Mathematical modeling of nitric oxide and mucus dynamics in the human lungs, using a phenomenological approach, to

Université libre de Bruxelles École polytechnique de Bruxelles

Service Transferts, Interfaces et Procédés (TIPs) Unité de Génie Chimique

thèse de doctorat

intitulée

Mathematical modeling of nitric oxide and mucus dynamics in the human lungs, using a phenomenological approach, to

provide new insights into asthma and cystic brosis.

Présentée en vue de l'obtention du titre de

docteur en sciences de l'ingénieur et technologie

par

Cyril Karamaoun

soutenue le 25 octobre 2017 devant le jury composé de

Gérard Degrez Examinateur (Université libre de Bruxelles, Bruxelles, Belgique) Julien Favier Examinateur (Aix-Marseille Université, Marseille, France) Benoît Haut Promoteur (Université libre de Bruxelles, Bruxelles, Belgique) Renaud Louis Examinateur (CHU Liège, Liège, Belgique)

Antoine Nonclercq Examinateur (Université libre de Bruxelles, Bruxelles, Belgique) Benoît Scheid Examinateur (Université libre de Bruxelles, Bruxelles, Belgique) Alain Van Muylem Promoteur (Hôpital Universitaire Érasme, Bruxelles, Belgique)

Année académique 2016-2017

Dans la vie, rien n'est à craindre, tout est à comprendre.

Marie Curie

Remerciements

L'écriture de remerciements après 4 ans de travail n'est paradoxalement pas chose aisée.

Je commencerai par remercier mes deux promoteurs, les professeurs Benoît Haut et Alain Van Muylem. Ils ont contribué, chacun à leur manière, à me faire progresser scientiquement et personnellement.

Benoît, ta rigueur au travail combinée à une éternelle bonne humeur m'ont très certainement amené jusqu'au bout de ce projet de thèse. J'ai appris beaucoup à tes côtés, tant sur la science des phénomènes de transport que sur la paternité et l'éducation de petits bouts de chou.

Alain, j'ai pu trouver en ta personne un interlocuteur de premier plan, tant profes- sionnellement que personnellement. Tu n'as jamais hésité à m'expliquer patiemment les subtilités de la physiologie respiratoire, qui étaient au début loin de ma compréhen- sion de néophyte. J'ai aussi découvert une personne très drôle et intéressée par mon avenir et mes ambitions personnelles, ce dont je te remercie.

Je tiens ensuite à remercier l'ensemble des membres du jury pour leurs remarques et questions pertinentes qui m'ont permis d'améliorer grandement la qualité de ce manuscrit.

Je tiens également à remercier le Dr. Benjamin Mauroy, de l'Université Côte d'Azur.

Notre collaboration scientique, que nous souhaitons fructueuse, m'a permis de ren- contrer une personne charmante et attentionnée, avec qui j'ai pris beaucoup de plaisir à travailler. Au plaisir de te revoir à Nice ou ailleurs !

Enn, je tiens à remercier particulèrement le Dr. Benjamin Sobac, avec qui j'ai partagé beaucoup de moments précieux. Benjamin, tu es quelqu'un à l'écoute de l'autre, avec qui j'ai passé des moments de franche camaraderie, de la rédaction du squelette de notre article à nos parties de tennis endiablées (ou nos visites répétées chez Makisu).

Comment pourrais-je ne pas remercier tous les membres du service Transferts, In- terfaces & Procédés (TIPs) de l'Université libre de Bruxelles, avec qui j'ai partagé tant de moments ? Merci, dans l'ordre et dans le désordre : à Youen, mon voisin de bureau, pour ses soirées crêpes et sa fameuse délicatesse avec la porte du labo; à Adrien, pour ses conseils sportifs et sa bonne humeur légendaire; à Bart pour m'avoir fait réviser ma connaissance de la langue de Vondel, mais pas que ça; à Javier, pour nos super parties de squash; à Charlotte (pour son rire), Loucine et Rosalie, les trois (plus si) nouvelles.

Bonne continuation à vous ! Merci également à Omar : la route a été longue depuis

REMERCIEMENTS

le Bénin en 2012 ! Merci à Odile pour son sourire (et son rire aussi), son ecacité et les cartes de visite. Merci aussi aux anciens et anciennes : Sacha, Pauline, Catherine, Julie, etc. Merci à Hosein pour son éclairage bienvenu sur les us et coutumes iraniens et les parties de FIFA. Un merci tout particulier à Pierre, Benoît S., Fred et Alex pour leur aide et conseils. Merci à Carinne, qui m'a accueilli à mes débuts. Enn, un pro- fond remerciement à Hervé et Céline, sans qui ce repère de scientiques qu'est le TIPs ne pourrait pas fonctionner ! Merci à Céline pour son ecacité et sa bonne humeur.

Et un tout grand merci à Hervé pour le temps (trop long pour un numéricien) passé à l'atelier, à s'occuper de mes projets farfelus à base de bois de palette et de mobilier pour chat.

Les remerciements commencent à se faire long, mais je ne pouvais pas omettre de mentionner tous mes amis et amies qui ont fait ce bout de chemin avec moi. Merci à Adrien (dit PDB Touf), avec qui j'ai partagé beaucoup de bons moments. Merci à Yegor (dit papa Igou), mon colloc' d'alors. Merci aux inséparables Gilles et Georges, les Dupont et Dupond de l'ULB, qui m'ont soutenu et fait rire tout au long de cette thèse. Merci à CDB Simon pour les sessions kicker et les pauses midi chez Mouss'.

Merci à Valentin pour son soutien. Je pense bien sûr aux autres inséparables, Maxime et Louis (les brollocs), dont les qualités d'orateur (surtout Louis) n'ont d'égales que leurs qualités d'amis (surtout ...). Merci à Ben Dum bien sûr, sauf pour nos ballades nocturnes d'Ixelles à Boitsfort. Merci à Hamza, Anicet et Alex Robert (oui oui) : ne changez pas les gars ! Merci à toute la bande : Margaux (la maman), Seb Grolier, Me.

Poelman, Alexia, Natacha, et les autres. Salutations fraternelles à Kawam et Goniaux, ainsi qu'à Carotte. Pour leur soutien et les bons moments passés avec eux, merci à Alizée, Marie Koto, Hélène et toute la bande des zintimes. Merci à Adam et Nathal'.

Merci aussi à Alexandra. Merci à tous les mauvais ambianceurs, ils se reconnaîtront.

Merci à Kiou et Victoria. Merci à Maxime et Gaetan, mes amis de toujours. Enn, merci à OL, sans qui je n'aurais jamais été au bout de ce chemin.

Pour terminer, je voudrais bien sûr remercier toute ma famille. Ils ont su, à leur manière, me donner le courage et la vigueur pour poursuivre l'aventure jusqu'au bout.

Merci à mes parents pour leur écoute et leurs conseils. Merci à mon frère pour ses

multiples touches d'attention. Merci au reste de ma famille, en particulier à Audrey

et Richard. Enn, salutations chaleureuses à ma grand-mère qui, du haut de son

grand âge, est toujours ravie de recevoir une petite carte postale de mes déplacements

professionnels. Je lui dédie ce travail.

Abstract

In this work, the general problematic of the transport phenomena in the lungs is addressed, with applications to lung diseases, in particular asthma and cystic brosis.

Regarding these two major diseases, the dynamics of the nitric oxide and the mucus layer in the lungs, respectively, are of particular interest. In asthma, it has been shown that the measurement of the exhaled concentration of nitric oxide can be used as a proxy for monitoring the disease status. In cystic brosis, which is a disease of the mucus layer, pieces of evidence show that the water content of the mucus is linked to the severity of the symptoms of the disease.

