Author's Accepted Manuscript
Validation of airway resistance Models for predicting pressure loss through anatomically realistic conducting airway Replicas of Adults and children Azadeh A.T. Borojeni, Michelle L. Noga, Andrew R. Martin, Warren H. Finlay
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S0021-9290(15)00207-9 http://dx.doi.org/10.1016/j.jbiomech.2015.03.035 BM7117
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Journal of Biomechanics
Accepted date: 27 March 2015 Cite this article as: Azadeh A.T. Borojeni, Michelle L. Noga, Andrew R. Martin, Warren H. Finlay, Validation of airway resistance Models for predicting pressure loss through anatomically realistic conducting airway Replicas of Adults and children, Journal of Biomechanics, http://dx.doi.org/10.1016/j.jbiomech.2015.03.035 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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Validation of Airway Resistance Models for Predicting Pressure Loss through
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Anatomically Realistic Conducting Airway Replicas of Adults and Children
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Azadeh A. T. Borojeni1,*, Michelle L. Noga2, Andrew R. Martin1’**, Warren H. Finlay1’**
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1.
Department of Mechanical Engineering, University of Alberta, Edmonton, Alberta, Canada T6G 2G8
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2.
Department of Radiology and Diagnostic Imaging, University of Alberta, Edmonton, Alberta, Canada
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*
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+1 780 492 2200
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**
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+1 780 492 4707; Fax: +1 780 492 2200
Corresponding author (during review and publication), Email: [email protected]; Tel: +1 780 492 2161; Fax:
Principal corresponding authors (post-publication), Email: [email protected], [email protected]; Tel:
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Keywords: Airway resistance; conducting airways; pressure drop; pressure loss; ventilation
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distribution
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Abstract
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This work describes in vitro measurement of the total pressure loss at varying flow rate through
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anatomically realistic conducting airway replicas of ten children, 4 to 8 years old, and five adults.
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Experimental results were compared with analytical predictions made using published airway
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resistance models. For the adult replicas, the model proposed by van Ertbruggen et al. (J. Appl.
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Physiol. 98:970-980,2005) most accurately predicted central conducting airway resistance for
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inspiratory flow rates ranging from 15 to 90 L/min. Models proposed by Pedley et al. (J. Respir.
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Physiol. 9:371-386,1970) and by Katz et al. (J. Biomechanics 44:1137-1143,2011) also provided
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reasonable estimates, but with a tendency to over predict measured pressure loss for both models.
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For child replicas, the Pedley and Katz models both provided good estimation of measured
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pressure loss at flow rates representative of resting tidal breathing, but under predicted measured
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values at high inspiratory flow rate (60 L/min). The van Ertbruggen model, developed based on
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flow simulations performed in an adult airway model, tended to under predict measured pressure
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loss through the child replicas across the range of flow rates studied (2 to 60 L/min). These
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results are intended to provide guidance for selection of analytical pressure loss models for use in
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predicting airway resistance and ventilation distribution in adults and children.
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1. Introduction
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Airway resistance is an important parameter used in describing the mechanics of breathing. The
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relationship between pressure loss and flow rate through the conducting airways influences work
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of breathing, and is widely used for diagnostic purposes in detecting and classifying respiratory
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diseases. In addition, variation in airway resistance between individual airways or groups of
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airways along different paths through the lung plays a role in determining the distribution of
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ventilation between different lung regions. Ventilation distribution in turn influences regional
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gas transport, mixing, and uptake, as well as transport and deposition of inhaled aerosols.
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Analytical models of airway resistance have been employed recently, for example, in modeling
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topographic distribution of ventilation in healthy lungs (Swan et al., 2012), in estimating
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contributions of peripheral airway resistance to ventilation distribution measured by PET
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imaging (Wongviriyawong et al., 2013), in predicting distribution of pressure through the airway
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tree during xenon anesthesia (Katz et al., 2012), and in optimizing algorithms used to predict
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alveolar pressure during mechanical ventilation (Damanhuri et al., 2014).
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Study of pressure loss over the conducting airways dates to the foundational work of Rohrer
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(1915). For a given flow rate, the difference in pressure between the airway opening (nose or
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mouth) and the alveoli was expressed as a summation of linear and quadratic terms, each term
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weighted by a dimensional constant.
