MODELING AND SIMULATION OF RESPIRATORY SYSTEM

Abstract

Respiration is the act of breathing, which is not possible without respiratory system. The respiratory system combines nose, throat, and lung. Lung 1 is the main organ, which performs the process of respiration. Airways inside lung bifurcate from the trachea to terminal airways (known as alveoli) in 23 generations and are collectively termed as tracheobronchial tree. Generations 0 to 15 take part to conduct air in other airways and generations 16 to 23 perform the process of gas exchange. When we inhale, gases such as nitrogen (78 %), oxygen (21 %), argon (0.965 %), carbon dioxide (0.04 %), helium other gases, and water droplets present in the air enter our lungs. The hemoglobin in red blood cells contains iron due to which oxygen is immediately absorbed whereas other gases remain inert with respect to the blood’s constituents, and are sent out during exhalation. Not only gases and water droplets but also other environmental particles with different shapes and sizes are inhaled with air and flow with air stream in different generations of the lung. Deposition of these particles causes various lung diseases such as asthma, chronic obstructive pulmonary disease, emphysema, fibrosis, stenosis, atherosclerosis, cancer, apnea, etc. Because of the structural change during respiration engineers, physicists, and mathematicians viewed lung as a mechanical system and performed theoretical, experimental and mathematical studies [39, 179, 204, 225] by applying sophisticated mathematical and computational methods over the past few decades. In respect of particle deposition inside lung generations, many studies [27, 79, 141, 224] have focused on various sizes and shapes particles. In most of the studies, microparticles have been considered, while some studies focused on nanoparticles (average size of particles are below 100 nm). But in real situations, many particles generated industrially are not necessarily spherical in shape. Many particles found in industries (cosmetic, textile, timber, material, grinding and polishing, etc), harvesting, coal mines, cigarette smoke, cooking smoke, incomplete combustion, organic fragments, fine hairs from plants and animals, and atmospheric dust differ in their flow characteristics and can 1Combines left and right lungs i be categorized as nonspherical nanoparticles. Due to the irregular shape and different flow characteristic of nanoparticles such as stretched spheroid, elongated or oblate, pollen, it is not yet possible to provide a single constitutive equation describing the properties of all shape nonspherical nanoparticles. Because of this fact, several theoretical and stochastic models of nonspherical nanoparticles have been proposed. To deal with the appropriate shape of nanoparticles, the ‘shape factor’ has attracted the attention of many researchers [62, 177, 218, 225]. This is because of the reason that the corresponding constitutive equation is comparatively less complicated and one can hope for the analytical and numerical solutions. In view of these facts, we consider the ‘shape factor’ of nonspherical nanoparticles in this thesis. In lung mechanics, one can use two types of elastic model i.e. linear or nonlinear. Many of the earlier studies have considered nonlinear elastic model, but some of the living tissues and cells are very soft and followed the linear postulate of elasticity that the stress tensor is directly proportional to the deformation tensor. When working on the pulmonary region of lung the elasticity is controlled by viscosity and named as viscoelasticity. In the pulmonary region, the gas is transferred not only by fluid movement but also by the mechanical movement of tissues. Several models [52, 139, 164] of viscoelasticity have been proposed in the literature. However, the simplest forms of the viscoelastic material are defined by Kelvin-Voigt which has attracted the attention of several researchers. So, the effect of viscoelasticity has been studied in the present work in order to show its effect on the flow of fluid through parenchymal tissues. Another way to makes the mathematical model more realistic is the use of the variable porous medium. Alveoli, tiny bubbles with an interconnected void, are supplies a porous region (alveolar region and parenchyma), where gas exchanges with the blood take place [42]. A few studies focused on the variable porosity of lungs. The interest in this topic has been stimulated, due to its three specifications: first, structural change in alveoli and lung volume during breathing; second, a number of alveoli changes with respect to age. Third, the irregular motion of the nonspherical nanoparticle around the individual alveoli responsible for deposition inside alveoli more effectively. So, variable porosity has an obvious influence on the flow pattern. Thus in the present work, studies are conducted involving variable porous media. Many researchers have focused on capillary flow by using conventional Krogh cylinder model [195, 204, 247] while a few have considered exchange of species in between capillary blood and surrounding tissue or intercapillary flow. For better understanding, the process of gas exchange from alveolar capillary to pulmonary capillary inclusion of intercapillary process is needed. ii Finally, the motivation behind the