Series in BioEngineering Clara Mihaela Ionescu The Human Respiratory System An Analysis of the Interplay between Anatomy, Structure, Breathing and Fractal Dynamics Series in BioEngineering For further volumes: www.springer.com/series/10358 Clara Mihaela Ionescu The Human Respiratory System An Analysis of the Interplay between Anatomy, Structure, Breathing and Fractal Dynamics Clara Mihaela Ionescu Department of Electrical Energy, Systems and Automation Ghent University Gent, Belgium ISSN 2196-8861 ISSN 2196-887X (electronic) Series in BioEngineering ISBN 978-1-4471-5387-0 ISBN 978-1-4471-5388-7 (eBook) DOI 10.1007/978-1-4471-5388-7 Springer London Heidelberg New York Dordrecht Library of Congress Control Number: 2013947158 © Springer-Verlag London 2013 This work is subject to copyright All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher’s location, in its current version, and permission for use must always be obtained from Springer Permissions for use may be obtained through RightsLink at the Copyright Clearance Center Violations are liable to prosecution under the respective Copyright Law The use of general descriptive names, registered names, trademarks, service marks, etc in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use While the advice and information in this book are believed to be true and accurate at the date of publication, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made The publisher makes no warranty, express or implied, with respect to the material contained herein Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com) “To raise new questions, new possibilities, to regard old problems from a new angle, requires creative imagination and marks the real advance in science” A Einstein As a result of the above thought, I dedicate this book to all those who are curious, critical, and challenging Foreword Fractional Calculus (FC) was originated in 1695 based on the genial ideas of the German mathematician and philosopher Gottfried Leibniz (1646–1716) Up to the end of the 19th century, this topic remained mainly abstract with progress centered in pure mathematics The application of FC started with Oliver Heaviside (1850– 1925), an English electrical engineer, mathematician, and physicist Heaviside applied concepts of FC in is operational calculus and electrical systems Nevertheless, FC remained a mathematical tool unknown for most researchers In the area of life sciences the first contributions are credited to the American scientists Kenneth Stewart Cole (1900–1984) and Robert Hugh Cole (1914–1990), who published several papers by the end of the 1930s They proposed the so-called Cole–Cole empirical model, which has been successfully applied up to today, in a large variety of tissues These pioneering applications of FC were apparently forgotten in the decades that followed There is no historical record, social event, or scientific explanation, for the ‘oblivium’ phenomenon Three decades later Bertram Ross organized the First Conference on Fractional Calculus and its Applications at the University of New Haven in 1974 Also, Keith Oldham and Jerome Spanier published the first monograph devoted to FC Again, these important contributions remained with FC focused on pure mathematics, but in 1983 the French engineer Alain Ousaloup developed the CRONE (acronym for ‘Commande Robuste d’Ordre Non Entier’) method, which is used since then in control and identification algorithms We can say that the modern era of application of FC in physics and engineering started there In 1998 Virginia Kiryakova initiated the publication of the journal Fractional Calculus & Applied Analysis We should mention the vision of Ali Nayfeh and Murat Kunt, editors-in-chief of journals ‘Nonlinear Dynamics’ and ‘Signal Processing’, respectively, that supported a sustained growth of the new–old field by means of several special issues In the area of biology and medicine the first book, authored by Richard Magin, was published in 2006 By 2004 a young researcher, Clara Ionescu, started an intensive work in modeling respiratory systems using FC I called her the ‘atomic woman’ given the intensity of her work that culminated with her Ph.D by the end of 2009 Clara continued improving the models, getting more results and publishvii viii Foreword ing her research This book formulates, in a comprehensive work, her vision on the application of FC in the modeling of respiratory systems I am certain that the book will constitute a novel landmark in the progress in the area and that its readers will be rewarded by new perspectives and wider conceptual avenues Porto May 2013 J.A Tenreiro Machado Preface The objective of the book is to put forward emerging ideas from biology and mathematics into biomedical engineering applications in general with special attention to the analysis of the human respiratory system The field of fractional calculus is mature in mathematics and chemistry, but still in infancy in engineering applications However, the last two decades have been very fruitful in producing new ideas and concepts with applications in biomedical engineering The reader should