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depending on airway dimensions and geometry. The utility of such an approach was clearly
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demonstrated by Otis and colleagues, who used the Rohrer airway resistance model in
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combination with experimental measurements to describe work of breathing (Otis et al., 1950),
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ventilation distribution (Otis et al., 1956), and effects of breathing different gas mixtures on
Values of these constants varied between subjects,
3
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airway resistance (Otis et al., 1949). These studies provided early insight into the functional
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relationship between pressure loss, flow rate, gas properties, and airway geometry. However,
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significant further understanding was gained through fundamental experiments and fluid
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dynamics analysis performed by Pedley and colleagues (1970). These authors presented an
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analytical model of pressure loss through conducting airways of the human bronchial tree based
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on measurements performed in idealized, symmetric airway bifurcations (Pedley et al., 1970).
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Pressure loss was found to exceed that predicted by assuming fully developed Hagen-Poiseuille
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flow, and a correction factor was proposed based on airway length and diameter, and the gas
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Reynolds number (Pedley et al., 1970).
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More recently, computational fluid dynamics (CFD) simulations have been performed to
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investigate pressure loss through idealized (e.g. Wilquem and Degrez, 1997; Comer et al., 2001;
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Kang et al., 2011; Katz et al., 2011) and anatomically realistic (e.g. van Ertbruggen et al., 2005;
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Gemci et al., 2008; Ismail et al., 2013) adult airway bifurcations. Previous authors have noted
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that, for adult airways, the Pedley model overestimates simulated pressure losses (Comer et al.,
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2001; van Ertbruggen et al., 2005; Katz et al., 2011; Ismail et al., 2013).
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alternative analytical models have been proposed (van Ertbruggen et al., 2005; Katz et al., 2011).
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In contrast, considerably less work has been done to model airway pressure losses for the
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conducting airways of children. The accuracy of using analytical models developed based on
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measurements or simulations performed in adult airway models in predicting pressure loss in
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children’s airways is presently unknown.
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Accordingly,
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In this paper, we describe in vitro measurements of the total pressure loss at varying flow rate
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through anatomically realistic conducting airway replicas of adults and children. The in vitro
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approach employed presents a unique opportunity to reduce the pressure-flow relationship
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through the three-dimensional, multi-branching, conducting airway geometry to the classic fluid
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mechanics problem of flow through a piping network comprising nonlinear resistances (White
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2010). Accordingly, analytical predictions of the total pressure loss through the airway replicas
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made using airway resistance models proposed by Pedley et al. (1970), van Ertbruggen et al.
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(2005), and Katz et al. (2011) are compared to measured experimental values. Our results are
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intended to provide guidance for selection of analytical models for use in predicting airway
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resistance in adults and children.
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2.
Methods
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2.1 Airway pressure drop measurement
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Fabrication of realistic adult and child conducting airway replicas has been explained in detail in
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previous work (Borojeni et al., 2014). In brief, conducting airways were identified using Mimics
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software (MIMICS 3D, Materialise, Ann Arbor, MI, USA) as regions of interest in CT-images of
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5 adults and 10 children. Subject information and average airway dimensions are shown in
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Table 1. A rapid prototyping technique (Invision® SR 3D printer, 3D systems, Rock Hill, SC,
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USA) was used to fabricate the conducting airway replicas from acrylonitrile butadiene styrene
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(ABS) plastic (P430, Stratasys, Eden Prairie, MN).
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5
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In the present work, pressure drop across the airway replicas was measured in vitro using a
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digital pressure gauge (HHP-103, Omega Engineering, Inc., Stamford, CT) with low range (0-
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498 Pa), high range (0-2590 Pa) and accuracy of ±0.2% full scale. The upstream pressure tap
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was open to atmosphere and the downstream pressure tap was positioned in the tubing that
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connected the outlet of an artificial thoracic chamber to a vacuum pump (Figure 1). Preliminary
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measurements were made with the chamber open to characterize the pressure drop across only
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the chamber outlet at flows ranging from 2 to 90 L/min. Constant flow rates were then drawn
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through the chamber with the replicas in place by operating a vacuum pump, with flow rates
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chosen in the range of 15 to 90 L/min for adult replicas and 2 to 60 L/min for child replicas. The
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desired flow rate was set using a needle valve and monitored by a digital mass flowmeter (TSI
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Model 4000 series, TSI Inc., Shoreview, MN, USA) downstream of the thoracic cylindrical
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chamber. The pressure drop across the airway replicas is reported below as the difference
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between the pressure drop measured with and without the airway replicas in place. Each
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experimental measurement was repeated three times.