In both conditions, however, these links are far from being obvious. Regarding nitric oxide, its exhaled concentration is not straightforwardly related to its concentration in the lungs. Regarding cystic brosis, the link between the disease and the alterations of the mucus content and dynamics is far from being understood. From these observations, several modeling approaches have emerged to complement the clinical measurements.

In this work, the dynamics of the nitric oxide and of the mucus layer in the lungs is studied from a modeling approach, following a phenomenological point of view.

Regarding the nitric oxide, a new model of its dynamics in the lungs is developed.

When compared to previous ones, the model presents multiple new features that allow for a better description of this dynamics, especially in the case of asthma exacerba- tion, characterized by the presence of mucus obstructions and bronchoconstriction.

From this model, the role of nitric oxide as a general marker of the bronchi caliber is suggested, in addition to its role in the asthma monitoring. Furthermore, based on clinical measurements of the exhaled nitric oxide concentration in cystic brosis pa- tients and the use of the new model, the role of the measurement of nitric oxide in the understanding and control of other lung diseases, such as cystic brosis, is evaluated.

Regarding the dynamics of the mucus layer in the lungs, a new analysis of the control of the mucus balance in the bronchial region of the lungs is presented in this work. Our approach is based on the combination of a balance equation for the mucus in an airway and a computational tool characterizing the evaporation of the mucus in the bronchial region. We show that this approach allows for new insights into the dynamics of the bronchial mucus and, more specically, on the mechanisms controlling the amount of mucus in an airway. The results are analyzed in order to bring interesting new perspectives for the understanding and the treatment of mucus pulmonary diseases, such as cystic brosis.

Altogether, this work demonstrates, applied to medical pathologies, the usefulness

of modeling approaches in giving a mechanistic view of the encountered problematic.

Scientic communications

PUBLICATIONS IN INTERNATIONAL PEER-REVIEWED JOURNALS C. Karamaoun, A. Van Muylem, B. Haut

Modeling of the nitric oxide transport in the human lungs Frontiers in Physiology 7:255 (2016)

Submitted publications

C. Karamaoun, B. Haut, A. Van Muylem

A new role of exhaled nitric oxide as a functional marker of peripheral airway caliber changes: a theoretical study

Journal of Applied Physiology - Under review

C. Karamaoun, B. Sobac, B. Mauroy, A. Van Muylem, B. Haut New insights in the control of the bronchial mucus balance PLoS Computational Biology - Under review

PUBLICATIONS IN PEER-REVIEWED CONGRESS ACTA BASED ON ABSTRACT

Poster communications

C. Karamaoun, A. Van Muylem, B. Haut

Simulation of airway calibre and inammation interaction by a new model of airway epithelium

European Respiratory Society Congress, September 2015, Amsterdam, the Netherlands, published in European Respiratory Journal 46 (suppl 59) (2015)

C. Karamaoun, A. Haccuria, A. Michils, B. Haut, A. Van Muylem Experimental and theoretical impact of hypertonic saline induction on exhaled nitric oxide

European Respiratory Society Congress, September 2017, Milan, Italy

SCIENTIFIC COMMUNICATIONS

Oral communications

C. Karamaoun, A. Van Muylem, B. Haut

Gas-liquid exchanges in the human lungs - analogy with chemical engineering

13 ^th International Conference on Gas-Liquid and Gas-Liquid-Solid Reactor

Engineering (GLS-13), August 2017, Brussels, Belgium

Contents

Remerciements 1

Abstract 3

Scientic communications 4

Glossary 9

Introduction 10

1 Literature review 13

1.1 The lungs: description and main characteristics . . . . 13

1.1.1 Lung anatomy . . . . 14

1.1.2 Breathing, ventilation and lung function . . . . 17

1.1.3 Air transport and respiratory gas exchange . . . . 20

1.2 Asthma and Nitric Oxide . . . . 21

1.2.1 Description of the asthma pathology . . . . 21

1.2.2 Nitric oxide: a link to asthma . . . . 22

1.3 Cystic brosis . . . . 23

1.3.1 Description of the cystic brosis pathology . . . . 23

1.3.2 The mucus layer in healthy and cystic brosis lungs . . . . 25

1.3.3 Nitric oxide in cystic brosis . . . . 27

1.4 Modeling the lungs . . . . 28

1.4.1 General modeling approaches . . . . 29

1.4.2 Modeling the lungs: the asthma pathology . . . . 30

1.4.3 Modeling the lungs: the cystic brosis pathology . . . . 31

2 Objectives 33

CONTENTS

3 Modeling of the nitric oxide transport in the human lungs 35

3.1 Introduction . . . . 36

3.2 Model development . . . . 39

3.2.1 Geometrical considerations . . . . 39

3.2.2 Air ow in the lungs . . . . 45

3.2.3 NO exchange . . . . 46

3.2.4 Transport of gaseous NO in the lumen . . . . 50

3.2.5 Model summary . . . . 51

3.3 Model capabilities . . . . 52

3.3.1 Healthy lungs . . . . 52

3.3.2 Dimensionless numbers . . . . 55

3.3.3 Unhealthy lungs . . . . 58

3.3.4 Model assumptions analysis . . . . 60

3.4 Conclusion . . . . 62

4 A new role of exhaled nitric oxide as a functional marker of peripheral airway caliber changes: a theoretical study 65 4.1 Introduction . . . . 65

4.2 Methods . . . . 66

4.3 Results . . . . 71

4.4 Discussion, Conclusion & Perspectives . . . . 75

5 Evaluation of the role of the Fe NO in the monitoring of CF: application to the control of the eciency of chest physiotherapy in CF patients 78 5.1 Introduction . . . . 78

5.2 Methods . . . . 79

5.2.1 Chest physiotherapy and Fe NO measurement . . . . 79

5.2.2 Modeling the NO dynamics in the lungs submitted to chest phys- iotherapy . . . . 81

5.3 Results . . . . 82

5.3.1 Experimental results . . . . 82

5.3.2 Modeling results . . . . 83

5.4 Discussion . . . . 84

5.5 Conclusion . . . . 85

CONTENTS

6 New insights in the control of the bronchial mucus balance 87 6.1 Introduction . . . . 87 6.2 Results and discussion . . . . 90 6.2.1 Mucus balance in an airway . . . . 90 6.2.2 Scale analysis of the mechanisms controlling the balance of the

bronchial mucus . . . . 92 6.2.3 Distribution along the bronchial tree of the amplitude of the

mechanisms controlling the amount of mucus in an airway . . . 95 6.2.4 Perspectives regarding the understanding and treatment of cystic

brosis . . . . 97 6.3 Computational tool for the characterization of the evaporation of the

mucus . . . . 98 6.4 Conclusion . . . 101

Conclusion 103

Appendix 108

List of Figures 112

List of Tables 114

Bibliography 115

Glossary

ASL Airway Surface Liquid ATP Adenosine Triphosphate BC Bronchoconstriction CF Cystic Fibrosis

CFTR Cystic Fibrosis Transmembrane Conductance Regulator CO 2 Carbon dioxide

COPD Chronic Obstructive Pulmonary Disease CT-scan Computer Tomography Scanner

DNA Deoxyribonucleic Acid

FENO Molar Fraction of Nitric Oxide in the Exhaled Air FEV 1 Forced Expiratory Volume in 1 second

FRC Functional Residual Capacity

Ha Hatta Number

Hb Hemoglobin

NO Nitric Oxide

NOS NO-Synthase

O 2 Dioxygen

PCD Primary Ciliary Dyskinesia PCL Periciliary Layer

Pe Peclet Number

ppb Part Per Billion

Re Reynolds Number

RH Relative Humidity

RV Residual Volume

TLC Total Lung Capacity

TV Tidal Volume

Introduction

From the rst days of the medicine and for a long period of time, the physiological role of the lungs remained totally misunderstood. Early descriptions, namely from Galen, interpreted the breathing lungs as a cooling organ, designed to evacuate the heat produced by the pumping heart. The development of the anatomy during the Renaissance allowed for a more detailed description of the lungs, but physicians lacked chemical and physical knowledge to fully understand the role of the lungs in the respi- ration process. Rise of the modern medicine in the last centuries enlightened the main roles of the lungs and the interdependence of the respiratory system with the other systems. Nevertheless, the race for a complete understanding of this organ is far from ending.

