D
EVELOPING COMPUTATIONAL METHODS TO
PREDICT THE FATE OF INHALED PARTICLES IN THE
LUNG
T
EODOR
E
RNGREN
A
THESIS SUBMITTED IN FULFILLMENT FOR THEDEGREE
M
ASTER OF SCIENCE IN BIOMEDICALENGINEERING WITH MASTERS IN
B
IOMEDICAL MODELLING AND SIMULATIONIN THE
D
EPARTMENT OF BIOMEDICAL ENGINEERINGL
INKÖPING’
SI
NSTITUTE OFT
ECHNOLOGYLIU-IMT-TFK-A–17/545—SE
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XAMINER: G
UNNARC
EDERSUNDAbstract
The respiratory system can be targeted by many different types of diseases, for example asthma and cancer. The drug delivery method by inhaling substances for treating diseases only started in the 1950s with the treating of asthma, considered also for many other diseases. Mathematical dosimetry models are used in drug development to predict the deposition of particles in the lungs. This prediction is not easily achieved experimentally, and therefore these mathematically models are of high importance. Monkeys are often used in the late stages of drug development due to their resemblance in humans. A good model for predicting the deposition pattern in monkeys is therefore useful in the development of drugs. However, there is at the moment no developed deposition model for monkeys. In this thesis both a static model and the first dynamic deposition model was developed using the data on the breathing pattern from respiratory inductance plethysmography (RIP) bands. This dynamic model provides regional and time resolved information on the particle deposition in the lungs of monkeys and can be used to get a deeper understanding of the fate of inhaled particles. This model can also determine inter-animals differences which have not been achieved before. An extensive implementation of these time resolved deposition models could be used to increase understanding about deposition in a variety of species and help the field to move forward.
Preface
This master thesis is for fulfillment of the degree master of science in biomedical engineering with master in biomedical modelling and simulation. The master thesis was conducted at AstraZeneca in Mölndal and consists of a literature study and development of different computational models in MATLAB.
Acknowledgements
Firstly I would like to thank AstraZeneca and especially my supervisor Elin Boger for the opportunity to conduct this thesis and all the support I have got during the thesis work. Elin’s support in both writing and problem solving have been invaluable. A big thank you to Steven Oag at AstraZeneca as well, for his input and help. Also thanks to my examiner Gunnar Cedersund and my supervisor William Lövfors for their support and guidance. Finally a big thank you to my friends and family that have served as my support group during the thesis. Without all of you the thesis result would not have been the same.
Contents
List of Figures 7 1 Introduction 9 1.1 Aims . . . 10 1.2 Limitations . . . 10 2 Theoretical background 11 2.1 Respiratory system . . . 11 2.1.1 Generating airflow . . . 122.1.2 Respiratory system of rhesus and cynomolgus monkeys . . . 13
2.1.3 Breathing parameters . . . 13
2.2 Depositions factors . . . 14
2.2.1 Impaction . . . 15
2.2.2 Sedimentation . . . 15
2.2.3 Brownian diffusion . . . 15
2.3 Respiratory inductance plethysmography . . . 16
3 Material and methods 17 3.1 Airway geometry . . . 17 3.2 Static model . . . 19 3.3 Dynamic model . . . 22 4 Results 27 4.1 Static model . . . 27 4.2 Dynamic model . . . 29
4.2.1 Regional deposition for 1 m particles . . . 29
4.2.2 Whole breathing cycle . . . 31
4.2.3 Deposition fraction for 2 µm particles over time . . . 34
5 Discussion and analysis 35 5.1 Static model . . . 35
5.2 Dynamic model . . . 38
5.3 Model comparison . . . 41
6 Conclusion 43
7 Future research 45
A Figures 47
List of Figures
2.1 A schematic picture of the airway structure . . . 12
2.2 A spirogram, showing the different breathing parameters, adapted from [9] . . . 13
2.3 A schematic picture of the major deposition mechanism, adapted from [25]. . . 15
2.4 An example of a summed filtered RIP signal acquired with dual-belt system. . . 16
3.1 A segmented breath from the RIP signal. . . 23
3.2 Schematic figures over the filtering. . . 24
3.3 A schematic picture of the output structure, with permission from the creator [26]. . . 25
4.1 Graphs over the regional deposition during inspiration and breath-hold. . . 27
4.2 Graphs over the regional deposition for expisation and the total deposition. . . 28
4.3 Total deposition fraction for different particle sizes. . . 28
4.4 Graphs over the total deposition for different regions of the lung. . . 29
4.5 Deposition fraction over time in different generations of the lung for 1 µm particles, during the inspiratory phase. . . 29
4.6 Deposition fraction over time in different generations of the lung for 1 µm particles, during breath holding. . . 30
4.7 Deposition fraction over time in different generations of the lung for 1 µm particles, during the expiratory phase. . . 30
4.8 Total deposition fraction over time in different generations for 0.1 µm particles. . . 31
4.9 Total deposition fraction over time in different generations for 0.5 µm particles. . . 31
4.10 Total deposition fraction over time in different generations for 1 µm particles. . . 32
4.11 Total deposition fraction over time in different generations for 2 µm particles. . . 32
4.12 Total deposition fraction over time in different generations for 3 µm particles. . . 33
4.13 Total deposition fraction for 2 µm particles over time. . . 34
4.14 Graphs over the fraction of particles that gets trapped in the nose and lungs. . . 34
A.1 Deposition fraction over time in different generations of the lung for 0.1 µm particles, during the inspiratory phase. . . 47
A.2 Deposition fraction over time in different generations of the lung for 0.5 µm particles, during the inspiratory phase. . . 48
A.3 Deposition fraction over time in different generations of the lung for 2 µm particles, during the inspiratory phase. . . 48
A.4 Deposition fraction over time in different generations of the lung for 3 µm particles, during the inspiratory phase. . . 49 A.5 Deposition fraction over time in different generations of the lung for 0.1 µm particles, during
breath holding. . . 49 A.6 Deposition fraction over time in different generations of the lung for 0.5 µm particles, during
breath holding. . . 50 A.7 Deposition fraction over time in different generations of the lung for 2 µm particles, during
breath holding. . . 50 A.8 Deposition fraction over time in different generations of the lung for 3 µm particles, during
breath holding. . . 51 A.9 Deposition fraction over time in different generations of the lung for 0.1 µm particles, during
the expiratory phase. . . 51 A.10 Deposition fraction over time in different generations of the lung for 0.5 µm particles, during
the expiratory phase. . . 52 A.11 Deposition fraction over time in different generations of the lung for 2 µm particles, during
the expiratory phase. . . 52 A.12 Deposition fraction over time in different generations of the lung for 3 µm particles, during
the expiratory phase. . . 53 A.13 Total deposition fraction for different particle sizes for different breaths. . . 53 A.14 Total deposition fraction for different particle sizes for different breaths,
with static breathing parameters. . . 54 A.15 Total deposition fraction for different particle sizes for different breaths
in the tracheobronchial (TB) region. . . 54 A.16 Total deposition fraction for different particle sizes for different breaths
in the tracheobronchial (TB) region, with static breathing parameters. . . 55 A.17 Total deposition fraction for different particle sizes for different breaths
in the pulmonary (PUL) region. . . 55 A.18 Total deposition fraction for different particle sizes for different breaths
in the pulmonary (PUL) region, with static breathing parameters. . . 56 A.19 Fraction of particles that gets trapped in the nose for different particle sizes
for different breaths. . . 56 A.20 Fraction of particles that gets trapped in the nose for different particle sizes
for different breaths, with static breathing parameters. . . 57 A.21 Fraction of particles that gets trapped in the lung for different particle sizes
for different breaths, after spontaneous nasal breathing . . . 57 A.22 Fraction of particles that gets trapped in the lung for different particle sizes
for different breaths, with static breathing parameters, after spontaneous nasal breathing . . . 58 A.23 Graphs from [3] for comparison with the models output . . . 58
Chapter
1
Introduction
The respiratory system can be affected by many different types of diseases, with different severity and targeting different parts of the respiratory system. Everything from chronic diseases as chronic obstructive pulmonary disease (COPD) and asthma, to infection such as pneumonia and also cancer. Many of these diseases can be very problematic for the patients if left untreated, and a good treatment can have a high impact on the quality of life of the patients.