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2.2 Analytical pressure loss model
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Measured values of pressure drop across the airway replicas were compared with predicted
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values made using three previously published models of airway resistance (Pedley et al., 1970;
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Katz et al., 2011; van Ertbruggen et al., 2005). For each replica, subject-specific geometry of the
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conducting airways was required as input in the analytical models of flow distribution and
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resistance.
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2.2.1 Determination of Airway Dimensions
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Diameters and lengths for all airway segments in each replica were measured from segmented
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CT images using a - post processing software package (MAGICS, Materialise, MI, USA). The
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length of a segment was defined as the centerline path length from one bifurcation to the next.
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The diameter was defined as the average of equivalent cross-sectional area diameters evaluated
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every 2 mm along the length of the segment.
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2.2.2 General Solution Procedure
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For convenience of calculation, a hydraulic resistance, R, is defined for a segment as R =
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∆h
(1)
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where Q is the flow rate through the airway and ∆h is a form of the head loss, expressed in
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dimensions of energy per unit mass as the ratio between pressure drop across the airway (∆p) and
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fluid density (ρ), i.e. Δh=Δp/ρ .
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Expressions for the head loss as a function of airway geometry, fluid properties and flow rate
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result from the application of specific airway resistance models. These will be provided in the
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following section. At present, Equation 1 is used to outline the solution procedure in general.
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For example, for the simple network of segments shown in Fig. 2 we wish to determine the
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fraction of the total flow entering airway 0 that flows through each branch, 1a and 1b. To do so,
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the resistances of branches 1a and 1b must be taken into account, as must be the resistances of all 7
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subsequent segments downstream. The equivalent hydraulic resistance of either branch can be
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determined as follows (White, 2010): eq, 1a = +
/
1
/
[ 2aa + 2ab ] (2)
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eq, 1b = +
/ [ 2ba
1
/
+ 2bb ] (3)
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In the tracheobronchial tree, the distal airways shown in Figure 2 will proceed to bifurcate into
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further downstream airway generations. It is noted that hydraulic resistances of these
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downstream segments (i.e. R2aa, R2ab, R2ba, or R2bb) may themselves be expressed as equivalent
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hydraulic resistance defined using these same principles.
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The head loss through any branch can then be expressed as
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∆hn = R eq,n Q2n (4)
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where the equivalent hydraulic resistance Req,n of that branch is formulated by generalization of
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equations 2 and 3 and is itself a function of Qn.
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Applying this procedure to a branching airway network results in a system of nonlinear
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equations. For a multiple-path network with n paths, a system of n equations is obtained. This
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system of equations can be solved iteratively provided that the head loss is the same across each
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path. This condition is met in the experiments described above given that all terminal branches
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supply the artificial thorax chamber. The head loss calculation for each path included a minor
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loss term accounting for dissipation of kinetic energy due to discharge of flow from terminal
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branches into the artificial thorax. This term was in all cases small compared with the total head
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loss.
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The iterative solution procedure starts with an initial guess for the flow rate in each branch. This
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was made assuming evenly distributed flow between daughter branches at all bifurcations. The
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flow rates through each branch at subsequent iterations may then be calculated according to:
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a
=
/
( eq,b )# (Qinlet ) /
/
[( eq,a )# + ( eq,b )# ] (5)
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where i+1 and i denote the current and previous iteration, respectively, and subscripts a and b
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indicate daughter airways. Equivalent hydraulic resistances are flow rate dependent and thus
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update on each iteration.
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After each iteration, flow rates along each path were compared with corresponding values from
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the previous iteration. The root mean square (RMS) difference in flow rates between successive
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iterations was used to determine convergence. The solution was deemed converged when this 9
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RMS difference fell below 0.006 L/min. The sensitivity of the total head loss to this criterion
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was trivial for values less than 0.006 L/min. For the converged solution, the total head loss
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across the airway tree was then calculated using the branch flow rates and resistances.