The lungs are the organ responsible for the respiration. Millions of years of evolution designed it to optimize the gas exchange between the air and the body, favoring the absorption of oxygen during inspiration and the elimination of carbon dioxide during expiration. Structure and functions of the lungs are thus intimately linked, in order to optimize the gas exchange. To maximize the exchange surface between the air and the body, the lungs evolved to an intricate branching structure with an enormous surface to volume ratio. Normal lungs weigh around 1 kilogram, for an intern volume of nearly 5 liters and an exchange surface close to 100 square meters [183]. The size of this surface makes it the largest barrier between the body and the outside world, partly explaining the lung sensibility to pollutants, allergens and infections. Furthermore, due to the complex geometry of the lungs, describing their precise structure is a hard task that still tackles the scientic community.

Due to this complexity, the involvement of multiple science elds is needed to uncover the remaining mysteries of this organ. From a medical point of view, there is a real urge to better understand, diagnose and cure lung pathologies, as the prevalence of some of them, such as asthma or lung cancer, is increasing. Asthma, for example, is thought to aect around 5 percent of the world population and cause 1 in 250 deaths worldwide [36].

Currently, major research elds such as physiology, molecular biology and medical

imaging are tremendously helping physicians in their daily practice, as well as from a

theoretical point of view. These researches are strongly helped by the development of

robust computing technologies. Informatics also propelled the development of a new

approach in medical science, and thus in pneumology: modeling. In a general way,

the purpose of modeling is to describe in simple terms the complexity of the studied

matter. Cartographers model the outer world by the mean of maps. Biologists use

INTRODUCTION

animal models such as mice or rats to apprehend the complexity of the human body.

Physiologists also use animal models as a proxy to understand the mechanisms of our body.

In particular, mathematical modeling is, obviously, a way of modeling that includes multiple mathematical features. This way of modeling applied to biomedical sciences is inspired by the old history of physics and chemistry, which early included a mathemat- ical point of view in their description of natural phenomena. Helped by the increasing computing power and by a more accurate picture of the lungs, mathematical modeling arose, in the second part of the 20 ^th century, as a new approach to improve the lung picture.

Several approaches can be taken in lung modeling, depending on which aspect needs to be emphasized. Breathing process, uid mechanics of the breathed air, oxygen and carbon dioxide exchange between the air and the tissues are few of many elds covered by modeling.

In this work, the focus is made on the modeling of transport phenomena occurring in the lungs. Precisely, the work focuses on the dynamics of nitric oxide (NO) and of the mucus layer in the lungs, and their links with two severe lung conditions: asthma and cystic brosis (CF).

Asthma is a pulmonary pathology with multiple origins. Patients suer from wheez- ing, cough and airway obstruction and inammation. As multiple forms of asthma exist, the eciency of the treatments depends on the accuracy of the diagnosis and the control of the disease. Pulmonary NO concentration has been shown to be increased in asthma patients. Based on this observation, the scientic community is struggling since many years to use the NO as an easy and objective marker of asthma, its control and its severity. A better understanding of the mechanisms of production, exchange and transport of NO is needed to full this requirement.

Cystic brosis is a severe disease caused by a genetic mutation that aects multiple organs of the body. In the lungs, it causes the mucus covering the bronchi to be thicker and more viscous than the mucus of healthy patients. This impairs its clearance from the lungs. Accumulated mucus blocks the bronchi and becomes a site of bacterial infection. Multiple treatments are currently developed or in use, such as hypertonic saline inhalation or chest physiotherapy, but no eective treatment has been developed so far. Nevertheless, lung transplantation can drastically extend life expectancy of CF patients [140]. As CF is a pathology of the bronchial mucus, a precise description of its dynamics is important in the understanding of this disease. In the bronchi, it seems that the water dynamics is of critical importance in controlling the characteristics and the balance of the mucus layer [90, 186]. Furthermore, other phenomena are thought to control the bronchial mucus balance. However, these underlying mechanisms are not fully understood. Thus, mathematical modeling could be of interest in the ne understanding of these mechanisms.

This work has two main objectives. The rst objective is to improve, by the way of

modeling, the understanding of the NO dynamics in the human lungs and the potential

roles of NO as a biomarker of the impairment of the lung function, in asthma and

other pathologies. The second objective is to assess the possibility of using modeling

INTRODUCTION

approaches to better apprehend the mucus dynamics in the lungs and its links with

the CF pathology. From these two objectives, multiple specic objectives have been

selected. These objectives are extensively described in Chapter 2.

Chapter 1

Literature review

In this chapter, a comprehensive review of the literature regarding the lungs, its diseases and the associated modeling approaches is established. The structure and functions of the organ are presented, in line with their particularities. Then, two pul- monary diseases, asthma and cystic brosis are reviewed from a biophysical point of view. Finally, the main modeling approaches regarding the lung dynamics are pre- sented, in regard with the two studied diseases.

1.1 The lungs: description and main characteristics

The main challenge of the lungs is to provide enough oxygen to all the cells of the body to allow cellular respiration and energy production ¹ . To match this requirement, dierent strategies have been selected through the evolution. In very small organisms, this challenge is not one. Diusion of oxygen through the cellular membranes is suf- cient to full the oxygen requirements. In larger organisms, diusion paths increase;

direct diusion is not sucient to provide enough oxygen to the tissues. Multiple so- lutions have arisen to full the oxygen requirements. One solution, as seen among the insects, is to bring the oxygen directly to the various organs via a network of aerial tubes going through the body. In animals with larger volumes, this direct gaseous diusion approach is not ecient enough. More ecient organs, such as gills and lungs, have emerged to solve this issue. These highly vascularized organs have the same purpose:

to optimize the gas exchange between the oxygen carrier uid (air or water) and the blood circulation. This is made possible by increasing the exchange surface and dimin- ishing the thicknesses of the uid-body interface. As this has to be done in a relatively small volume, these organs have evolved to highly compact and ramied structures, with a high surface to volume ratio. Note that, besides the respiratory function, the lungs have other minor roles such as the metabolization of some compounds, blood ltration and blood reservoir [183].

1 Cellular respiration is the biochemical process through which glucose reacts with oxygen to pro-

duce carbon dioxide and water. This reaction also produces energy which is stocked in the form of

Adenosine Triphosphate (ATP), the chemical source of energy of the body.

1.1 The lungs: description and main characteristics

1.1.1 Lung anatomy

Anatomically, the lungs are a large organ in the thorax, protected by the rib cage.