Inhaling substances for treating diseases has only begun to be used in modern medicine. In the 1950s asthma started to be treated by inhaled drugs and inhalation is now considered for a number of lung conditions and other diseases. Modern discoveries have led to a rise of interest of systemic delivery of drugs by inhalation [6]. Monkeys or nonhuman primates have been used to investigate the benefits and risks of inhaled exposure to different compounds in humans. Due to similarities between humans and monkeys, some biological responses found in monkeys can be expected to occur in humans as well. Prediction of drug exposure and deposition is hard to determine experimentally and therefore it is of interest to investigate the potential usage of mathematical dosimetry models [3]. However, there is at the moment no developed deposition model for monkeys. Such a model could be used to predict the drug deposition for preclinical studies and also to see if monkeys can be used as a predictor for the deposition in human lungs. Such models needs to account for and handle the complexity of the respiratory system and especially the lungs.
The lungs are a part of the complex respiratory system, which is crucial for all mammals survival. In collaboration with the cardiovascular system, it does not only supply the body with oxygen, but also eliminates the by-product carbon dioxide from the body. The respiratory system consists of the nose, pharynx, larynx, trachea, bronchi and the lungs. The bronchi and the lungs are subdivided into primary bronchi, secondary bronchi, tertiary bronchi, bronchioles, terminal bronchioles, respiratory bronchioles, alveolar ducts, alveolar sac and alveoli [9]. All this builds up a rather messy tree structure.
In modelling, the structure is simplified and divided into smaller parts called generations. These generations of the respiratory tree starts in the nose and ends in the alveoli, and are indexed from 1 to ∼25. Parts of these generation indices are coupled together to form three main regions, the extrathoractic region (ET), tracheobronchial region (TB) and alveolar region (AL). The ET region reaches from the nose down to the start of the trachea and are defined as generation 1. The TB region reaches from the trachea all the way down to the bronchioles, e.g. generation 2-18. The bottom of the lung tree is called the AL region which reaches from the terminal bronchioles to the alveoli e.g. generation 19 -∼25 [5].
As of today a conventional way to calculate the predicted dose in an inhalation study is given by:
Dose = DdeposcAVttfmin (1.1)
where Ddeposis the deposition fraction,e.g. the fraction of the inhaled particles that gets trapped in the
chosen region, cAis the particle concentration in the inhaled air, Vtis the tidal volume, t is the exposure
time and fminis the breathing frequency [7]. The values of the tidal volume and breathing frequency are
mean values of the exposure time and are measured by, for example, RIP bands (respiratory inductance plethysmography). The deposition fraction is in the best case calculated according to species, body weight, particle size distribution and average breathing pattern based on empirically derived equations, but are sometimes only taken from a table. This way of calculating the dose assumes a fixed breathing pattern and it can thus significantly deviate from the truth. The tidal volume differs between each breath and the breathing frequency varies as well. Clearly, the deposition fractions are dependent of tidal volume and flow rate, which means that they will also vary from breath to breath.
It would be of interest to use the time resolved flow signal from RIP bands and thereby make the the deposition modelling time resolved. This would give a more precise prediction of the dose and also enable tracking of the deposition in the monkey over the exposure time. An extension of this kind, would closer resemble the ideal situation and thus provide a prediction of the particle deposition. CFD (computational fluid dynamics) would also be an attractive option, as such modelling approach would account for different flow profiles in the airways.
1.1
Aims
The aims of this master thesis are :
- To develop a computational model for the particle deposition in the lungs of monkeys.
- To integrate the volumetric flow data from RIP bands to extend the model and make it time resolved.
1.2
Limitations
The models will be developed in MATLAB and using a variety of MATLAB’s toolboxes. This means that the models will be dependent of MATLAB to be used. In the models, no clearing factors such as mucociliary clearance will be accounted for.
Chapter
2
Theoretical background
In order to construct these deposition model in both man and monkeys (mainly rhesus (Macaca Mulatta) and cynomolgus (Macaca Fascicularis) monkeys [12]) one need to have a deep understanding on how the respiratory system functions. If one understands the structure and the functionality of the upper and lower respiratory tract, one can then construct suitable models to enable prediction of the particle deposition. The particles will travel down into the lungs by the inhaled air and on must understand how the air behaves in the lungs. The lungs reside in the thoracic cavity and are separated by, for example, the heart. Because of the hearts placement, the left lung is approximately 10 % smaller than the right. The most peripheral part of the lungs is the alveoli, which is the site of gas exchange in the lungs. At the alveoli the oxygen in the air diffuse into the blood and the carbon dioxide diffuses from the blood to the lungs [9].
2.1
Respiratory system
The respiratory system helps the body to contain homeostasis by enable the exchange of oxygen and carbon dioxide between the air, blood and cells in the body. By doing this, it also contributes to a stable pH level in the body fluids. The respiratory system reaches all the way from the nostrils down to the alveoli. It is usually divided into two main parts; 1) the upper respiratory system which includes the nose, nasal cavity, pharynx with their associated structures, and 2) the lower respiratory system, which includes the larynx, trachea, bronchi and the lungs. The different parts of the lower respiratory system are divided in a variety of sub-parts as can be seen in Figure 2.1 [9].
Figure 2.1: A schematic picture of the airway structure
The main function of the upper respiratory system is to warm, moisten and filter the incoming air, which is mainly preformed in the nasal cavity. When air is inhaled through the nostrils, it passes by skin lined with coarse hairs, which is designed to capture large particles in the air. While the air is traveling through the upper respiratory tract, it is warmed by the blood in the capillaries in the airways. Throughout the air’s path down towards the lungs mucus is secreted from the goblets cells that help moisten the air, but also traps dust particles. The trapped particles are transported to the pharynx by the cilia where it can be ei-ther swallowed or spat out [9]. The air will then continue down the trachea and down throughout the lungs. At the carina, the trachea divides into the right and left primary bronchus, which connects the trachea to the lungs. The right bronchus is more vertical, shorter and wider than the left one, which means that an inhaled object is more prone to enter the right bronchus than the left. From these primary bronchus the airways branches out more and more, as shown in Figure 2.1 [9]. Throughout the airways branching there are mucous and cilia that help trapping foreign objects and transporting them up to the pharynx for removal.
The particles travels to all part of the lung by the airflow in the lungs and this airflow is of course driven by the ventilation of the lungs by the act of breathing. The process of breathing is a rather complex mechanism in the body with many different contributors which work together in order to get air in and out of the lungs.
