Modeling of the Lung Impedance Using a Fractional-Order Ladder Network With Constant Phase Elements

The self similar branching arrangement of the airways makes the respiratory system an ideal candidate for the application of fractional calculus theory. The fractal geometry is typically characterized by a recurrent structure. This study investigates the identification of a model for the respiratory tree by means of its electrical equivalent based on intrinsic morphology. Measurements were obtained from seven volunteers, in terms of their respiratory impedance by means of its complex representation for frequencies below 5 Hz. A parametric modeling is then applied to the complex valued data points. Since at low-frequency range the inertance is negligible, each airway branch is modeled by using gamma cell resistance and capacitance, the latter having a fractional-order constant phase element (CPE), which is identified from measurements. In addition, the complex impedance is also approximated by means of a model consisting of a lumped series resistance and a lumped fractional-order capacitance. The results reveal that both models characterize the data well, whereas the averaged CPE values are supraunitary and subunitary for the ladder network and the lumped model, respectively.

[1]  Robin De Keyser,et al.  A Theoretical Study on Modeling the Respiratory Tract With Ladder Networks by Means of Intrinsic Fractal Geometry , 2010, IEEE Transactions on Biomedical Engineering.

[2]  Cevdet Acar,et al.  Convergence and periodic solutions for the input impedance of a standard ladder network , 2007 .

[3]  D E Olson,et al.  Models of the human bronchial tree. , 1971, Journal of applied physiology.

[4]  B. Suki,et al.  Forced oscillatory impedance of the respiratory system at low frequencies. , 1986, Journal of applied physiology.

[5]  C. Sparrow.  The Fractal Geometry of Nature , 1984 .

[6]  E. Weibel.  Mandelbrot’s Fractals and the Geometry of Life: A Tribute to Benoît Mandelbrot on his 80 th Birthday , 2005 .

[7]  W. Marsden.  I and J , 2012 .

[8]  S. Gheorghiu,et al.  Is the Lung an Optimal Gas Exchanger , 2005 .

[9]  S. Gheorghiu,et al.  Gas Diffusion through the Fractal Landscape of the Lung: How Deep Does Oxygen Enter the Alveolar System? , 2005 .

[10]  Damian Craiem,et al.  A fractional derivative model to describe arterial viscoelasticity. , 2007, Biorheology.

[11]  Verne C. Fryklund,et al.  What systems analysis? , 1981, Nature.

[12]  A. Oustaloup.  La dérivation non entière , 1995 .

[13]  E. Gabryś,et al.  Fractal models of circulatory system. Symmetrical and asymmetrical approach comparison , 2005 .

[14]  J. Hildebrandt.  Comparison of mathematical models for cat lung and viscoelastic balloon derived by Laplace transform methods from pressure-volume data. , 1969, The Bulletin of mathematical biophysics.

[15]  Zoltán Hantos,et al.  Changes in Respiratory Mechanics During Cardiac Surgery , 2003, Anesthesia and analgesia.

[16]  Z. Hantos,et al.  Mechanical impedances of lungs and chest wall in the cat. , 1992, Journal of applied physiology.

[17]  Z Hantos,et al.  An improved forced oscillatory estimation of respiratory impedance. , 1982, International journal of bio-medical computing.

[18]  E. Oostveen,et al.  The forced oscillation technique in clinical practice: methodology, recommendations and future developments , 2003, European Respiratory Journal.

[19]  Robin De Keyser,et al.  A Novel Parametric Model for the Human Respiratory System , 2003, Modelling and Simulation.

[20]  Robin De Keyser,et al.  Mechanical Properties of the Respiratory System Derived From Morphologic Insight , 2009, IEEE Transactions on Biomedical Engineering.

[21]  Jacques Jossinet.  Elementary electrodynamics. , 2008, Technology and health care : official journal of the European Society for Engineering and Medicine.

[22]  J. Bates.  A Recruitment Model of Quasi-Linear Power-Law Stress Adaptation in Lung Tissue , 2007, Annals of Biomedical Engineering.

[23]  E. Gabryś,et al.  Blood flow simulation through fractal models of circulatory system , 2006 .

[24]  Thomas F. Coleman,et al.  An Interior Trust Region Approach for Nonlinear Minimization Subject to Bounds , 1993, SIAM J. Optim..

[25]  Damian Craiem,et al.  Arterial viscoelasticity: a fractional derivative model , 2006, 2006 International Conference of the IEEE Engineering in Medicine and Biology Society.

[26]  Isabel S. Jesus,et al.  Development of fractional order capacitors based on electrolyte processes , 2009 .

[27]  Ionut Muntean,et al.  A Simulator for the Respiratory Tree in Healthy Subjects Derived from Continued Fractions Expansions , 2009 .

[28]  Béla Suki,et al.  Fluctuations and power laws in pulmonary physiology. , 2002, American journal of respiratory and critical care medicine.

[29]  B. Suki,et al.  Airway geometry and wall mechanical properties estimated from subglottal input impedance in humans. , 1994, Journal of applied physiology.

[30]  Jacques Jossinetca,et al.  Elementary electrodynamics , 2008 .

[31]  Aaas News,et al.  Book Reviews , 1893, Buffalo Medical and Surgical Journal.

[32]  J. Fredberg,et al.  Input impedance and peripheral inhomogeneity of dog lungs. , 1992, Journal of applied physiology.

[33]  Robin De Keyser,et al.  A Model of the Lungs Based on Fractal Geometrical and Structural Properties , 2009 .

[34]  R. Riedi,et al.  Origin of Fractal Branching Complexity in the Lung , 2000 .