The lungs are supported by the diaphragm, which is a skeletal muscle separating the thorax from the abdomen. The lungs are divided in two main parts: the right and left lungs. The right lung itself is divided in three lobes, the smaller left lung is divided in two lobes, due to presence of the hearth on the left side of the thorax.

From the nose and the mouth, the inspired air is conducted through the pharynx and larynx to the trachea, a long cartilaginous tube which divides itself in two primary bronchi (i.e. ducts through which air is transported), one for each lung. These primary bronchi divide themselves in a nearly dichotomous way into more and more bronchi.

These bronchi become shorter and thinner after each subdivision, up to the so-called terminal bronchi, often called terminal bronchioles the term bronchioles is usually preferred to the term bronchi for these smaller ducts. The terminal bronchioles subdi- vide in the respiratory bronchioles, which are garnished with alveoli. Alveoli are thin budding pseudo-spheric structures, each one of them being a basic respiration unit, as seen in Figure 1.1. The respiratory bronchioles eventually subdivide in the alveolar ducts, totally garnished by alveoli, as seen in Figure 1.2. The anatomic region from the respiratory bronchioles to the alveolar sacs is called the acinar region. Each acinus has a grape-like structure as depicted in Figure 1.1, and is considered as a fundamental unit for the respiratory gas exchange between the air and the tissues. A generation collectively refers to all the lung elements (airways and alevoli) at the same level of subdivision in the bronchial tree. Thus, the trachea is the generation 0, the two pri- mary bronchi are the generation 1, and so on up to the end of the acini at generation 23. Note that it is an average value, as the number of generations up to the alveolar sacs depends on individuals, varying from about 18 to 30 [182].

In the literature, the term airway is often equally used to designate a bronchus, although an airway should be dened as a general conducting duct through which the air is transported (trachea, bronchus, ...). In this work, the term bronchus is preferred to designate the anatomic element, and the term airway to designate a more general air duct.

The bronchial region

The bronchial region of the lungs is composed of the 17 rst generations. It is also called the conductive zone, because no respiratory gas exchange occurs. The bronchi in this region are also called conductive bronchi. Morphometric measurements showed that, after subdivision, the daughter bronchi are not often symmetric one to an other, with small variations in bifurcation angles, lengths and diameters. Nevertheless, in homogeneous models such as Weibel's one [181], one of the approximations is that the daughter bronchi are symmetric, allowing to model the lungs as a dichotomic symmetric tree. This approach is convenient for modeling purpose. Sketch of Weibel's homogeneous model is represented in Figure 1.2.

At each subdivision, the diameter and the length of the bronchi is reduced. However,

as the number of individuals pipes increase, the cumulative volume and the ow cross-

1.1 The lungs: description and main characteristics

Trachea

Bronchioles

Respiratory bronchioles

Acinus

Alveoli

Figure 1.1: Sketch of the lungs, depicting dierent levels of details of the lung anatomy.

The bronchi divide in bronchioles that are progressively garnished with alveoli, where the respiratory gas exchange occurs. Adapted from [162]

sectional area of each generation increase in an exponential way, as seen on Figure 1.3.

It is interesting to note that a homothetic ratio can be used to describe the evolution of several parameters of the bronchial tree. Based on Weibel's data, the homothetic ratio is close to 2 ^{− 1 3} or ≈ 0.79 . Thus, the diameter of a daughter branch d ₁ is related to the diameter of its parent branch d ₀ as d ₁ = 2 ^{− 1 3} d ₀ [182].

The conductive bronchi do not contribute to the respiratory gas exchange, and thus collectively represent what is called the anatomic dead space, or conductive zone.

The volume of the dead space varies among individuals and measurement techniques, but is around 150 ml [107, 183]. In this zone, the air is mainly transported through convection. At the contrary, in the last 7 generations, called the respiratory zone (see further), the air is mainly transported through diusion. Indeed, in these generations, due to the tiny dimensions of the last generations, the characteristic times of diusion are far smaller than the convection characteristic.

The tissues structure of the bronchi vary depending on the generation. Nevertheless all the bronchi share the same characteristic cylindrical layered structure. The inner part of each bronchus, through which the air is transported, is called the lumen. It is surrounded by a layer of internal tissue called the epithelium. It is made of a fairly monocellular layer composed of a various types of cells. Among these, the Clara cell is a type of non-ciliated cell found in the bronchial epithelium that is characterized by its secretion role [184]. Some cells produce mucus ² , which is a gel-like substance lining the apical surface of the epithelium, where inhaled pathogens and dust are trapped. Some of the epithelial cells carry cilia on their apical side, while others do not. The cilia are

2 Namely the goblet cell, which is a type of cell found in various epithelia that possesses multiple

roles, including the secretion of the mucus lining the epithelium [138].

1.1 The lungs: description and main characteristics

Conduc ting (br onchial) z one Respir at or y (alv eolar) z one

Figure 1.2: Graphic representation of the Weibel's symmetric model of the lungs, con- stituted of 24 successive generations. Repro- duced from [183].

Figure 1.3: Evolution of the total cross- sectional area of the bronchi and alveoli, calculated from Weibel's data. Reproduced from [183].

able to beat in a synchronous way, propelling the mucus from the distal bronchi (far from the trachea) to the proximal bronchi (close to the trachea). Once at the top of the trachea, the mucus goes into the larynx where it is generally swallowed or coughed [42].

Patches of cilia covering the epithelium can be seen on Figure 1.4. Figure 1.5 sketches a classical bronchial epithelium, depicting various cell types encountered. Epithelium is supported by the basal membrane, itself surrounded by a layer of smooth muscles, as seen on Figure 1.6. By contracting, these muscles can tune the opening of the bronchus.

This phenomenon is called bronchoconstriction or bronchodilation depending on the state of the smooth muscles. It has to be noted that the large airways, such as the trachea and the rst bronchi, are also supported by cartilaginous rings or plates, which give them some rigidity.

The respiratory zone

The respiratory zone roughly comprises the last 7 generations, where respiratory gas exchange between the air and the blood occurs (i.e. acinar region). From the respiratory bronchioles, alveoli start to bud along the airways, up to the alveolar ducts which are nearly totally covered in alveoli. Finally, the airways of the last generations are totally covered by the alveolar sacs, which form a tiny grape-like structure where alveoli occupy all the volume. All these pseudospherical alveoli, whose diameter is around 200 µ m [124], form an exchange surface of about 50-100 square meters and occupy a volume of about 2.5-3 liters, thus the major part of the lungs volume [183].

As seen on Figure 1.7, alveolar epithelium is extremely thin, and tightly surrounded

1.1 The lungs: description and main characteristics

Figure 1.4: Microscopic picture of bronchial epithelium cilia. Reproduced from [75].

Figure 1.5: Sketch of a transverse section of the airway epithelium. Several types of cells can be observed, including mucus secretory cells. Reproduced from [25].

by blood capillaries, forming a gas-exchange barrier as thin as 0.3 µ m, optimizing the gas transfer from and to the blood.

The alveoli are not covered in mucus and cilia. Immune system cells such as macrophages are thus the rst line of defense against inhaled pathogens and parti- cles that could deposit inside the alveoli. Captured by macrophages, alien body are then removed through the blood circulation or the lymphatic system [183].

1.1.2 Breathing, ventilation and lung function

At inspiration, the diaphragm and the intercostal muscles contract. It induces an

expansion of the thorax. The lungs expand in return, forcing the inspiration of fresh

1.1 The lungs: description and main characteristics

Figure 1.6: Microscopic image of the airway epithelium. Reproduced from [97]. C:

Clara cells. G: goblet cells. L: lumen. Arrows: basal membrane.