2.1.1 Generating airflow
The breathing is driven purely by pressure differences between the atmospheric pressure and the pressure in the lungs. In order to achieve this pressure difference and allow air to flow in and out of the lungs, the muscles surrounding the lungs work to change the volume of the lungs, which thus leads to changes in the pressure in the lung. The main contributor to the increased lung volume is the diaphragm, which is a dome-shaped muscle that basically is the floor of the thoracic cavity. During inhalation the diaphragm contracts and by doing that it flattens, which increases the lung volume and thus decreases the pressure. Not only the diaphragm is contracting during inhalation, the external intercostals also contract and elevate the ribs [9].
In order to empty the lungs during exhalation, the body has to decrease the lung volume and there-fore increase the pressure. The exhalation begins when the contracted muscle at inhalation starts to relax. Due to the elasticity of the muscles, they will spring back to their original shape and position and by doing that the lung volume decreases. This process is active during relaxed breathing and does not require any real effort from the body. It is only at forceful expiration that the abdominal muscles and the intercostals start working to increase the expiratory flow rate, which might be needed when playing a wind instrument for example [9].
2.1.2 Respiratory system of rhesus and cynomolgus monkeys
Detailed information about the airways and the breathing pattern of maraque monkey are very sparse. But similarities between the maraque and humans have been established, for example the bifurcation of the airways. The number of airway generations is roughly the same in the majority of mammalian species, but the actual bifurcation is rather unique in primates, including humans. The airways is branching out at 45 degrees and are almost uniformly in size, and is called dichotomous branching [13]. Other similarities has been found, in both architectural, morphological and development patterns. The lungs of non-human primates are also more similar in number of airway generations, number of alveoli and the type of the distal airways than any other laboratory animals [14].
2.1.3 Breathing parameters
At rest, an adult human breathe with a frequency of about 12 breaths/min and the inhaled and exhaled volume are approximately 500 mL. This volume is called the tidal volume (Vt) and if one multiply
this volume by the breathing frequency, one get the minute volume which is the total inhaled/exhaled volume during a minute. The Vt is individual and varies a lot between different persons, but also in
the same person at different times. To investigate these parameters and also to measure other relevant volumes of the lung, one can use a spirometer to produce a spirogram, which can be seen in Figure 2.2 [9].
These additional parameters are related to forceful breathing, e.g. when putting more effort into the breathing to inhale/exhale more air. The extra air that a person is able to inhale under a breath is called the inspiratory reserve volume (IRV) and by adding this to the Vtone get the inspiratory capacity (IC). With
the same logic applied, the volume a person can exhale is called the expiratory reserve volume (ERV) and if adding this to the IC one get the vital capacity (VC) of the lungs. The volume that resides in the lungs after a person has exhaled the maximum amount of air possible, is the residual volume (RV). This volume is impossible to exhale due to the subatmospheric intrapural pressure that keeps the alveoli slightly inflated. By adding up all these volumes of the spirogram, one get the total lung capacity (TLC), which is the maximum volume of the lung [9].
All of these parameters will impact the number of particles deposited in different parts of the airways. But to understand how and where the particles will end up in the lungs, one need to have an understanding on how the particles deposit and what influences these deposition factors.
2.2
Depositions factors
Predicting the fate of inhaled particles in the lung is a multidisciplinary task that includes solving a complex physical problem within a biological system. With the use of mathematical equations and accounting for biological factors, one can describe the deposition of particles in the lungs. The biological system in question, the lungs, are described by building up a morphology consisting of cylindrical tubes that branches out and represents the different airway generations. The respiratory parameters are of great importance, because they decide the flow rate and velocity of the inhaled air and therefore also the particles’ velocity in the airways. Deposition models should strive to achieve two main goals: firstly, the assumptions made should be as anatomically and physiologically realistic as possible. Secondly, they must allow numerical or analytical solutions to the mathematical expressions describing the air flow patterns, deposition and the biological system [5].
Aerosols suspended in inhaled air will be subject to different physical mechanisms, which will re-sult in the aerosols leaving the airstream of the inhaled air and finally deposit in the surrounding area. The three main mechanisms ( which can be seen in Figure 2.3) that will affect the aerosols and contribute to the deposition are 1) Brownian diffusion, 2) sedimentation due to gravity, and 3) impaction due to inertial forces. Other factors like deposition due to phoretic forces, electrical charge and cloud settling may occur for specific aerosols and conditions [5].
Figure 2.3: A schematic picture of the major deposition mechanism, adapted from [25].
2.2.1 Impaction
Impaction is mostly present in the upper airways due to the higher air velocities compared to the peripheral lung regions [5]. Impaction comes from when a particle sticks to its original trajectory in the airways, instead of following the curvature of the actual airway. For this phenomenon to occur, the particles momentum needs to be high enough for the centrifugal forces to force the particle out from the turbulent airstream and impact on the nearby airway walls. This is mostly present in the first 10 generations of the airways where the speed is sufficiently high and the flow is predominately turbulent [15]. Impaction also occurs more for larger particles and for fast breathing, due to the increase in air flow velocities [5].
2.2.2 Sedimentation
The slower air velocities in the distal regions of the lung lead to longer residence times. As the longer residence times allow the particles with sufficiently high masses to deposit due to gravitational forces, de-position due to sedimentation will predominantly occur in the last five generations of the bronchioles [15]. Clearly, it also follows that a slower breathing pattern will increase the deposition due to sedimentation as the decreased air velocity leads to longer residence times [5].
2.2.3 Brownian diffusion
The particle deposition due to diffusion is the main deposition mechanism in the lower airways and in the alveoli. The particles are moving around randomly in the inhaled air and are depositing on the walls of airways, this motion of the particles is called Brownian motion [16]. Brownian motion is a stochastic process of particle motion suspended in a fluid. The particle, which is much larger than an air molecule, will hit the air molecules repeatedly and will then appear the be moving randomly in the air. Diffusional deposition is more prominent for smaller particles with a diameter of 0.5 µm or less. If the particles are even smaller, down to nm size, the particles will also be trapped in the upper airways.
2.3
Respiratory inductance plethysmography
RIP is a non-invasive method to measure lung volumes and is performed by measuring the movement of the chest and abdomen induced by breathing. Many breathing parameters can be extracted using RIP, such as respiratory rate, tidal volume, peak inspiratory/expiratory flow and work of breathing index.
The RIP system consists of an elastic band with a coiled wire inside. This belt is worn either around the chest or abdomen (single-band system) or both (dual-band system). An alternating current is applied to the coiled wire, which will create a magnetic field. The RIP system works of the principle from Faraday’s law [21], that a current through a loop will generate a magnetic field orthogonal to the orientation of the loop. The change of area of the enclosed loop will then generate an opposing current proportional to the change of area of the loop according to Lenz’s law ("The direction of the induced current is such as to create a magnetic field which opposes the change of magnetic flux" [21]). When breathing, the raising and lowering of the chest/abdomen will result in a change of the cross-sectional area of the subject’s body and therefore the enclosed area of the loop. This change in area will thus change the magnetic field that is generated by the loop and the change will then induce an opposing current in the wire that can be measured. This is usually done by measuring the change in frequency of the applied alternating current [20].
