.9 
12  0.32  0.080  20.79  20.79  0.563  35.5 
13  0.26  0.072  33.56  33.56  0.542  17.85 
14  0.20  0.062  50.13  50.13  0.514  9.19 
15  0.20  0.056  81.56  81.56  0.493  5.65 
16  0.16  0.050  130  130  0.470  2.84 
17  0.14  0.046  249  217  0.451  1.49 
18  0.12  0.043  530  385  0.439  0.72 
19  0.1  0.041  1140  683  0.34  
20  0.08  0.037  3069  1155  0.16  
21  0.06  0.038  6810  2324  0.059  
22  0.06  0.036  15015  4244  0.033  
23  0.06  0.036  30155  8516  0.016 
Ω_{i, 0} and
The total gas volume in healthy lungs at rest (
For lungs at rest and not impacted by BC,
As mentioned previously, in unhealthy lungs, for instance in asthmatic patients, an epithelial layer can be coated with a significant mucus layer (Farmer and Hay,
BC can only occur in generations 0–18. Indeed, the lateral wall of an airway in generations 19–23 is totally composed of alveoli and the alveoli are not surrounded by muscles. If BC occurs in an airway, the muscles and epithelial layers volumes in this airway are conserved. As a consequence, the epithelial layer wrinkles (see Figure
For lungs at rest and after their possible alteration by BC, Ω_{i}(
The conservation of the muscles, epithelial and mucus layers volumes implies that the following equations must hold for any lungs and for 0 ≤
α_{i} is a measure of the extent of a possible constriction in generation i and is defined as follows:
α_{i} is necessary equal to zero for
Combining Equation (2) and Equation (1) gives:
Equation (3) are only valid if radicands are positive, which gives the maximal possible value of α_{i}, written α_{i, max}:
Values of α_{i, max} are given in Table
β_{i} is defined as 1−
β_{i} = 1 logically implies α_{i} = α_{i, max}.
As mentioned previously, the alveoli are not surrounded by muscles and they cannot by coated by mucus. Hence, the inner volume of the alveoli in a generation of unhealthy lungs is the same as the inner volume of the alveoli in this generation in the corresponding healthy lungs. Therefore,
The total gas volume in lungs at rest and after their possible alteration by BC (
A respiratory cycle can be divided in 3 phases: inspiration (duration:
In normal breathing conditions, preinspiratory and postexpiratory lungs volumes are equal. Therefore, the following equality must hold:
During inspiration and expiration, it is usually assumed that the length of each airway does not change. Therefore, the following equations can be written at any time:
During an inspiration phase, as
where θ =
Dil is the ratio between the total inspired air volume and the initial gas volume in the lungs.
For 0 ≤
The alveolar part of the lungs begins approximately at generation 17. However, only a few alveoli are found in generations 17 and 18. Hence, during an inspiration phase and for 0 ≤
If the alveoli are considered as hemispheres,
with
During a breathhold phase, as
where θ =
Consequently, the following equations can be written during a breathhold phase:
During an expiration phase, as
where θ =
Consequently, the following equations can be written during an expiration phase:
At any time,
In each airway, an axial coordinate
The air flow in lungs can be considered as incompressible (Paiva and Engel,
Using Equation (6), this mass balance equation can be rewritten in a dimensionless form, for each respiratory phase:
where
The integration of Equation (22) gives:
It appears that
In the lungs, the major part of the NO production is thought to arise from various types of epithelial cells (Dillon et al.,
Note that NO has a great affinity with the blood hemoglobin (Hb), far more than O_{2} or even CO. Trapped by the Hb, NO becomes unavailable, so that its blood concentration can always be considered as null (Tsoukias and George,
According to Van Muylem et al. (
where
We first examine the case of an airway with its epithelial layer coated with mucus.
A schematic representation of the NO transfer inside the layers of an airway wall is presented in Figure
In order to write NO transport equations in the layers of an airway wall, two assumptions are made. The first assumption is that the transport of NO inside these layers can be assumed quasisteady (i.e., the time derivative terms in the transport equations can be set to zero). The second assumption is that the curvature of the layers can be neglected when establishing these transport equations. This assumption is valid if
where
In the mucus layer, at the interface with the lumen, the NO concentration is assumed to be at equilibrium with the NO concentration in the lumen. Hence, the following equation can be written:
where λ_{t:air} is a thermodynamic equilibrium constant, calculated from the Henry's constant of the NO in water, the soft tissues here being approximated as having the same chemical properties as water. At 37°C and 1 atm,
The other boundary conditions completing Equation (25) are:
According to these transport equations,
where, according to the units of
Solving the transport Equation (25) with boundary conditions 26 and 27 allows obtaining the following expression for
where
Ha_{i} and Hã_{i} compare a reaction characteristic time (
It can be observed in Equation (29) that, similarly to
It is important to highlight that, at the contrary to
In the case of an airway without mucus, a similar reasoning allows obtaining the following equation to calculate
As mentioned previously, the alveoli are being closely separated from each other. Therefore, the axial transport in an airway occurs only through its lumen. According to this, a mass balance for the gaseous NO over a slice of the airways in generation i leads, considering convective and diffusive transport, and considering Equations (6) and (21), to the following equation:
where D_{NO, air} is the diffusion coefficient of NO in air.
Using Equations (7), (8), (12), (16), and (23), this transport equation can be rewritten in its dimensionless form, for each phase of a respiratory cycle:
These equations are completed by boundary conditions expressing the continuity of the gaseous NO concentration and diffusion flux at the junctions between generations. The NO diffusion flux is assumed null at the end of the last generation. During inspiration, the gaseous NO concentration at the mouth (beginning of the first generation) is set to zero. During breathhold and expiration, the NO diffusion flux is assumed null at the mouth (pure convective transport).
Pe_{i} appearing in these equations is defined as:
Pe_{i} is a dimensionless number, usually called the Péclet number, defined for each generation. Pe_{i} is the ratio of a characteristic time of the axial transport of gaseous NO by convection in generation i (
The model developed in this paper is characterized by several input parameters. These include geometrical parameters of the healthy lungs at rest (
0.0200  cm  Ochs et al., 