Figure 1.7: Microscopic image of the alveolo-capillary region. C: capillary. EC: ery- throcyte or red blood cell. EP: epithelium. Reproduced from [183].

air inside the lungs, due to the pressure drop. Inspiration is thus, even at rest, an active process. At the contrary, during expiration, relaxation of the muscles induce an elastic recoil of the lungs to their initial volume, and the expiration of the inhaled air. Expiration is thus, at rest, a passive process. During exercise, both inspiration and expiration are active processes, where secondary muscles such as the abdominal muscles help the respiratory muscles.

The ventilation designates the advection of air as it enters or exits the lungs during

1.1 The lungs: description and main characteristics

the breathing process. Physiologically, the ventilation is measured as a ow rate, often expressed in ml/s. At rest, the average inspired or expired volume, called the tidal volume, is about 500 ml. Nevertheless, to obtain the eective inspired volume, it is thus important to take the dead space volume into account. Therefore, the eective inspired volume, the one bringing the oxygen to the alveoli, is about 350 ml per inspiration.

With 15 breaths per minute, the ventilation ow at rest is about 5250 ml / min. This rate drastically increases during exercise to provide enough oxygen to the body.

Evaluation of the lung function is primarily based on the measurement of the pul- monary volumes and ow rates. This measurement is performed through spirometric tests, using a device called a spirometer, able to measure the respiratory volumes and ow rates. A representation of the dierent pulmonary volumes is seen on Figure 1.8.

At rest, the total lung volume oscillates around an average lung volume. The magni- tude of this oscillation is called the tidal volume (TV). At the end of rest expiration, the remaining volume in the lungs is called the functional residual capacity (FRC).

From this volume, forced expiration decreases the volume of the lungs up to a minimal value where the residual air can not be expelled: the residual volume (RV). From this value, forced inspiration increases the lung volume up to its maximal volume, called the total lung capacity (TLC). The dierence between TLC and RV is called the vital capacity, or VC. Note that the value of the RV, being absolute, can not be measured with spirometry but, if needed, is accessible through other techniques. Measurement of respiratory ow rates also allows for a better characterization of the lung function, through various indicators such as the the forced expiratory volume in 1 second (FEV 1 ).

Patient's FEV 1 , as the other volumes and ow rates, is often expressed in terms of its relative value compared to the average FEV 1 value for the same age and gender.

Volume

Time TV

VC

RV

FRC TLC

Figure 1.8: Lung volumes graph analysis as obtained by spirometry. See text for the

abbreviations. Adapted from [176].

1.1 The lungs: description and main characteristics

1.1.3 Air transport and respiratory gas exchange

Dierent mechanisms take place in order to transport the air through the lungs.

First, at inspiration, due to the pressure dierence between the lungs and the atmo- spheric pressure, the air is moved at relatively high speed through the upper part of the bronchi down the lungs. For example, in the trachea and at rest, the maximal air speed is about 5 m/s. As the total ow cross-sectional area increases when going deeper in the lungs, the air regularly slows down up to the end of the alveolar sacs.

A comparison of the magnitude of the axial convective transport to the magnitude of the axial diusive transport of a gaseous specie in an airway is possible through the use of the Péclet number, noted Pe. This number evaluates the ratio of a characteristic diusive time to a characteristic convective time:

Pe = ^L

2

D S L

Q

= Q L S D = v L

D (1.1)

where L is the length of the airway (in m), S its ow cross-sectional area (in m ² ) and Q is the volumetric ow rate in the airway (in m ³ s ⁻¹ ). v = ^Q S (in m s ⁻¹ ) is thus the average air velocity. D is the diusion coecient of the studied specie in the air (in m ² s ⁻¹ ).

An axial transport dominated by convection corresponds to Pe ≫ 1 : the characteristic diusive time is way larger than the characteristic convective time; convection is the main transport process. At the contrary, Pe ≪ 1 corresponds to an axial transport dominated by diusion (see Section 3.2.4 for further details).

In the lungs, convective and diusive axial transports coexist. Convection domi- nates in the conductive bronchi (Pe ≫ 1 ), while diusion dominates in the respiratory zone (Pe ≪ 1 ). Depending on the diusing gas, around the entrance of the acini (Table 3.1) [126].

Respiratory gas exchange between the air and the tissues occurs in the respiratory zone. During the gas exchange process, oxygen from the fresh inspired air diuses through the alveolar-capillary membrane to the blood. A part of it is dissolved in the plasma. The main part is linked to the hemoglobin protein contained in the red blood cells. These high-oxygen red blood cells are transported through the blood circulation to the main organs, where they release their oxygen and are loaded with carbon dioxide produced from cellular respiration. Carbon dioxide is partially dissolved into the plasma (5-10%) and mainly transported by the red blood cells under the form of bicarbonate or carbamino compounds [183]. Carbon dioxide is transported from the organs to the lungs, where it is released in the alveoli and exhaled during expiration.

The oxygen and carbon dioxide exchange is driven by partial pressure dierence

between a high pressure compartment and a low pressure one, through a permeable

membrane. During inspiration, the oxygen partial pressure is high in the alveoli and

low in the blood, which favors its diusion to the blood. At the contrary, carbon

dioxide partial pressure is high in the blood and low in the alveoli, which favors its

diusion to the alveoli. During expiration, the alveolar air loaded in carbon dioxide is

evacuated, preparing the lungs for a new respiratory cycle.

1.2 Asthma and Nitric Oxide

1.2 Asthma and Nitric Oxide

1.2.1 Description of the asthma pathology

Asthma is one of the most prevalent disease in the world, with an estimated 5%

of the world's population suering from it, with increasing prevalence over the years.

Yearly, around 1 in 250 deaths worldwide is probably caused by asthma [36]. Facing these numbers, there is a real urge to better understand, diagnose and cure this disease.

One way to fulll these objectives is to use biomarkers of the disease as a proxy to apprehend its complexity.

Asthma is a multifaceted pathology for which it is hard to give a straightforward denition. It is characterized by multiple lung symptoms that can severely aect the patients. Clinically, physicians observe that asthma patients often suer from wheez- ing, cough and chest tightness. At the physiological level, asthma is often marked by a decrease in lung function and bronchial hyper-reactivity, linked to general air- way inammation [36]. These symptoms are unfortunately quite vague, and could be linked to other lung diseases, such as chronic obstructive pulmonary disease (COPD).

Overlapping symptoms often exist between these two aections.

The bronchial hyper-reactivity (or hyperresponsiveness) observed in asthma is mainly due to allergic exacerbation, a phenomenon called atopy. Atopy is a general body propensity to allergic disease such as asthma or allergic eczema. This explains why allergens such as pollens and house dust mite faeces are common triggers for asthma crisis, along with other nonallergic triggers such as exercise, cigarette smoke or air pollution [36]. Bronchial hyper-reactivity causes the smooth muscles surrounding the bronchi to contract, reducing the bronchial diameter in a phenomenon called bron- choconstriction (BC), which can be temporary or even permanent. Depending on its level and extent, BC can severely aect the respiratory function.

In order to better diagnose asthma and give patients the more adapted treatment, clinicians need to identify specic markers accurately linked to asthma pathology. Sev- eral tests and markers are nowadays used in clinical practice.

First of all, clinicians will look up for characteristic symptoms of asthma, such as wheezing, chest tightness or breathlessness. Other atopic conditions like allergic eczema can be a clue too. Along with these observations, spirometric tests are often made to assess the lung function of the patient, although it can be normal in asthma patients.