The signals generated both from the chest and abdomen can either be presented independently or as a mathematical summation between the two. To handle amplitude differences, the signals are usually normalized before the summation. The summed RIP signal is a good measurement of the subject’s breathing, a typical RIP signal can be seen in Figure 2.4, but different factors may affect the quality of the signal. For instance how firmly the belt is attached can play an important role, if too tight the belt itself can restrict the breathing and therefore the area change of the loop. If the belt is too loose the belt can start sliding and the two bands can even overlap. Also the belt placement is of great importance, if placed down at the hips for example, very little change in area will occur when breathing [20].
Time [s] 0 5 10 15 20 25 Volume[ml] -60 -40 -20 0 20 40 60 80 Filtred RIP-signal
Chapter
3
Material and methods
The approach of building the models started off by defining a geometry and with that starting point trying to find expressions for the deposition of drug particles/aerosols in the different generations of the lung tree. Due to lacking data from the deposition in different lung generations quite a few approximations had to be done. All approximations in the models were either confirmed by literature or were deemed to be reasonably coherent with anatomical and physiological knowledge of the airways. From here on monkey will refers to both rhesus and cynomolgus monkeys, because literature shows no major difference between the species. In all the equations, SI units is assumed unless stated otherwise.
3.1
Airway geometry
In the models, the airway geometry of the monkey and the deposition in the nose were given from [3]. Unfortunately this article was the only one that provided a full description of the airways and the article only included data from one monkey (six months old male weighing 1.79 kg). This meant that the lung geometry had to be scaled according to body weight (BW) and breathing parameters to be able to handle monkeys of different sizes. The breathing parameters used were total lung capacity (TLC), functional residual capacity (FRC), tidal volume (VT), upper respiratory tract (URT) minute volume and breathing
frequency. The breathing frequency was determined to be independent of BW and sex and was set to be 39 breaths/min. The other parameters were scaled according to BW with the following equations:
Minute volume = −0.44051 + 3.8434Log(BW ) (3.1)
For males with a BW ≤ 4 kg.
Minute volume = −2.5302 + 1.7744BW − 0.15866BW2 (3.2)
For females with a BW ≤ 4 kg.
Minute volume = 1.9108 − 24.7378e−1.2479BW (3.3)
For both sexes with a BW > 4 kg.
TLC = −51.304 + 104.02BW − 3.6788BW2 (3.4)
FRC = −52.593 + 68.651BW − 2.2103BW2 (3.5) For both sexes with a BW ≤ 15 kg.
VT = −23.818 + 26.093BW − 2.1946BW2 (3.6)
For males with a BW < 5.5 kg.
VT = −61.578 + 45.391BW − 4.3842BW2 (3.7)
For females with a BW < 4.5 kg.
VT =
1000 ∗ Minute volume
Breathing frequency (3.8)
For males with a BW ≥ 5.5 kg and for females with a BW ≥ 4.5 kg [3].
The lungs are modeled like cylindrical tubes, with a given length and diameter representing the air-ways and each airway is then branching out to two daughter branches in adjacent generations. An "airway" refers to one cylinder in the model, with a given length and diameters depending on the airway generation. If summing up the volume of each cylindrical shaped airway in the model, calculating them with the corresponding length and diameter, the total calculated volume would be be far less than the real volume. This is because the cylindrical volume only makes up a part of the total lung volume, as the alveoli volume is unaccounted for by this cylindrical airway structure. The alveolar volume is therefore added to the volume of the last seven generations [3]. The method of adding the alveolar volume was developed by Weibel [24] and is dependent on the number of alveoli in the given generation. After adding the alveolar volume in a way that the accumulative volume of the lungs equals TLC, the volumes are subsequently scaled to reflect the lungs at FRC, because a typical breath starts at FRC [24]. This was done for the monkey lung tabulated in [3] in order to create a "normalized" lung or "lung zero". The produce of adding and scaling was then preformed again to re-scale "lung zero" according to BW.
To further scale the morphology to be more coherent to reality the length and diameters are scaled with a factor fARLV given by:
fARLV = F RC + 0.5VT T LC 1 3 (3.9) and the volumes were scaled by α, according to:
α = F RC + VT
F RC (3.10)
The scaling fARLV is used because the dimensions of the airways given in [3] are acquired at, or near,
TLC with alveolar volumes added. This will result in a morphology that will not correspond to a realistic respiratory lung volume. By scaling the dimensions by fARLV and α, you get an average respiratory lung
volume over an entire breathing cycle. Also the URT was subtracted from the ingoing tidal volume to account for the volume trapped in the nose and will therefore not effect the lung deposition.
3.2
Static model
The static model was developed by building up a lung morphology using the data from the article by [3], and implement the scaling according to BW with the equations given in section 3.2.1 and adding the corresponding alveolar volumes [3]. To get a more realistic lung volume over an entire breathing cycle the airway morphology was subsequently scaled with fARLV and α. When a suitable morphology had
been created, the particle loss in the nose was modeled. Due to the high flow velocity in the nose, the two dominating factors for particle loss are inertial impaction for particle sizes of 1µm > and Brownian diffusion for particles < 1µm.
The particle loss due to Brownian diffusion was given by Yeh et al [10] and then modified by [3] to enable scaling according to BW:
ηd= 1 − e −13.3 S/V S0/V0 −0.219 D0.543Q−0.219 (3.11) where ηdis the particle loss efficiency due to Brownian diffusion in the nose, D is the particles diffusion
constant in cms2 and Q is the flow rate in litersmin . S/V is the surface-to-volume ratio of the BW of the monkey that is evaluated. The S0/V0 is the S/V ratio of the measurements done by [10] (a monkey with a
BW of 8.5 kg) and both ratios is calculated by equation:
S/V = 6.23 + 30.306e−0.2658BW (3.12)
The particle loss due to inertial impaction was given by Kelly et al [11], and yet again the equation was modified to enable BW scaling [3]:
ηi = 1 − e − 3.227∗10−4 S/V S0/V0 ρd2Q !2.162 (3.13) where ηiis the deposition efficiency due to inertial impaction, S/V and S0/V0is calculated in the same
way as for the Brownian diffusion but S0/V0was calculated for a monkey with a BW of 10 kg from [11].
ρ is the particle density given incmg3, d is the particle diameter in µm and Q is the flow rate in
cm3
s [11].
Even though the two different particle loss processes occur primarily for different particles sizes, it is safe to assume the net particle loss, e.g. the fraction of particles that gets trapped in the nose, can be written as the sum of the two particle loss processes [3]:
ηnet = ηd+ ηi (3.14)
where ηnetis the total deposition efficiency in the nose.
The ηnetcan also be written as [11]:
ηnet = 1 −
Cout
Cin
(3.15) where Cinis the concentration of the air inhaled through the nostrils and Coutis the concentration that
will travel down into the respiratory tree. By rearranging Equation 3.15 one can calculate Cout:
This calculated Coutis then used as input in the dose calculations in the lungs.