0.217  cm^{2} s^{−1}  Van Muylem et al., 

3.3 × 10^{−5}  cm^{2} s^{−1}  Tsoukias and George, 

δ_{E, 0}  0.0015  cm  Farmer and Hay, 
δ_{M, 0}  0.0030  cm  Farmer and Hay, 
γ  2.545 × 10^{4}  cm^{3} mol^{−1}  Adapted 
2.001  s^{−1}  Adapted from (Tsoukias and George, 

λ_{t:air}  1.64 × 10^{−6}  molNO cm^{−3}  Adapted from (National Research Council (U.S.), 
5.17 × 10^{−12}  molNO cm^{−3} s^{−1}  Adapted from (Tsoukias and George, 

3.167 × 10^{−6}  mlNO s^{−1}  Pietropaoli et al., 

1558  cm^{3} s^{−1}  Pietropaoli et al., 

500  ml s^{−1}  Kerckx and Van Muylem, 

−50  ml s^{−1}  American Thoracic Society and European Respiratory Society, 

2  s  Kerckx and Van Muylem, 

0  s  Kerckx and Van Muylem, 

20  s  American Thoracic Society and European Respiratory Society, 
The values of
Once the values of the input parameters of the model defined, the geometrical properties of the lungs at rest and during respiratory cycles can be successively determined using the equations presented previously. Then, the transport Equations (33), (34), or (35) are solved for each generation, depending on the considered respiratory phase, with
In this section, the model is used to discuss some features of the NO transport in healthy lungs. In particular, it is checked that the model is able to reproduce experimental information available in the literature.
The developed model can be used to simulate a respiratory cycle, for a given set of the model parameters. Using the simulation results, the time evolution of the F
A first characteristic of the pulmonary NO transport is related to the time evolution of the F
A second characteristic of the pulmonary NO transport is related to the link between the expiration flow rate and the value of the F
A third characteristic of the pulmonary NO transport is related to the link between the duration of a breathhold phase and the time evolution of the F
Several authors also point out different characteristics of the gaseous NO concentration profile in the lungs during a classical respiratory cycle. In the 2–3 last generations, the gaseous NO concentration remains almost homogeneous and constant during the entire cycle, at a value of approximately 2–3 ppb (Pietropaoli et al.,
The use of the model to simulate a respiratory cycle with a short breath hold phase (2 s) allows highlighting an interesting feature of gas diffusion in the lungs. In Figures
It is also interesting to use the model [including the solutions of Equation (25) with boundary conditions 26 and 27, given in
Three important dimensionless numbers appear in the model: two Hatta numbers [see Equation (30)] and the Péclet number [see Equation (36)].
Two Hatta numbers (Ha) are defined for each generation. A Ha number compares a characteristic time of NO consumption in a tissue composing an airway wall (epithelial layer or muscle layer) to a characteristic time of transport by diffusion in this tissue. The two Ha numbers introduced in this work are proportional to the epithelial layer thickness and to the muscles layer thickness, respectively. In healthy lungs, these thicknesses are the same in each generation, and they experience variations during a respiratory cycle. Using the model and the data given in Tables
A Péclet number (Pe) is defined for each generation. It compares a longitudinal convective characteristic time to a longitudinal diffusion characteristic time. As shown in Table
In this section, the model is used to discuss some features of the NO transport in unhealthy lungs.
Several authors showed that the F
To analyze with our model the NO transport in unhealthy lungs in which BC has occurred (and with the possible presence of a mucus layer coating the walls of the airways in some generations), several parameters are introduced in order to compare these unhealthy lungs and the corresponding healthy lungs experiencing a same respiratory cycle (i.e., the parameters listed in Tables
In Figures
It can be observed in Figure
It can be observed in Figure
It can be observed in Figure
In Figures
In conclusion, the use of our model shows that the relation between BC and F
As mentioned previously, in order to determine Equations (29) and (31), allowing to express
the NO transport equations in the layers can be written as 1D transport equations in a cartesian coordinate system;