But for unclear cases, clinicians are in need for more specic markers of asthma.

Among these markers, the presence of an airway inammation with eosinophils strongly suggests asthma [177]. Eosinophils are a type of white blood cells normally not present in the lungs. Airway inammation is triggered by stimuli such as allergen or exercise which induce, through a cascade of inammatory reactions, the recruitment of eosinophils, among other cells, on the site of inammation. These recruited cells release substances that also contribute to BC and, at long exposure, structural change in the airways, in a process called airway remodelling. Recently, physicians started to collect eosinophils from blood but also from induced sputum ³ in their daily practice

3 Sputum is the name of an expectorate mucous substance that is supposed to contain materials

1.2 Asthma and Nitric Oxide

to assess the presence and type of asthma [47]. However, although the presence of esosinophilic inammation gives strong evidence for asthma, it is not specic to this disease, as eosinophils can be found in other patients, for example in COPD patients [36].

1.2.2 Nitric oxide: a link to asthma

An other important marker of the airway inammation is the exhaled nitric oxide (NO). Nitric oxide is a small, labile, ubiquitous gaseous molecule produced in nearly every organ of the body that occupies multiple roles, from neurotransmission [17] to vasodilation [130]. NO also acts as a direct and indirect cytotoxic agent in immune response against pathogens [119, 120]. In the lungs, it has been shown that NO acts as a bronchodilatant and as a marker of the inammation, due to its anti-microbial activity [118].

By structure, NO possesses an unpaired electron that makes it a free-radical molecule.

This reactive molecule has a short half-life time and reacts, under biological conditions, with metals and other free radicals [164]. As a small aqueous-soluble molecule, NO easily diuses through the tissues and cellular membranes, explaining its multiple roles in cell signalling [98].

NO is produced in various cell types, including epithelial cells, by a group of enzymes collectively called NO-synthases (NOS) [18]. Three similar enzymes, called isoforms, do have a NO-synthase activity: the endothelial eNOS, the neuronal nNOS and the inducible iNOS. These three isoforms produce NO under the same molecular mech- anism, but have dierent levels of expression. It is commonly admitted that eNOS and nNOS are constitutively produced, while iNOS is only produced in reaction to an inammatory trigger [45]. This distinction is in fact partly incorrect, as it has been shown that a constitutive activity in iNOS can be possible, for example in airway ep- ithelial cells [54]. It would rather be more correct to express a distinction in terms of activation mechanisms, as eNOS and nNOS activities are regulated by the intracellular concentrations of calcium (Ca ²⁺ ) while iNOS, once produced, seems to be constantly activated [45].

Regarding the respiratory system, NO has been shown to be produced in various types of tissues and cells. The major part of the NO production arises outside the lungs, namely in the nose and the sinuses, where measured NO concentration are far greater than in the bronchi [34]. Nevertheless, NO production do occur in the lungs, in the bronchial zone [152] as well as in the broncho-alveolar zone [95, 147, 167].

Regarding asthma, it has now been demonstrated for years that the molar fraction of NO in the exhaled air (written Fe NO ) is higher in asthmatic patients than in healthy patients [3, 88, 132]. This high level of exhaled NO has even been shown to be linked to higher NO concentrations in the airways [87]. Furthermore, the Fe NO seems to be linked to the severity of asthma, being higher in severe asthma than in moderate asthma [19, 159]. As NO production is generally increased in case of inammation,

that are representative of the composition of the inside lung. It is generally induced by inhalation of

a saline solution [161].

1.3 Cystic brosis

NO seems to be a good biomarker for linking asthma diagnostic and management to the inammatory status of the patient.

For all these reasons, the measurement of the Fe NO is more and more often included in the asthma diagnostic procedure, as well as for asthma management [37]. However, even if it is a good indicator, especially combined with eosinophils measurements [73], a high Fe NO is not straightforwardly linked to asthma [81]. Nowadays, in clinical practice, these two markers are often combined to infer the presence and severity of asthma with increased accuracy [144].

Regarding the use of the Fe NO measurement alone, clinical evaluations of its reli- ability tend to show that it is not sucient in itself to correctly diagnose asthma and track the evolution of this disease. Although its combination with other biomarkers, such as eosinophils measurements, improve its reliability [106], the Fe NO measurement does not give in itself enough insights in the internal NO dynamics in the lungs. Thus, in addition to its measurement, a complete evaluation of the lung function of the pa- tients is often recommended. In order to better interpret the Fe NO dynamics, the use of modeling approaches, further described, has arisen (see Section 1.4.2). Modeling results allow for a deep understanding of the pulmonary NO dynamics, for example to mimic asthma events such as inammation, bronchoconstriction or mucus accumulation.

1.3 Cystic brosis

1.3.1 Description of the cystic brosis pathology

Cystic brosis (CF) is a genetic disease that aects all types of population, with the highest birth prevalence in white population (around 1:2500) [105]. This disease aects several organs in the body, with the major symptoms occurring in the lungs.

In this organ, due to the genetic defect, the composition and thus the displacement of the mucus layer covering the bronchi is altered ⁴ , leading to major pulmonary chronic infections and impairments in the lung function. A better description of this mucus layer and of its interactions with the tissues and the air in the lungs could thus help in a better understanding of this disease. The total prevalence of CF was around 7 per 100,000 in 2004 in the EU, with similar numbers in the US. Nowadays, the median age of death is around 25 years in Europe. It has been drastically increasing over the last decades. Current survival age is around 40-50 years in developed countries [105].

CF is a complex autosomal ⁵ recessive disease due to variable genetic mutations on the CF transmembrane conductance regulator (CFTR) gene, located in the seventh chromosome. This gene codes for the CFTR protein, which is mainly expressed in the apical membrane of epithelial cells. The CFTR protein acts as a passive anionic chan- nel, mainly transporting chloride ion (Cl ⁻ ) across the membrane following an osmotic pressure gradient [5, 26, 148]. This transport is not selective to Cl ⁻ , because other neg- atively charged ions can be transported by CFTR [69], such as bicarbonate [92, 135].

4 In link with the other name of CF: mucoviscidosis.

5 The genetic defect is located on a autosome, a non-sexual chromosome.

1.3 Cystic brosis

Defect in CFTR protein causes the multiple symptoms of the CF pathology. Among those, gastrointestinal and reproductive organs failures, elevated salt concentration in sweat and respiratory failure are observed. The latter defect is the most common cause of more than 95% of CF mortality [16].

Linking the genetic defect to the pathophysiology of CF is still poorly understood, and constitutes a debated question among the scientic community. CF impacts at least two important lung defense mechanisms: mucus clearance and antimicrobial activity [160]. This leads to mucus accumulation that impairs the lung structure and function (see Figure 1.9) and provokes chronic inammation and infections, possibly leading to death.

Figure 1.9: Age-dependency of the median value of the FEV 1 in CF patients. Repro- duced from [105].