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2.2.3
Airway Resistance Models
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2.2.3.1 Pedley Model
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Pedley and colleagues (1970) determined the pressure drop through idealized symmetric models
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of bronchial airways based on measurement of changes to inspiratory flow profiles. They
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derived an expression for the ratio relating the actual pressure drop through an airway section to
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that which would be predicted assuming fully-developed Hagen-Poiseuille flow. This ratio was
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given by:
205 /
1.85
, $= * + . 4√2
(6)
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where D and L are airway diameter and length, and Re is the flow Reynolds number:
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+ =
4 0,1 (7)
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where 1 is the kinematic viscosity of the fluid (air). 10
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The pressure drop is then calculated as: ∆2pedley = $∆2Hagen-Poiseuille (8)
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215
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where, ∆2Hagen-Poiseuille =
128µ- 0,? (9)
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where µ is the fluid dynamic viscosity.
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Accordingly, the expression for hydraulic resistance, R in a given airway branch for the Pedley
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model is as follows:
pedley =
128 ∙ 1.85 - A.B A.B * . + √20 ,? , (10)
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where ρ is fluid density.
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Note that in analyzing the realistic airway replicas, the value of Z was less than 1 at some distal
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airways where airway diameter was considerably smaller than the airway length and Reynolds
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number was small. This is not a physically realistic result as the dissipation of energy in realistic
11
226
airways cannot be less than dissipation of energy in the ideal Hagen-Poiseuille flow. Therefore,
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the value of $ was replaced by 1 in these cases.
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2.2.3.2 Modified Pedley Model (van Ertbruggen model)
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van Ertbruggen and colleagues ( 2005) used CFD to simulate flow through an anatomically-
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based model of the airways extending from the trachea to the segmental bronchi. This model
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combined morphometric data data from Horsfield et al. (1971) with orientation of branching
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planes based on bronchoscope and CT images. Simulated pressure drop through each airway
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generation in the model was compared by van Ertbruggen and colleagues ((2005) to values
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predicted using the Pedley model described above. Pressure drop predicted using the Pedley
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model was consistently higher than simulated values. The authors therefore proposed a modified
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form of equation 8, as follows:
∆2ModiDied-Pedley
/ 32µ-G
, = E * + . -
,
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(11)
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where values of γ for generations 0-3 are 0.162, 0.239, 0.244 and 0.295 respectively (van
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Ertbruggen et al., 2005).
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Using these values for γ, the expression for hydraulic resistance, R, in a given airway branch for
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the modified Pedley model is: modiDied Pedley =
512 - A.B E ( ) 0 ,? , + A.B
12
(12)
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2.2.3.3 Katz Model
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Katz and colleagues (2011) presented a model for calculating pressure drop through the
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conducting airways based on concepts of major and minor head losses.
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The expression for hydraulic resistance, R, in a given airway branch for the Katz model is:
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Katz =
8
0 ,? L
[M
+ N] , (13)
249
250
where M is the friction factor, assumed to be 64/Re for Re < 2000 (laminar flow). Above this
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value, flow is assumed turbulent and the friction factor is calculated using the Blasius correlation
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for smooth tubes: M=
0.316 + A.B
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(14)
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The minor loss coefficient (N) in Equation 13, is calculated using the following correlation based
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on CFD simulations of flow through idealized symmetric airway bifurcations (Katz et al., 2011): N=
Q +S(log +) + ,(log +) + T + R (15)
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13
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Values for the constants A through E in Equation 15 are provided for each tracheal-bronchial
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lung generation by Katz and colleagues (2011).
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For each of the three airway resistance models, prediction of measured pressure drop through the
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adult and child airway replicas was assessed using Lin’s concordance correlation coefficient, ρc
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(Lin 1989). In the present work, reported values of the concordance correlation coefficient
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indicate goodness of fit of plotted predicted vs. measured data to the identity line, with higher
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values indicating better fit and ρc = 1.0 indicating a perfect fit.
264
265
3. Results
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Tracheal Reynolds numbers for adult and child subjects are presented in Table 2. At an
268
equivalent volumetric flow rate, Reynolds numbers were higher in child than in adult replicas,
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owing to the smaller tracheal diameters (and thus higher flow speeds) of the child replicas.