The rest of the airway system (from the trachea and down to the alveoli) were modeled by using components from the models described by Lee et al [4] and Schmid et al [7]. The basic idea is to calculate a probability that a particle will be deposited in a given lung generation, if not deposited in a generation, the particle will simply exit the lung during exhalation. The probability of deposition will depend on three different factors; 1) diffusion of particles, 2) inertial impaction, and 3) gravitational sedimentation in the airways. Together these factors contribute to a probability given by:
Pi = 1 − (1 − DIFi)(1 − IM Pi)(1 − SEDi) (3.17)
The calculations for the gravitational sedimentation (SEDi) was given by [7]:
SEDi=
2
π(2(1 −
2/3)1/2− 1/3(1 − 2/3)1/2+ arcsin(1/3)) (3.18)
where is given by:
= 3vgticosφi 4Di
(3.19) where ti is the mean residence time of the air in given airway generation i, φiis the angle that forms
between the tube in the given airway and the gravity of the earth. Di is the diameter of the airways
in generation i and vg is the settling velocity of a particle due to gravitation and is described by
Equa-tion 3.20 [4]:
vg =
ρpd2gC(d)
18η (3.20)
where ρp is the inhaled particle density, d is the particle aerodynamic diameter, g is the gravitational
acceleration constant, η is the viscosity of air at ambient conditions. The difference from the equation in [4] is the use of a sin term instead of cos, that is due to the rat’s lung orientation with respect to the gravitation field of the earth and C(d) is the Cunningham correction factor given by Equation 3.21 [4]:
C(d) = 1 + λ d 2.514 + e−0.55dλ (3.21) where λ is the mean free path of air molecules at ambient conditions [4].
The deposition due to inertial impaction in the airways (IM Pi) was described as:
IM Pi = 0.768θiStk (3.22)
where Stk is the Stokes number of the flow given by Equation 3.23 and θi is the bend angle, which is
given by Equation 3.24 [23]. Stk = ρ0d 2u iC(d) 9ηDi (3.23) where ρ0is the unit particle density and ui is the flow velocity in the given airway generation.
θ = Li
4Di
The deposition due to Brownian diffusion during inhalation and exhalation (DIFi) is given by [4]:
DIFi= 1 − 0.819e−14.63µ− 0.0976e−89.22µ− 0.0325e−228µ− 0.0509e−125.9µ
2/3
(3.25) where µ is given by Equation 3.26 for inhalation and exhalation.
µ = DmolLi uiD2i
(3.26) where Dmolis the Brownian diffusion constant calculated by Equation 3.27, Liis the length of the airway
in a given generation [4] and uiis the mean velocity of the air in a given airway [7].
Dmol =
kT C(d)
3πηd (3.27)
kis the Boltzmann constant and T is the temperature in Kelvin [4]. For breath holding, the deposition due to Brownian diffusion is given by:
DIFi = 1 − e
−5.784kT Ct
6πµdi2D2i (3.28)
where t is the breath-hold time in a given airway.
By using these factors for deposition, one can calculate the probability that a particle will deposit in a given generation during the three different parts of the breathing cycle; 1) inhalation, 2) breath holding and 3) exhalation according to Equation 3.17. With these probabilities for deposition, one can calculate the deposition fractions for the different parts of the breathing cycle.
The deposition fraction during inhalation (DE(i)in) is given by:
DE(i)in = fiPiin imax
X
j=i
Vj (3.29)
where i denotes the airway generation, V the total volume of the airway generation, imax is the last ventilated airway generation, e.g. the last generation penetrated by the tidal volume and the factor f is given by [4]: fi = i−1 Y j=1 (1 − Piin) (3.30)
The factor f is the fraction of the aerosols in the inhaled air that travels to a given generation without being deposited [22]. Deposition during breath holding (DE(i)bh) is given by [7]:
DE(i)bh= fi(1 − Piin)PibhVi (3.31)
And finally the deposition during exhalation (DE(i)ex) is given by:
DE(i)ex = fi+2Vi+1Piex+ Piex imax X j=i+2 fj+1Vj(1 − Pjb) j−1 Y l=i+1 (1 − Plex) (3.32)
By summing up these deposition factors for all generations of the lung and for the three different parts of the breathing cycle, one can get a total deposition factor (DEi) for the lung for that particle size [4]. This
DEi = DEiin+ DEibh+ DEexi (3.33)
By using the calculated deposition factors, one can estimate the deposited dose in the lungs during a selected time period. The estimated dose is dependent on the concentration that enters the lungs, the deposition factor calculated in Equation 3.33 and the minute volume. The estimated dose is calculated according to Equation 1.1 [7], which also can be written as:
Dose = DE ∗ Cout∗ M inutevolume (3.34)
3.3
Dynamic model
The basic idea of the dynamic extension of the deposition model was to account for differences in the breathing pattern during the exposure time. Because the deposition of the lungs is highly dependent on the breathing pattern, it is safe to assume that the deposition will vary with varying breathing parameters. Changes in flow rate and tidal volume will have a big impact on the deposition fraction. The static model does not account for asymmetry in the breathing, e.g. if the inhalation is faster than the exhalation etc. In order to make the model dynamic, one need measurements of the breathing pattern to use as input to the model. RIP bands are often used to measure the breathing in monkeys during inhalation studies and the measurements from RIP bands are used in this model to make it dynamic. The dynamic model is actually the static model that is run over and over again with different breathing parameters as input for each run. Such RIP data have been collected by AstraZeneca with emka RIP band system and processed with thier iox2 software, the data was then imported into MATLAB. The data needed to be calibrated and the calibration coefficient was extracted from emka’s software ECGauto.
The relevant breathing parameters to be extracted were the tidal volumes for inhalation and exhala-tion, breath holding time and flow rate for inhalation and exhalation. To enable extraction of these parameters, the signal had to be processed and filtered to reduce noise and artifacts. The signal has a sample frequency of 200 Hz, but the relevant information, the breathing, has a frequency around 1 Hz. This means that a lot of the samples carries the same information. In order to avoid unnecessary calculations, the signal was decimated/down sampled to a frequency of 40 Hz.
To account for some movement artifacts and baseline wandering of the signal, the signal was high-pass filtered using a high-order high-high-pass filter with an empirically decided cut-off frequency to get a satisfactory result, e.g a result which minimized the noise without loosing any information. The built in convolution function filtfilt in MATLAB was used to ensure that no phase-shift was introduced in the signal. The same procedure was used to remove the high frequency noise that is partly introduced due to movement of the monkey but also due to bad placement and/or movement of the bands, e.g. sliding down. The high-pass filtered signal was filtered again by a high-order low-pass filter with an empirically decided cut-off frequency which produced a satisfactory result, e.g a result which minimized the noise without loosing any information. Yet again the MATLAB function filtfilt was used to ensure that no phase-shift was introduced in the signal. An example of a filtered signal can be seen in Figure 2.4.
The idea behind the extraction of breathing parameters can be seen in Figure 3.1 which shows a typical breath from the RIP data. The result of the filtering processes can be seen in Figure 3.2a and 3.2b.
Samples 0 200 400 600 800 1000 Volume[ml] -80 -60 -40 -20 0 20 40 60 80 High-pass filtering
(a) An example of the high-pass
filtering of the RIP signal, the red signal is after filtering and the blue is before the filtering.
Samples 0 200 400 600 800 1000 Volume[ml] -80 -60 -40 -20 0 20 40 60
80 Low pass filtering
(b) An example of the low-pass
filtering of the RIP signal, the red signal is after filtering and the blue is before the filtering.
Figure 3.2: Schematic figures over the filtering.