thesis is to make the mathematical model of lung mechanics more realistic, by incorporating the idea of variable porosity of lung, shape factor of nonspherical nanoparticle, viscoelasticity of parenchymal tissue, inter capillary gas transfer from alveoli to the pulmonary capillary. Numerical simulation of the problems is carried out by using conventional explicit finite difference method [188]. Results are displayed in the form of graphs in Origin (8.0) and behavior of fluid particles are visualized by images generated in MATLAB R2013 and R2016. The whole work of the thesis is divided into nine chapters and chapter-wise summary of the thesis is as follows: Chapter 1 is introductory in nature and gives a brief account of the general theory of lung mechanics, disease, human respiratory system, air flow in capillaries, deposition mechanism of various sizes and shapes nanoparticles, mathematical modeling and numerical solutions in biology. It presents the current status of the field, motivation to the investigations, a review of the available literature and tools/techniques used in the study. It also includes objective and brief ideas about various concepts used in the forthcoming chapters. In Chapter 2, we investigate the flow dynamics of needle prolate nanoparticle from generations 5-16 under periodic permeability of airways with oscillatory boundary conditions. An appropriate one-dimensional unsteady momentum equation in the cylindrical polar coordinate system is used by incorporating the idea of the shape factor of needle prolate nanoparticles. Filtration efficiency of the lung from generations 5-16 is also calculated using appropriate biofilter model. The effect of various physical parameters, such as mean permeability of media (K0), the aspect ratio of particle ( ), the orientation of particle with respect to the flow stream, Reynolds number (Re), and frequency of oscillation (f) are analyzed on the flow dynamics of air, particles and filtration efficiency of lung. Results show that the aspect ratio of a particle causes an increment in drag force and decrement in pressure gradient; and for parallel orientation velocity of particles increases than perpendicular orientation. Additionally, we obtained that the filtration efficiency of lung varies inversely with the value of mean permeability. In Chapter 3, we presented an age-based study of the human lung from childhood to the age of 30 by considering the growth of lung is caused by a progressive increment in a number of alveoli and calculated its effect on airflow dynamics and filtration efficiency of the human lung. Incorporating the idea of filtration through lung with respect to age biofilter model is extended for variable porous media by assuming that the porosity of lung varies with the number of alveoli and their surface area. Additionally, transportation and filtration properties of nanoparticles of various shapes during iii inhalation are calculated by using particle shape factor with and without orientation of the particles. Generalized Navier Stokes’ equation is used for flow dynamics of viscous air and Newton second law of motion is applied to study the flow of particles. Effect of aspect ratio ( ), the orientation of particles with respect to the flow stream, Darcy number (Da), and porosity ( ) are studied. Results indicate that the filtration efficiency of lung decreases as age increases from childhood to 30 years; additionally, nonspherical nanoparticles with high aspect ratio’s will take a longer time to be filtered from the lung as compared to spherical nanoparticles. In Chapter 4, we use a mathematical model to study the effect of homogeneous porosity of the alveolar region on the flow dynamics of viscous air by considering a pulsatile pressure gradient due to periodic breathing of respiratory system. Two-dimensional momentum equations are used with Darcy law to show the flow through porous media. An extensive quantitative study is performed through a finite difference technique to solve the governing equation. Effect of Reynolds number (Re), Darcy number (Da), porosity ( ) are studied on air and particle velocities inside the alveolar region. Results show that porosity is an active factor for deposition of nanoparticles and the fraction of particles trapped in the alveolus increases by decreasing the Darcy number. In Chapter 5, we consider lung as a porous permeable medium to build a mathematical model of early-stage emphysema. The computational geometry utilized in this study spans one generation of alveolar duct attached with alveoli. Two-dimensional generalized equation of momentum is used to study the flow of air and equation of motion for elongated shaped nanoparticles. Variable porosity is used by incorporating the idea of change in volume during ventilation together with a shape factor of elongated shaped nanoparticles. Various parameters such as inlet Reynolds number (Re), media porosity ( ), Darcy number (Da), breathing rate (f), and particle shape factor (Sf ) are varied to study the condition of the emphysematous lung. Results demonstrated that during inhalation, breathing stress increases and the deposition of particles is smaller due to the rupture alveoli in an emphysematous lung as compared to a healthy lung. In Chapter 6, we focused on the flow through an axisymmetric constricted artery of the