The rst line of CF treatment comprises prophylactic use of antibiotics to ght

against chronic infections in CF lungs. In case of an exacerbation of the symptoms,

higher doses of antibiotics are usually given to the patients in order to reduce the

spread of the infection, and partially restore lung function [39]. In addition, a bat-

tery of molecules are used in daily treatment, each one targeting a precise impaired

function in CF lungs. In particular, dornase alfa is a commonly used molecule which,

by cleaving the DNA fragments present in the mucus layer, contributes to reduce the

mucus viscosity [103] and to improve the general lung function [78]. Recently, it has

been shown that inhalation of a hypertonic saline solution contributes to improve the

general lung function in CF patients, supposedly increasing the mucus hydration and

thus its clearance [35, 41]. In addition, new innovative therapies, still in development,

are based on directly targeting the CFTR channel, either by the use of molecules that

partially restore its function, either by the use of gene therapy that aims at replacing

directly in the cells the defective CFTR gene by a correctly expressed gene [39]. Be-

sides from these molecular or pharmaceutical approaches, airway clearance techniques,

dedicated to mobilize and evacuate the mucus, are used on a regular basis to control

and improve lung function. Although these techniques are commonly used and thought

to be eective, there is a need for controlled studies to evaluate and compare their ab-

solute and relative eects [114]. As a last resort, the need for lung transplantation has

1.3 Cystic brosis

to be evaluated but, if successful, transplantation can drastically improve the quality of life and life expectancy of the patient.

1.3.2 The mucus layer in healthy and cystic brosis lungs

Defects in the mucus layer, especially in the lungs, are characteristic of the CF pathology, in link with many of the observed symptoms. Mucus has thus been exten- sively studied, in healthy and CF individuals. However, the bronchial mucus dynamics is still largely unknown, partly because there is a lack of biophysical data about the mucus. Thus, there is a need for new tools to better describe this dynamics.

Mucus is a complex gel-like substance that covers the epithelium in multiple organs of the body, such as the lungs, the gastrointestinal tract or the vagina. This mucus layer possesses multiple roles, among which tissue protection against foreign pathogens and particles and lubrication [96]. At the chemical level, mucus is constituted of a network of large entangled proteins, the mucins, which form by hydration a thick polymeric substance [94, 174]. In addition to water and mucins, mucus is composed of other macromolecules, such as DNA from cellular debris, lipids and other proteins.

Furthermore, various levels of salts have been measured in mucus. Mucins accounts from 2 to 5% of the mucus weight, with the other molecules and macromolecules accounting each for less than one percent. Water predominates with more than 90%

in weight [96].

The physical properties of the mucus layer are highly dependent on its chemical com- position. From a rheological point of view, mucus is a highly viscous non-Newtonian uid, whose viscosity varies in a complex non-linear way along with the applied shear rate. More precisely, the mucus viscosity is a nonlinear decreasing function of the ap- plied shear stress (shear thinning uid) ⁶ . It has been shown that the mucus viscosity is highly dependent on its chemical composition, as even slight variations in mucins, DNA or salts compositions can greatly inuence the viscosity of the mucus [96, 101], and thus its ability to be cleared o.

In the lungs, the mucus is heterogeneously produced, with the vast majority of the production arising in the large bronchi, while there is few mucus producing cells in the small bronchi (diameter ≤ 1-2 mm) [74]. Under the microscope, the bronchial mucus appears as a thin layer covering the bronchial epithelium. As depicted in Figure 1.10, the mucus layer is covering the cilia present on the apical membrane of the epithelial cells. Mucus is propelled by the beating cilia from the distal bronchi to the central bronchi ⁷ , up to the proximal ones, which is at the basis of to the pulmonary mucocil- iary clearance. Numerous studies have been conducted to understand the interaction between the cilia and the mucus layer. Nowadays, a complex picture do appear, with the mucus layer divided in two superposed layers, collectively called the airway surface liquid (ASL). The upper layer lies on the lower one, a thin layer bathing the cilia,

6 More precisely, mucus can be considered as a Bingham uid: mucus behaves as a solid under a threshold shear stress, then ows like a liquid at high shear stress.

7 The term central collectively designates the intermediate bronchi located in the bronchial tree be-

tween the proximal and the distal bronchi. It roughly corresponds to the bronchi located in generations

7 to 14.

1.3 Cystic brosis

called the periciliary layer (PCL). The role and precise composition of the PCL is still debated, but it seems that it consist of a gel substance that tightly covers the cilia [21]. The thickness of the PCL having the same dimensions than the length of the cilia (≈ 7 µ m), it forms a lubrication layer into which the cilia are beating, with their tips probably in contact with the basal part of the upper layer, allowing to propel the entire ASL (see Figure 1.10).

The variability of some physical properties of the mucus layer is still a debated ques- tion. It seems that the mucus velocity varies along the bronchial tree, with an increased velocity in the proximal bronchi and a decreased velocity in the distal bronchi [136].

Experimentally, the observed mucus velocity is about 10 ^µ _s ^m . Smith et al. suggest an average value of about 40 ^µ _s ^m over the bronchial tree [156]. Furthermore, the thickness of the ASL seems to depends on the generation, with a possible proportionality with the bronchial diameter [188].

Figure 1.10: Sketch of the airway epithelium, depicting dierent cell types, the cilia and the airway surface layer. Adapted from [42].

In CF patients, it is clear that the reduced mucociliary clearance is linked to CFTR

defect. However, it is still debated if this impaired mucus clearance is the main factor

of the high level of infections observed in CF [53]. Nevertheless, this hypothesis is more

and more accepted, as this is suggested by comparison with other genetic diseases, for

example in primary ciliary dyskinesia (PCD), which is a disease where the cilia beating

function is altered [102]. Although CFTR is correctly expressed in PCD, PCD patients

get pretty the same range of lung infections as observed in CF patients, but not exactly

1.3 Cystic brosis

[16, 53]. Anyway, there is still a scientic debate about the mechanisms explaining how the absence of a functional CFTR channel induce the observed defects in mucociliary transport. There has been some controversies regarding the origin of this impaired mucus transport [50, 53, 108]. Nevertheless, nowadays there seems to have a consensus about the so-called low-volume hypothesis, preferred by the scientic community [31], which is presented in this work.

Following the low-volume hypothesis, CFTR defect in CF, which modies the ionic transport of chloride but also sodium ions, implies a lack of hydration of the mucus [53, 76, 94]. This hypothesis is supported by the observation that, in vitro, the bathing of epithelial cells by a hypertonic saline solution contributes to a good hydration of the mucus [35]. It is hypothesized that this water depletion modies the physical properties of the mucus layer, making it more viscous, impairing its elimination by the mucociliary transport or by cough [94, 96]. However, this hypothesis has been challenged by few experimental studies that show that CF mucus seems to possess the same rheological properties as normal mucus [141, 146].

The low-volume hypothesis is thus a seducing model for the pathophysiology of CF. However, aside from the reduced mucociliary clearance, it does not totally explain why the mucus layer in CF patients is subject to chronic bacterial infections. Some observations show that the loss of CFTR function partially reduces the bicarbonate secretion by the epithelium [157], which potentially acidies the ASL in CF compared to normal [23]. Stoltz et al. thus suggest that this acidication reduces the ASL antibacterial activity, contributing to infections [160].

1.3.3 Nitric oxide in cystic brosis

Due to the presence of inammation in CF airways, one would expect an increase of the Fe NO value in CF patients. However, the Fe NO value seems to be decreased in CF patients [51, 125], or at least to have the same order of magnitude [46, 58]

than the Fe NO value of healthy individuals. These results are not straightforward and this tendency has not been perfectly demonstrated, especially in children. Numerous hypotheses have thus been suggested to explain this surprising low Fe _NO value, which can be grouped in three categories: increased diusion barrier, reduced NO production or increased NO consumption.

The increased diusion barrier could be due to a thicker and denser mucus in CF.