find the book a revelation of the latest trends in modeling and identification of the human respiratory parameters for the purpose of diagnostic and monitoring Of special interest here is the notion of fractal structure, which tells us something about the biological efficiency of the human respiratory system Related to this notion is the fractal dimension, relating the adaptation of the fractal structure to environmental changes (i.e disease) Finally, we have the dynamical pattern of breathing, which is then the result of both the structure and the adaptability of the respiratory system The distinctive feature of the book is that it offers a bottom-up approach, starting from the basic anatomical structure of the respiratory system and continuing with the dynamic pattern of the breathing The relations between structure (or the specific changes within it) and fundamental working of the system as a whole are pinned such that the reader can understand their interplay Moreover, this interplay becomes crucial when alterations at the structural level in the airway caused by disease may require adaptation of the body to the functional requirements of breathing (i.e to ensure the necessary amount of oxygen to the organs) Adaptation of the human body, and specially of the respiratory system, to various conditions can be thus explained and justified in terms of breathing efficiency The motivation for putting together this book is to give by means of the example chosen (i.e the respiratory system) an impulse to the engineering and medical community in embracing these new ideas and becoming aware of the interaction between these disciplines The net benefit of reading this book is the advantage of any researcher who wants to stay up to date with the new emerging research trends in biomedical applications The book offers the reader an opportunity to become aware of a novel, unexplored, and yet challenging research direction ix x Preface My intention was to build a bridge between the medical and engineering worlds, to facilitate cross-fertilization In order to achieve this, I tried to organize the book in the traditional structure of a textbook A brief introduction will present the concept of fractional signals and systems to the reader, including a short history of the fractional calculus and its applications in biology and medicine In this introductory chapter, the notions of fractal structure and fractal dimension will be defined as well The second chapter describes the anatomy of the respiratory system with morphological and structural details, as well as lung function tests for evaluating the respiratory parameters with the aim of diagnosis and monitoring The third chapter will present the notion of respiratory impedance, how it is measured, why it is useful and how we are going to use it in the remainder of the book A mathematical basis for modeling air-pressure and air-flow oscillations in the airways is given in the fourth chapter This model will then be used as a basis for further developments of ladder network models in Chap 5, thus preserving anatomy and structure of the respiratory system Simulations of the effects of fractal symmetry and asymmetry on the respiratory properties and the evaluation of respiratory impedance in the frequency domain are also shown Chapter will introduce the equivalent mechanical model of the respiratory tree and its implications for evaluating viscoelasticity Of special importance is the fact that changes in the viscoelastic effects are clearly seen in patients with respiratory insufficiency, hence markers are developed to evaluate these effects and provide insight into the monitoring of the disease evolution Measurements on real data sets are presented and discussed Chapter discusses models which can be used to model the respiratory impedance over a broad range of frequencies, namely ladder network model and a model existing in the literature, for comparison purposes The upper airway shunt (not part of the actual respiratory system with airways and parenchyma) and its bias effect in the estimated values for the respiratory impedance is presented, along with a characterization on healthy persons and prediction values Measurements on real data sets are presented and discussed Chapter presents the analysis of the breathing pattern and relation to the fractal dimension Additionally, a link between the fractal structure and the convergence to fractional order models is shown, allowing also a link between the value of the fractional order model and the values of the fractal dimension In this way, the interplay between structure and breathing patterns is shown A discussion of this interplay points to the fact that with disease, changes in structure occur, these structural changes implying changes in the work necessary to breath at functional levels Measurements on real data sets are again presented and discussed Chapter introduces methods and protocols to investigate whether moving from the theory of linear system to nonlinear contributions can bring useful insight as regards diagnosis In this context, measuring frequencies close to the