pulmonary region to study the condition of stenosis. Theory of dust particles suspended in gas is applied on blood flow through the artery, where the ‘particles’ represent ‘cells’ suspended in plasma. The flow is governed by two dimensional Navier-Stokes’ equations by including Darcy-Forchheimer drag force caused by non-Darcian effect. The material of the artery is approximated as a linear elastic and simplest rheological equation that includes viscosity and elasticity (considered lung as a Voigt body) is used. Effect of various parameters, such as Reynolds number (Re), Forchheimer number iv (Fs), Darcy number (Da), aspect ratio ( ), shape factor (Sf ), porosity ( ), aerodynamic diameter (dp), bulk compression ( ) of elasticity, shear ( ) and bulk ( ) coefficients of parenchymal viscosity are obtained on the radial and axial velocities of blood and particles graphically. We found that the fluid (blood) and particle (cells suspended in plasma) velocities along both the axes (axial and radial) increase by increasing Reynolds number, the pulsating amplitude, aspect ratio, and porosity of walls. While by increasing Forchheimer number, velocities of blood and particles in both the axes decreases gradually. The present analysis is also indicate that the viscoelasticity of walls are affected by the amplitude of pulsatile flow of blood and for a large value of amplitude, the viscoelastic effect decreases. In Chapter 7, we analyzed the behavior of fluid flow through parallel walls, where both walls are porous and flow is induced by the oscillation of walls and sinusoidal pressure gradient. Concept of generalized Couette flow is applicable to model the flow of mucus through oscillatory walls, where porous walls are partially filled with fluid (mucus). Mathematical modeling of the flow of viscous fluid is done by using two-dimensional generalized momentum equations and flow of particle is done by using Newton equation of motion. Finite difference method is used to solve the problem numerically. Effect of wall oscillation frequency (f), wall porosity ( ), pressure (p), and particle aspect ratio ( ) are calculated to make various hypotheses related to clearance of mucus. Result show that high frequency of breathing is applicable to clearance of mucus from the airways. In Chapter 8, we worked on the effective area average concentration and dispersion coefficient associated with the unsteady flow, to understand the dispersion in the fibrosis-affected lung. We assumed that the tube wall (i.e. alveolar or pulmonary capillary wall) is thicker than its normal size due to fibrosis and chemical species may go through linear first-order kinetic reactions, one is reversible phase exchange with the wall material and other is irreversible absorption into the tube wall. By considering diffusivity as a function of thickness, the dispersion can be calculated by the distribution of concentration of gas along a tube. Mathematical modeling is done by using diffusion equation; and effects of various dimensionless parameters e.g., the Damkohler number (DA), phase partitioning number ( ), dimensionless absorption number (􀀀), thickness and permeability of wall are observed. Numerical simulation shows that the diffusion rate through the respiratory wall is decreased significantly as the thickness of wall increases, while it increased with the increment in the porosity of wall, the concentration of species increased when the tube wall thickness increases; which cause reduction in the spread of the species; additionally, the dispersion coefficient achieves the steady-state values in a very short time when 0 < Damkohlar number 1 and absorption rate < 1, while for 1 < v Damkohlar number 20 and absorption rate =1 long time dispersion is achieved. In the last Chapter 9, we present a mathematical model of spatial and temporal variations in inhaled gas partial pressure within the tissue and concentric blood capillary. Gas exchange between capillary blood and surrounding tissue is take place through the combined effect of convection and diffusion. We assumed capillary material is absorptive and reactive and the gas could bear linear first-order kinetic reactions, one is reversible among capillary material and other is irreversible into the surrounding tissue. Partial pressure distribution of gas between immobile (tissue) and mobile (blood) phase (through the interface) is calculated with the help of Aris method of moments. Effect of various non-dimensional parameters e.g., the Damkohler number (DA), phase partitioning number ( ), dimensionless absorption number (􀀀) are analyzed. The inferences we have drawn are (i) partial pressure of gas decreases inside tissue by increasing the value of porosity, (ii) by increasing breathing rate, partial pressure of gas first decreases and after some time it increases with breathing rate inside blood, (iii) under the consideration of tissue porosity or permeability, the convection speed does not change but the diffusion of gas increases from tissue, which causes increment in pressure gradient in mobile phase. Hence, these results can be helpful to optimize the reactive condition of excessive permeability of tissue due to aging, emphysema, asthma, and tuberculosis. The thesis, finally, ends with the future research directions regarding studies done in this thesis, appendices, and bibliography.