To be able to extract the relevant breathing parameters from the filtered RIP signal, the MATLAB command findpeaks was used to find the points in the signal that are marked by the circles and crosses in Figure 3.1. The circles and crosses denote the start and end of inhalation of a breath, and start and end of the exhalation of a breath. By looking at the amplitude differences between the start of inhala-tion/exhalation and the end of inhalainhala-tion/exhalation one can calculate the tidal volume going in and out the lungs. The idea behind the calculation of tidal volumes can be seen in the spirogram shown in Figure 2.2.
To extract the flow rate of the inhalation and exhalation, a gradient vector between the start and end of inhalation/exhalation was created and the flow rate was set as the mean of the gradient vector. The last parameter to be extracted was the breath holding time between inhalation and exhalation. In order to calculate the breath holding time, the difference between the ten adjacent samples from both sides of the peak at end of inhalation were investigated. As long as the difference between two adjacent samples was less than 2 mL, the signal were considered to be flat. Meaning that there is no change in volume of the lungs, e.g. the subject is holding its breath. By measuring the number of samples when the signal is flat, one can decide the breath holding time by converting the number of samples to seconds.
Some error handling were implemented to account for when the calculated breathing parameters took unreasonable values. This was achieved by checking if either the flow rates or tidal volumes were unreasonably high or low. If one of theses parameters were outside the defined normal range of values, the whole breath associated with that parameter was deemed bad and replaced with a predefined standard breath. This was to ensure that no unrealistic asymmetry of a breath occurred.
When the relevant breathing parameters had been extracted from the RIP signal, these were used as input to the rest of the model. Firstly the nose deposition was calculated, and the calculations were preformed in the same way as in the static model. Two loss factors were calculated, one depending on impaction (Equation 3.13) and one depending on diffusion (Equation 3.11), and the outgoing concentration was subsequently calculated according to Equation 3.16. The nose deposition, or nose filtering, is highly dependent on flow rate, which means that the filtering efficiency of the nose will vary for the same particle size, but with different breathing parameters. By running the nose algorithm for each breath with the corresponding flow rate, and for all simulated particle sizes, the generated output from the nose filtering will be a 2D-array with outgoing concentration for each breath and for each particle size. This array was later used to predict the deposited dose in the lungs.
The deposition fraction is clearly dependent on the airway geometry and the airway scaling of TLC and FRC according to BW is the same as in the static model. However, since the other breathing pa-rameters that are used as input to the model are varying between each breath, the airway geometry will be scaled slightly different for each breath. This is evident from the equations describing the scaling factors (Equation 3.9 and Equation 3.10), where the breath-varying parameter Vtis used as input. All
these different scaling factors are used in the scaling for each breath. Also the tidal volumes needs to be adjusted to account for the part of the tidal volume that gets trapped in the URT. The variation in lung geometry because of the variation of breathing parameters will of course have even more impact on the deposition fraction. Not only will the RIP data introduce variations in the lung geometry between breaths, but also, the variation in breathing pattern will effect the impaction, sedimentation and diffusion in the lungs. Both the tidal volume and the flow rate influence the deposition mechanism. The tidal volume decides how far the particles will penetrate in the lungs, and the flow rate decides the particles velocity in the airways.
When all the scaling factors and breathing parameter vectors were established, all of them were used as input to the actual deposition fraction calculation. The deposition calculation itself is more or less identical to the static model, but now the algorithm is run over and over again for each breath and each particle size. Each simulated particle size is run through the model with the varying breathing parameters and scaling factors for the corresponding breath. This means that the static model is run for each breath and each particle size with the corresponding breathing parameters. Hence, this results in a rather complex output. Rather than to get one deposition fraction for each part of the breathing cycle, and one total for each generation of the lung tree, the resulting output instead becomes several outputs for each breath. The basic structure of the output can be seen in Figure 3.3.
The deposition algorithm will generate three 3D-arrays, one for each part of the breathing cycle, with deposition fractions. The rows in the arrays corresponds to a lung generation, the columns corresponds to each breath and finally the different pages of 2D-arrays correspond to the different particle sizes simulated. These arrays are what makes the model dynamic. Since the model works breath by breath, one can use these arrays and reduce the Equation 1.1 to the following:
Dose = DDepos(t)CA(t) (3.35)
By incorporating the time dependence in the concentration and in the deposition fraction, there is no need of having the Vt, exposure time and fminin the dose calculations. By subsequently manipulating these
3D-arrays, one can choose which parts of the lung or at which time one wants to look at the deposition fraction.
Chapter
4
Results
This section contains a selection of outputs from both models. The outputs have been chosen to emphasize the differences between the static and the dynamic model. Also, a partial validation of the output will be presented. The deposition fractions in the graphs are assumed to be with endotracheal breathing unless stated otherwise. Endotracheal breathing means that the particle loss in the nose is not accounted for, the model only considers air going in to the trachea. The other mode of breathing is spontaneous nasal breathing which is the most common way of breathing in monkeys, meaning that the breath through their nose and particle loss in the nose is accounted for.
4.1
Static model
The output from the static model was simulated with a BW of 1.79 kg and with breathing parameters scaled according to BW. The BW was chosen to resemble the monkey, which was used to generate the morphology data in [3]. By doing that, one can validate the output by comparing the results from that paper.
Figure 4.1 and 4.2 show regional deposition fractions for a variety of particle sizes. The graphs show the fraction of particles of a given sizes that will deposit in a given generation. Figure 4.3 and 4.4 show the total deposition for a spectrum of particles sizes for the whole lung, the TB region and PUL region.
Lung generation 0 5 10 15 20 25 Deposition fraction 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09
0.1 Deposition during inspiration
0.1 µm 0.5 µm 1 µm 2µm 3µm
(a) Deposition fraction during inspiratory phase in the different lung generations.
Lung generation 0 5 10 15 20 25 Deposition fraction 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035
0.04 Deposition during breath holding
0.1 µm 0.5 µm 1 µm 2µm 3µm
(b) Deposition fraction during breath holding in the different lung generations.
Lung generation 0 5 10 15 20 25 Deposition fraction 0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018
0.02 Deposition during expiration
0.1 µm 0.5 µm 1 µm 2µm 3µm
(a) Deposition fraction during expiratory phase in the different lung generations.
Lung generation 0 5 10 15 20 25 Deposition fraction 0 0.02 0.04 0.06 0.08 0.1
0.12 Total deposition during the breathing cycle 0.1 µm 0.5 µm 1 µm 2µm 3µm
(b) Total deposition fraction
in the different lung generations. Figure 4.2: Graphs over the regional deposition for expisation and the total deposition.
Particle diameters [m] 10-8 10-7 10-6 10-5 Deposition fraction 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
1 Deposition factor for different particle diameters
Particle diameters [m] 10-8 10-7 10-6 10-5 Deposition fraction 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Deposition factor for different particle diameters, tracheobronchial (TB) region
(a) Total deposition fraction for different
particles sizes in the tracheobronchial (TB) region.
Particle diameters [m] 10-8 10-7 10-6 10-5 Deposition fraction 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35
Deposition factor for different particle diameters, pulmonary (PUL) region
(b) Total deposition fraction for different particle sizes in the pulmonary (PUL) region. Figure 4.4: Graphs over the total deposition for different regions of the lung.
4.2
Dynamic model
The outputs from the dynamical model were simulated with a BW of 3 kg, because it is similar to the average weight of the monkeys used in the study were the RIP data was extracted from. It was also simulated with a BW of 1.79 kg and with static breathing parameters from the static model to highlight the differences between the models. Figure A.1 to Figure 4.12 show the regional deposition for different particle sizes over time, during the different parts of the breathing cycle.