breathing of the patient is more useful than measuring frequencies outside the range of tidal breathing This also implies that viscoelasticity will be measured in terms of nonlinear effects The nonlinear artifacts measured in the respiratory impedance, are then Preface xi linked to the viscous and elastic properties in the lung parenchyma Measurements on real data sets are presented and discussed Chapter 10 summarizes the contributions of the book and point to future perspectives in terms of research and diagnosis methods In the Appendix, some useful information is given to further support the reader in his/her quest for knowledge Finally, I would like to end this preface section with some words of acknowledgment I would like to thank Oliver Jackson for the invitation to start this book project, and Ms Charlotte Cross of Springer London for her professional support with the review, editing, and production steps Part of the ideas from this book are due to the following men(tors): Prof Robin De Keyser (Ghent University, Belgium), Prof Jose-Antonio Tenreiro Machado (Institute of Engineering, Porto, Portugal), Prof Alain Oustaloup (University of Bordeaux1, France) and Prof Viorel Dugan (University of Lower Danube, Galati, Romania) Clinical insight has been generously provided to me by Prof Dr MD Eric Derom (Ghent University Hospital, Belgium) and Prof Dr MD Kristine Desager (Antwerp University Hospital, Belgium) I thank them cordially for their continuous support and encouragement Further technical support is acknowledged from the following Master and Ph.D students throughout the last decade: Alexander Caicedo, Ionut Muntean, Niels Van Nuffel, Nele De Geeter, Mattias Deneut, Michael Muehlebach, Hannes Maes, and Dana Copot Next, I would like to acknowledge the persons who supported my work administratively and technically during the clinical trials • For the measurements on healthy adult subjects, I would like to thank Mr Sven Verschraegen for the technical assistance for pulmonary function testing at the Department of Respiratory Medicine of Ghent University Hospital, Belgium • For the measurements on healthy children, I would like to thank Mr Raf Missorten from St Vincentius school in Zwijnaarde, Principal, for allowing us to perform tests and to Mr Dirk Audenaert for providing the healthy volunteers I would also like to thank Nele De Geeter and Niels Van Nuffel for further assistance during the FOT (Forced Oscillations Technique) measurements • For the measurements on COPD patients: many thanks to Prof Dr Dorin Isoc from Technical University of Cluj-Napoca and to Dr Monica Pop for the assistance in the University of Pharmacy and Medicine “Iuliu Hatieganu” in ClujNapoca, Romania • For the measurements on asthmatic children, I would like to thank Rita Claes, Hilde Vaerenberg, Kevin De Sooner, Lutje Claus, Hilde Cuypers, Ria Heyndrickx and Pieter De Herdt from the pulmonary function laboratory in UZ Antwerp, for the professional discussions, technical and amicable support during my stay in their laboratory • For the measurements on kyphoscoliosis adults, I would like to thank Mrs Hermine Middendorp for the assistance with the Ethical Committee request; to Philippe De Gryze, Frank De Vriendt, Lucienne Daman, and Evelien De Burck Useful Notes on Fractional Calculus 203 The common formulation for the fractional integral can be derived directly from a traditional expression of the repeated integration of a function This approach is commonly referred to as the Riemann–Liouville approach: t f (τ ) dτ dτ = n t (n − 1)! (t − τ )n−1 f (τ ) dτ (A.11) n which demonstrates the formula usually attributed to Cauchy for evaluating the nth integration of the function f (t) For the abbreviated representation of this formula, we introduce the operator J n such as shown in: J n f (t) = fn (t) = t (n − 1)! (t − τ )n−1 f (τ ) dτ (A.12) Often, one will also find another operator, D −n , used in place of J n While they represent the same formulation of the repeated integral function, and can be seen as interchangeable, one will find that the us of D −n may become misleading, especially when multiple operators are used in combination For direct use in (A.11), n is restricted to be an integer The primary restriction is the use of the factorial which in essence has no meaning for non-integer values The gamma function is, however, an analytic expansion of the factorial for all reals, and thus can be used in place of the factorial as in (A.2) Hence, by replacing the factorial expression for its gamma function equivalent, we can generalize (A.12) for all α ∈ R, as shown in: J α f (t) = f∞ (t) = (α) t (t − τ )α−1 f (τ ) dτ (A.13) It is also possible to formulate a definition for the fractional-order derivative using the definition already obtained for the analogous integral Consider a differentiation of order α = = 2; α ∈ R+ Now, we select an integer m such that m − < α < m Given these numbers, we now have two possible ways to define the derivative The first definition, which we will call the Left Hand Definition is f (n) = D α f (t) = dm dt m f (t), dm dt m t f (τ ) (m−α) (t−τ )(α+1−m) m−1