NO being produced in the epithelium, this would impair its transfer to the airways, and thus the Fe _NO value. In parallel, it has been suggested that this reduced Fe _NO value could be induced by a reduced activity of the NOS, either because they would lack of substrate (L-arginine) to produce NO, either because their activity would be reduced in CF airways. It remains unclear wether this reduced activity could be a direct consequence of the mutation or an indirect consequence via the presence of inammation and infections. Finally, it has been suggested that chronic bacterial infection could result in an increased production of bacterial enzymes able to react with the NO, hence consuming a part of the NO production [33].

Hofer et al. measured that, in twelve CF patients, Fe NO was lower than in controls.

1.4 Modeling the lungs

Furthermore, they observe lower bronchial NO ows in CF compared to controls, but similar alveolar NO concentrations. This seems to indicate that the lower Fe NO value is maybe linked to a reduced bronchial NO component, in line with the diusion barrier hypothesis [61]. Indeed, it has to be remembered that mucus is present in the bronchi but not the alveoli. In line with this hypothesis, it has been shown that treatment with inhalation of dornase alfa in a limited set of patients seems to be positively correlated with a Fe NO increase [52, 61]. This suggests that dornase alfa, modifying the rheological properties of the mucus, inuence its interaction with NO and thus supports the diusion barrier hypothesis.

On the other hand, in line with the reduced production hypothesis, Kelley & Drumm show that iNOS expression, although constitutively expressed in normal subjects, ap- pears to be reduced in CF bronchial epithelial cells [82]. This would imply a reduction in Fe NO value, but also in the antibacterial NO activity, which could contribute to promote infections.

Finally, regarding the increased consumption hypothesis, it has been observed that there is no sensible dierence in NO bronchial ow in CF patients with or without chronic infection. This seems to indicate that NO consumption by bacteria or bacterial enzymes would not be the major cause of the Fe NO reduction in CF [61].

No matter the hypothesis, it has to be noted that CF patients with lower general lung function seem to have lower Fe NO values, indicating a link between the measure- ment of the Fe _NO in CF patients and pulmonary exacerbation [51].

Nevertheless, to the best of our knowledge, no model of the pulmonary NO dynamics has been used to analyze the Fe NO behaviour in the case of CF.

1.4 Modeling the lungs

To better understand an organ, the best way is to look inside it. The rst bron- choscopies have been performed at the end of the 19 ^th century. Since then, exible bronchoscospy with cameras have been developed, allowing a better view of the rst generations of the lungs. Nowadays, physicians can have a good insight of the airways up to the 4-5 ^th generation [100]. Due to the shrinking diameters of the bronchi, it is merely impossible to look further with actual technology.

Of course, other new well-mastered technologies such as X-rays or computer to- mography (CT-scan) allow to obtain a better image of the lungs, with the specic characteristics of the studied individual. But these advanced techniques, due to resolu- tion problems, lack of a clear view regarding the small bronchi and the respiratory part.

Furthermore, they are often time- and computer-consuming. Even though good images

of the lungs can be obtained nowadays, they lack of a comprehensive understanding of

the lung function. Altogether, these stumbling blocks fostered the development of var-

ious lung models able to better understand the mechanisms underlying lung function.

1.4 Modeling the lungs

1.4.1 General modeling approaches

Lung modeling took an major step in the early 60's with the publication of the rst accurate lung morphometric data by Weibel [181]. In this exhaustive work, Weibel presented precise measurements on nearly all aspects of the lung morphometry: number of generations, length and diameter, size and number of alveoli, etc. At the reading of Weibel's paper, one can evaluate the tremendous work performed to get these data properly. It is thus no surprise why these data are still used with condence today.

Note that other precise data have been obtained by other authors, such as data from Horseld & Cumming [65, 66, 67, 131]. In this work, the focus is made on Weibel's data.

From these various data, dierent models have been developed to describe the lungs and the pulmonary gas dynamics. All these models have been developed in a try to complement the clinical and physiological measurements, in order to give new insights into diseases. At the morphometric level, the main dierence between the various models is about the symmetry of the bronchial tree. The symmetry hypothesis is at the basis of the so-called homogeneous models. Nevertheless, these models do not fairly represent all the complexity and asymmetry of the lungs. At the opposite, in heterogeneous models, a level of asymmetry is introduced, which make them more complicated [181]. Because of its simplicity and its fair accuracy, the homogeneous modeling approach has been used in this work.

As the pulmonary function is complex, so is the related modeling. The lungs being the organ responsible for the physiological respiration, the rst models to emerge were interested in describing the transport and exchange of the respiratory gases between the air and the tissues, in the bronchi and the alveoli. General phenomenological ap- proaches emerged that analyzed these convection-diusion based transfers [28, 126].

The description of the transport of respiratory gases has then been extended the de- scription of the transport of non-respiratory gases such as N 2 [128], NO or even water vapor [163]. In parallel, the general air transport in the bronchi and alveoli is often analyzed through a uid dynamics approach, characterized by the solving of the Navier- Stokes equations in a more or less accurate geometry of the lungs. This approach is useful in studying the velocity proles of the air in the bronchi [22] or the path of inhaled particles in the bronchi [112, 169].

Regarding the modeling of the respiratory or non-respiratory gas dynamics, the rst developed lung models were based on the so-called two-compartments or pipe-balloon models [112]. In these models, which can take various expressions, the bronchi are collectively grouped in a single rigid tube, that connects to the larynx at one extremity and to the alveoli to the other. The alveoli are represented as an inating sac or balloon, as seen on Figure 1.11. This simple model is able to reproduce the major characteristics of the respiratory dynamics and gas exchange. However, it masks a part of the complexity of the lungs. For example, axial diusion is not taken into account in these models (see Section 3.3) [151, 171].

On the other hand, other modeling approaches can emerge from Weibel's data.

Based on the homogeneous modeling approach, all airways of the same generation can

be considered as identical. In that way, all the airways of the same generation can be

1.4 Modeling the lungs

Conducting airways

Respiratory airways + alveoli

Figure 1.11: Illustration of the pipe- balloon model.

Cumula ted cr oss-sec tion Airways

Alveoli

Distance from alveoli

Figure 1.12: Illustration of the trumpet- shape model. Total cross-sectional area is plotted again generation number. Mouth is at right. Alveolar cross-sectional surface is in dashed. Adapted from [126] and [83].

replaced by a single airway whose ow cross-sectional area is equal to the sum of the cross-sectional areas of all the airways of the generation. Thus, as the cumulative cross- sectional area increases with the generation number (see Table 3.1), and as the bronchi divide in a symmetric dichotomous way, Figure 1.12 illustrates a characteristic feature of the homogeneous model: the trumpet-shape. It clearly represents the exponential increase of the exchange surface of the lungs as the bronchi divide themselves.

The alveolar compartment is represented in the same way, with all the cumulated cross-sectional area of the alveolar sacs being added to the cross-sectional area of the airways in the last generations, as depicted in Figure 1.12.

The trumpet-shape representation has been preferred to the two-compartments ap- proach in this work due to its simplicity and accuracy, when compared to experimental data.

1.4.2 Modeling the lungs: the asthma pathology

Modeling approaches of the pulmonary function in the case of asthma are multiple.

In this work, we use, as stated before, an approach based on transport phenomena to describe the unhealthy lungs. More precisely, as NO is a pertinent biomarker linking the pulmonary function to its disease status, we focus on developing an accurate model of the NO transport in the lungs, able to link the measured Fe _NO to a representative pulmonary status.

Models of the NO transport in the lungs have been developed, and can roughly be

separated in two groups. The rst group of models are based on the two-compartments