4.2.1 Regional deposition for 1 m particles
25 20
Deposition fraction in different generations for 1 µm particles, during the inspiratory phase
15 10 Lung generation 5 0 25 20 15 Time [s] 10 5 0.03 0.025 0.02 0.015 0.01 0.005 0 0.035 0 Deposition fration 0 0.005 0.01 0.015 0.02 0.025 0.03
Figure 4.5: Deposition fraction over time in different generations of the lung for 1 µm particles, during the inspiratory phase.
25 20
Deposition fraction in different generations for 1 µm particles, during breath holding
15 10 Lung generation 5 0 25 20 15 Time [s] 10 5 2 4 6 8 0 0 ×10-3 Deposition fration ×10-3 0 1 2 3 4 5 6 7
Figure 4.6: Deposition fraction over time in different generations of the lung for 1 µm particles, during breath holding.
25 20
Deposition fraction in different generations for 1 µm particles, during the expiratory phase
15 10 Lung generation 5 0 25 20 15 Time [s] 10 5 0.03 0.025 0.02 0.015 0.01 0 0.005 0 Deposition fration 0 0.005 0.01 0.015 0.02 0.025
Figure 4.7: Deposition fraction over time in different generations of the lung for 1 µm particles, during the expiratory phase.
4.2.2 Whole breathing cycle
25 20
15
Total deposition fraction in different generations for 0.1 µm particles
Generation 10 5 0 25 20 15 Time (s) 10 5 0.07 0.02 0.06 0.05 0.08 0.01 0 0.04 0.03 0 Deposition fration 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07
Figure 4.8: Total deposition fraction over time in different generations for 0.1 µm particles.
25 20
15
Total deposition fraction in different generations for 0.5 µm particles
10 Lung generation 5 0 25 20 15 Time [s] 10 5 0.01 0.015 0.02 0.025 0 0.005 0 Deposition fration 0 0.005 0.01 0.015 0.02
25 20
15
Total deposition fraction in different generations for 1 µm particles
10 Lung generation 5 0 25 20 15 Time [s] 10 5 0.03 0.05 0.02 0.01 0 0.06 0.04 0 Deposition fration 0 0.01 0.02 0.03 0.04 0.05
Figure 4.10: Total deposition fraction over time in different generations for 1 µm particles.
25 20
15
Total deposition fraction in different generations for 2 µm particles
10 Lung generation 5 0 25 20 15 Time [s] 10 5 0 0.02 0.04 0.06 0.1 0.12 0.14 0.08 0 Deposition fration 0 0.02 0.04 0.06 0.08 0.1 0.12
25 20
15
Total deposition fraction in different generations for 3 µm particles
10 Lung generation 5 0 25 20 15 Time [s] 10 5 0.15 0.1 0.05 0 0 Deposition fration 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14
4.2.3 Deposition fraction for 2 µm particles over time
Figure 4.13 and Figure 4.14 shows deposition fraction over time for 2 µm particles. Both the total deposition fraction (e.g. the sum of the nose and lung deposition) and also the fraction of particles that gets trapped in the nose and in the lung.
Time [s] 0 5 10 15 20 25 Deposition fraction 0.7 0.72 0.74 0.76 0.78 0.8 0.82 0.84 0.86 0.88
0.9 Total deposition fraction
Figure 4.13: Total deposition fraction for 2 µm particles over time.
Time [s] 0 5 10 15 20 25 Fraction 0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24
0.26 Fraction of particles deposited in the nose
(a) Fraction of 2 µm that gets stuck in the nose over time.
Time [s] 0 5 10 15 20 25 Fraction 0.6 0.62 0.64 0.66 0.68 0.7
0.72 Fraction of particles deposited in the lung
(b) Fraction of 2 µm that gets stuck in the lung over time. Figure 4.14: Graphs over the fraction of particles that gets trapped in the nose and lungs.
Chapter
5
Discussion and analysis
Both models are capable of modeling the particle deposition in a sufficient way, but the two approaches yield different results. Therefore an evaluation gives a deeper understanding of the output and how the output is generated, and also if the output and the models behavior is coherent with reality. Because of the usefulness mathematical dosimetry models in drug development and the increased usage of inhalation as a drug delivery method, a well developed model can be ground breaking. With a dynamical extension of the deposition models on can get more precise prediction of the particle deposition and therefore a better understanding.
5.1
Static model
The static model is the more conventional way of modelling particle deposition and this kind of models are already implemented for different species such as [4][7] for rodents. The logic used in these models has also laid the foundation for the models created specifically for monkeys. The model describe all parts of the lung as cylindrical tubes, which the particles deposit in, either by impaction, sedimentation or diffusion. Since the models in [4] and [7] are built upon deposition of particles in cylindrical tubes, it follows that the same equations with some modification can be implemented for deposition in monkeys as well. The equations are well known and heavily cited, which strengthens the use of them in the model.
The model is more or less divided into two parts. One part that calculates the concentration of par-ticles in the air that goes down in the the trachea. Meaning that the algorithm calculates the filtering efficiency of the nose, i.e. the number of particle that gets trapped in the nose. The two factors for particle loss in the nose, diffusion and impaction, are both empirically derived by simulating the airflow in casts of the URT of monkeys. For the studies that evaluated the URT deposition in monkey, an animal with a BW of 12 kg [11] and one with 8.5 kg BW [10] was used for the impaction and diffusion study, respectively. As the constants in the Equations 3.11 and 3.13 are empirically derived, the BW scaling implemented from [3] might not be as accurate when interpolating down to a BW of around 3 kg for example. The two factors that impact the filtering in the nose are dominant for different particle sizes. Diffusion is present for particles < 1 µm and inertial impaction is present for particles > 1 µm. But there is a "grey area" where both the diffusion and impaction is very low and the nose filtering are very low in total, e.g. the majority of the particles will travel down to the lungs. The "grey area" is roughly between 0.1 µm - 2.5 µm.
BW scaling of the lung morphology. The same goes for the breathing parameters such as TLC, FRC etc. Because of the limited sample size, the breathing parameters were only measured in 121 monkeys, the regression analysis has limited strength. Also when constructing the airways in the model, the alveolar volumes are added according to Weibel’s method [24] and then it is assumed that the branching pattern and alveolating pattern is the same in monkeys as in humans. This assumption is fairly good because similarities has been documented between the lungs in maraques and humans [14]. This limitation in the morphology and scaling to other BW is one of the Achilles’s heel of the model. But as of today there are not any more available data and one has to rely on this kind of interpolation and the assumption of similarities in humans, and therefore this is the best solution at the moment. The nose filtering has the same limitation as the equations need to be scaled to a lower BW, the same problems occur when scaling the breathing parameters. Since the regression analysis was performed on a data set collected from 121 monkeys with a BW < 4 kg, the scaling if the breathing parameters might be less accurate at higher BWs.
Despite these limitations in the airway morphology and scaling, the model provides satisfactory re-sults and the model’s behavior corresponds to the expected behavior in reality. This can be highlighted in Figure 4.1 and 4.2. As one can see in the graphs the larger particles deposit in the upper generations and the smaller particles in the lower generations. This is an expected behavior because impaction is more present in the upper generations, and larger particles are also more prone to impact in the upper bifurcations. With the same logic, one can expect that the smaller particles are more prone to deposit in the lower generations since diffusion is the predominating deposition factor in the lower generations and also the smaller particles are very prone to diffuse.