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Comparison of measured versus calculated pressure drop through the adult replicas is made in
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Figures 3 for the three airway resistance models employed. These comparisons indicate there is,
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in general, reasonable agreement between measured and predicted values, for all three resistance
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models. It can be further observed that the experimental data are in best agreement with the
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predicted values made using the modified Pedley model.
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Comparison of measured versus calculated pressure drop through the child replicas is
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demonstrated in Figures 4-6. Here the Katz model and original Pedley model displayed better
14
278
agreement with measured values than did the modified Pedley model, though all three models
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had a tendency to under predict measured data for the highest measured values of pressure drop,
280
corresponding to higher flow rates through the replicas.
281 282
In calculating the pressure drop across the replicas, the distribution of flow through the
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branching airways is also calculated. Calculated values of the fraction of flow through the right
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lung are presented in Table 3. For both adults and child replicas, a greater fraction of the
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inhalation flow passes through airway of the right lung. The standard deviation between replicas
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was higher in the child group than in the adult group (0.15 versus 0.03 on average, respectively).
287 288 289
4. Discussion
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In the present work, we measured pressure loss at varying flow rate through conducting airway
292
replicas of adults and children. Measurements were compared with values predicted using
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analytical airway resistance models previously developed based on measurements or simulations
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performed in adult conducting airway geometries. For the present five adult airways, the airway
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resistance model proposed by Pedley and colleagues (1970) tended to over-predict measured
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pressure loss. This is consistent with the results of previous CFD studies comparing predictions
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of conducting airway pressure loss made using the Pedley model to simulated values (Comer et
298
al., 2001; van Ertbruggen et al., 2005; Katz et al., 2011; Ismail et al., 2013). Of the three
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resistance models evaluated, the modified Pedley model proposed by van Ertbruggen et al.
300
(2005) demonstrated closest agreement with our measured adult data. These authors modified
301
the original Pedley model by proposing alternative coefficients for predicting the pressure loss 15
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through each proximal lung generation. The modified coefficients were based on pressure losses
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determined by CFD simulation of laminar flow through a single, representative, anatomically-
304
based adult airway geometry. The good agreement of predicted pressure losses made using their
305
model with our data in five different adult airway replicas, at tracheal Reynolds numbers
306
exceeding 10,000 in the highest case (Table 2), was therefore not expected a priori. However,
307
van Ertbruggen et al. (2005) argue that below Reynolds number averaging 8,000 the flow regime
308
through conducting airways can be considered laminar or ‘lightly’ transitional, but not turbulent.
309
Again with reference to Table 2, for a strong majority of cases we studied tracheal Reynolds
310
numbers in the adult replicas that fell below 8,000.
311
angle (and branching asymmetry) are not explicitly accounted for in the modified Pedley model,
312
and surely existed between our five adult replicas, Kang et al. (2011) found only a weak
313
influence of branching angle on simulated pressure loss through a model airway bifurcation.
Furthermore, while variation in branching
314 315
For the child airway replicas, predicted and measured pressure losses were in closer agreement
316
for the Katz model and the original Pedley model than for the modified Pedley model. Poorer
317
agreement of the modified Pedley model with the child versus adult data may be due to
318
differences in central conducting airway geometry between children and the single adult
319
geometry used in creating this model, that are not captured explicitly in Equation 12 (recalling
320
that model coefficients were fit to simulation data obtained in a representative single adult
321
airway geometry).
322 323
All three models tended to under predict measured pressure loss at the highest flow rate studied
324
in children (60 L/min), where tracheal Reynolds numbers averaged over 10,000. Indeed, for
16
325
replicas from the youngest children studied (ages 4 and 5), the pressure loss at 60 L/min was
326
sufficiently high that such an inspiratory flow rate is likely to be achieved in vivo by the
327
corresponding child subjects only near maximum inspiratory effort (Vilozni et al., 2009). At
328
lower flow rates, corresponding to lower pressure loss, the original Pedley model was the most
329
accurate of the three models studied. Pedley et al. (1970) based their model on measurements
330
made in idealized symmetric models of branching airways, where Reynolds number ranged from
331
~100 to ~1,000. Tracheal Reynolds numbers in our child replicas were on average within this
332
range for the two lowest flow rates studied (2 and 5 L/min).