the time derivative of the NO concentration in the layers can be neglected in these equations, i.e., the NO transport in the layers can be considered as being quasisteady.
The first assumption gives accurate results for a given generation if the radius of the lumen of the airways in this generation is at least one order of magnitude larger than the thicknesses of the layers composing the airways walls. According to the data mentioned in Tables
The second assumption gives accurate results for a given respiratory cycle if the characteristic times of the NO transport by diffusion in the layers composing an airway wall (
In order to check more precisely if these two assumptions are appropriate, we have simulated numerically the NO transport in the epithelial and muscles layers of an airway in a given generation in lungs at rest, in response to a sudden increase (from 0 to 5 ppb) of the NO concentration in the lumen. The airway wall was considered as being a hollow cylinder (i.e., the transport equations in the layers composing the airway wall were written in cylindrical coordinates, assuming axisymmetry) and the time derivative in the transport equations were considered. Data given in Table
In this work, a new model of the NO transport in the human lungs is presented and used. It belongs to the family of the socalled morphological models and it is based on the morphometric model of Weibel (Weibel,
The model is based on a geometrical description of the lungs, at rest and during a respiratory cycle, coupled with transport equations, written in the layers composing an airway wall and in the lumen of the airways.
It has been checked that the model is able to reproduce experimental information available in the literature. The model has been used to discuss some features of the NO transport in healthy and unhealthy lungs. The simulation results were analyzed, in order to give new insights into the NO transport in the human lungs, especially when BC has occurred in the lungs. For instance, it has been shown that BC can have a significant influence on the NO transport in the tissues composing an airway wall. BC increases the NO exchange flux density between the epithelial layer and the lumen of an airway (due to the increase of the epithelial and muscles layers thicknesses) but decreases the total flux of NO from the epithelial layer to the lumen of an airway (due to the decrease of the exchange surface). It has also been shown that the relation between BC and F
CK, BH, and AV designed the main characteristics of this article. CK and BH constructed the new proposed model based on the one of AV. CK and BH wrote the article.
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.
The authors gratefully acknowledge financial support of ESA and BELSPO (ESAESTECPRODEX arrangement 4000109631).
The Supplementary Material for this article can be found online at:
The solutions of Equation (25) with boundary conditions 26 and 27 are:
Bronchoconstriction
Concentration, ml/cm^{3} or mol/cm^{3}
Inner diameter of the alveoli, μm
Diffusion coefficient, cm^{2}/s
Ratio between the total inspired air volume and the initial gas volume in the lungs, 
Ratio of the volume of gas in a generation to the total volume of gas in the lungs, when they are at rest, 
Hatta number, 
Hatta number, 
Hemoglobin
NO exchange flux density, ml/(cm^{2}.s)
kinetic constant,
Length of the airways in a generation, cm
Dimensionless number appearing in Equation (29), 
Nitric oxide
Péclet number, 
Total alveolar NO production rate, ml/s
Volumetric NO production rate in an epithelial layer, mol/(cm^{3}.s)
Gas flow rate, ml/s
Distance between the center of an airway and the surface of a layer, cm
Exchange surface, cm^{2}
Time, s
Total alveolar consumption rate, cm^{3}/s
Total gas volume in the lungs, ml
Coordinate, cm
Axial coordinate in an airway, cm
Tissue thickness, cm
Dimensionless axial coordinate in an airway, 
Equilibrium constant, mol/cm^{3}
Correcting coefficient, cm^{3}/mol
Dimensionless time, 
Total crosssectional area in a generation, cm^{2}
Total flow crosssectional area in a generation, cm^{2}
At rest
Airway
Alveolus
Diffusion
Epithelium
Generation i
Lumen
Muscle
Mucus
Maximal value
Nitric oxide
Tissue
Total (sum over the 24 generations)
Breathhold
Expiration
Inspiration