Another thing that is worth mentioning is the high differences in deposition fraction in the different parts of the breathing cycle. These differences are mainly caused by the different deposition factors that dominate different parts of the breathing cycle. During inhalation, impaction will dominate the upper generations due to the high air velocities and relatively many bifurcations. As the air travels down in the lung tree, the air velocity decreases and therefore the deposition due to impaction will decrease as well. The decreasing air velocity will also increase the deposition due to gravitational sedimentation. This is because the streamlines in the airways will be weaker meaning that the gravitational force will be able to pull the particles down towards the walls of the airways. Also at increasingly lower generations, the airway diameter decreases which means that the distance the gravitational force need to pull the particle will be shorter and shorter. When the air reaches the most distal airways, diffusion will be the main deposition mechanism and therefore one can see that the deposition fractions are higher for smaller particles because they are more prone to diffuse.
The deposition during breath-hold is a lot lower than for inhalation and the regional deposition pattern is different as well. The regional deposition pattern can be seen in Figure 4.1b and one thing to mention is that almost no particles deposit in the upper generations. This is because in the upper generations, impaction is the dominating deposition mechanism and during breath-holding the air stands still in the airways. This means that the particles will not flow by the bifurcations, therefore no deposition will occur due to impaction. Therefore sedimentation and diffusion are the only factors that influence the deposition. As a result, the deposition during breath-hold is highly dependent on the tidal volume and the breath holding time. The tidal volume decides how deep the air will penetrate in the lungs and since diffusion and sedimentation is more present in lower generations, the deposition during breath-hold will increase with higher penetration. Also with longer breath-holding time, the deposition will increase because of the particles have more time to sediment and diffuse before the air turns and the exhalation start. Clearly, another factor contributing to the lower deposition during breath-hold and exhalation, it that many particles have already deposited during the inspiratory phase. With that exception, the same logic applies to exhalation, e.g. larger particles are more prone to deposit in the upper generations and vice versa.
Another common representation of the output is to look at the total deposition fraction over a spec-trum of particle sizes. As can be seen in Figure 4.3, this motivates the selection of particle sizes for the simulations shown in Figure 4.1 and 4.2. In the particle size region of 0.1 - 3 µm, the deposition fraction changes a lot and therefore it is more interesting to look at, this is highlighted in Figure A.13. Also in a typical inhalation study in monkeys, the size range of the inhaled particles is roughly between 1 - 3 µm. Different deposition mechanism are expected to be dominant for different particle sizes, which clearly was reflected in the models output. Because as the particles get smaller, they are more prone to diffuse. Hence, as the particles get smaller , the deposition increases and this can be seen for 0.1 µm in Figure 4.2b. The same logic applies for larger particles, because larger particles are more prone to impact and therefore the deposition fraction will increase as the particle size increase, this can be seen for 3 µm in Figure 4.2b.
To validate the model, Figure 4.4 was generated showing the the deposition fraction for different regions of the lung for endotracheal breathing. This Figure, when compared to the Figures A.23 from [3], it shows good agreement. The differences might be caused by the different modelling approach that was used in the article, where they use mass conservation equations. This comparison serves as validation of the static model, i.e. it follows the expected behavior.
5.2
Dynamic model
As mentioned before, the dynamic model is basically the static model run over and over again with varying breathing parameters. This means that the particles will exhibit identical behaviours in the two models. The key difference is that the dynamic output has a third dimension; time. Rather than to have a static output for a given generation and particle size, the deposition factor will vary over time. As one can see in Figure A.1, the overall shape of the 3D-plot is similar to the ones found in Figure 4.1a. As stated previously, the crucial difference is the variation in time, which can be seen in the 3D-plots in the appendix. The many peaks that are shown in the graph are caused by the varying breathing parameters, which are derived from the RIP data. These peaks depend on parameters such as tidal volume that decides how deep the air will penetrate in the lung and therefore the deposition in the lower generations. The peaks that can be seen in the upper generations in Figure A.2 depend on the flow rate, as a higher flow rate will increase the deposition due to impaction in the upper generations. One interesting thing to highlight is that if there is a peak in the upper generations, there is also a valley in the lower generations. This is because that higher flow rate will increase the impaction, but also decrease the sedimentation and diffusion because less particles will reach the distal airways.. Also the deposition fraction is higher in the dynamic model, mostly because in the static model the breathing parameters are derived from anethezised animals. Anaesthesia is know to alter the breathing pattern [3] and these animals are expected to breath slower and not take as big breaths as animals that are awake and also in a stressful situation. Also the dynamic output where simulated with a BW of 3 kg, which can contribute to the differences.
During breath-holding one can clearly see the impact of different breathing parameters, in this case the breath-holding time. In the Figures that show the breath-holding, e.g. Figure 4.6 and the ones in teh appendix A.8, one can see a distinct peak at the same location for all particle sizes. It is because the breath-holding time is longer in that particular breath and therefore the deposition fractions are higher for that breath. If one looks at the amplitude of the breath-holding graphs, the deposition fraction is very low and the breath-holding will not have as much impact on the total deposition. Nevertheless these graphs highlight the differences over time of the deposition fraction.
During exhalation, the same behavior seen in the static model for different particle sizes and differ-ent generations. The shape of the curves in Figure 4.7 are similar to the shapes found in Figure 4.2a, but with the same variations as can be seen during inhalation. As one can see in these graphs, peaks and valleys occur because of the varying breathing parameters from the RIP data. The same pattern observed during inhalation can also be found during exhalation. Breaths with high peaks in lower generations have a valley in the upper generations and vice versa, and this occurs for the same reasons as for inhalation. Also when looking at the amplitude of the exhalation graphs, one can see that amplitude here is lower than for inhalation, but higher than for breath-holding. This means that the main contributor to the total deposition fraction is the deposition which comes from the inhalation phase. This does not mean that the exhalation and breath-holding is redundant, they still contribute to regional deposition and should not be neglected. One can still see the contributions from exhalation and, for some particles sizes, the contribution from breath-holding in the graphs for total deposition, Figure 4.8 to 4.12. One can see that there is constructive interference between the different parts of the breathing cycle and together builds up the shape of the total deposition.
Clearly it is of interest in the dynamical model to check the total deposition fraction for endotracheal breathing for different particle sizes, this can be seen in Figure A.13. The surface of the 3D-plot is very uneven and this is caused by the varying breathing parameters. This is in line with the behaviour shown in the regional deposition graphs, where the deposition varies with time. It is thus logical that the output in Figure A.13 follows the same pattern. For the same reason as in the static model, it is of interest to divide the deposition fraction into the TB and PUL region to enable comparison of the output to the corresponding output generated in [3]. The overall shapes of the graphs in Figure A.15 and A.17 are very much alike the one found in [3], but with some amplitude differences for the different particles sizes. This difference is primarily dependent on two things: 1) like mentioned before, the model used to generate the output in [3] use equations based on mass conservation to calculate the deposition fraction, 2) the output in the article was simulated using breathing parameters from anesthetized animals, while in Figure A.15 and A.17 the breathing parameters were derived from the RIP data. Still the results from the paper and the dynamic model are similar and the output from the model is deemed reasonable.