333
relatively close agreement between our measured data and predictions made using the Pedley
334
model for low pressure loss (where flow rate and Reynolds number were also low) and lack of
335
such agreement for higher pressure loss.
This very likely explains the
336 337
In calculating the total pressure loss through the airway replicas using each of the three airway
338
resistance models, the distribution of flow through the replicas is also obtained. The fraction of
339
flow directed to the right lung side of the replicas was found to average 0.58 in the adult replicas,
340
with little variation with flow rate or between replicas.
341
distribution of ventilation to the right lung observed in vivo (Lumb 2010), especially given that
342
for the intact, healthy, in vivo lung, ventilation distribution is determined more by downstream
343
tissue compliance than by airway resistance (Swan et al., 2012). Ostensibly, the dimensions of
344
the central conducting airways act to reinforce, rather than oppose, this ventilation distribution.
345
For the child replicas, the fraction of flow directed to the right lung side of the replicas
346
(averaging 0.63) did not vary significantly from that for the adult replicas. However, variation
347
between replicas was considerably greater for child than for adult replicas. This variation is
17
This value is consistent with the
348
potentially attributable to a greater variation in airway geometry for the child replicas, given that
349
these geometries were obtained from images of children aged 4-8 years, and therefore at
350
different stages of physical development.
351 352
A limitation in the present work is the absence of any upper, extrathoracic airway upstream from
353
our conducting airway replicas. In this regard, our replicas are similar to the original single and
354
multiple bifurcation geometries used to develop the three analytical resistance models studied
355
herein. None of these models was based on measurements or simulations done in airway
356
geometries that included the upper airway.
357
performed by Xi et al. (2008), the upper airway most significantly influences flow features
358
within the trachea, whereas flow in subsequent downstream airway branches is much less
359
affected. The accuracy of the resistance models in predicting in vivo pressure loss through the
360
trachea, directly downstream from the upper airway, was not assessed in the present study.
As observed in numerical flow simulations
361 362
5. Conclusion
363
Pressure loss was experimentally measured at varying flow rate through anatomically realistic
364
conducting airway replicas of adults and children. Measured values were compared with values
365
predicted using published analytical airway resistance models. The modified Pedley model
366
proposed by van Ertbruggen et al. (2005) provided close agreement with measured central
367
conducting airway resistance in adult replicas across a wide range of inspiratory flow rates, from
368
15 to 90 L/min. In contrast, in child airway replicas, the Pedley et al. (1970) model most closely
369
matched the experimental data.
370
18
371 372
6. Acknowledgements
373
Andrew Grosvenor and Jamie Schmitt from the Institute for Reconstructive Sciences in Medicine
374
(iRSM) are acknowledged for their technical help in fabricating the replicas. Greg Wandzilak
375
from PET/CT center, University of Alberta, Stollery Children’s Hospital is gratefully
376
acknowledged for providing CT scans. Travis MacLean and John Chen are appreciated for their
377
help in measuring pressure drop in the child replicas.
378 379
7. Conflict of Interest
380
The authors report no conflict of interest.
381
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Gemci, T., Ponyavin, V., Chen, Y., Chen, H., Collins, R., 2008. Computational model of airflow in upper 17 generations of human respiratory tract. Journal of Biomechanics. 41(9), 2047-2054.
Ismail, M., Comerford, A., Wall, W.A., 2013. Coupled and reduced dimensional modeling of respiratory mechanics during spontaneous breathing. International Journal For Numerical Methods In Biomedical Engineering. 29, 1285-1305. Kang, M-Y, Hwang, J., and Lee, J-W., 2011. Effect of geometric variations on pressure loss for a model bifurcation of the human lung airways. Journal of Biomechanics. 44, 1196–1199.
Katz, I. M., Martin, A. R., Feng, C. H., Majoral, C., Caillibotte, G., Marx, T., Bazin, J.-E., and Daviet, C. 2012. Airway pressure distribution during xenon anesthesia: the insufflation phase at constant flow (volume controlled mode). Applied Cardiopulmonary Pathophysiology, 16, 5-16. Lin, IL. 1989. A concordance correlation coefficient to evaluate reproducibility. Biometrics, 45, 255-268.
20
426 427 428
Lumb, A.B., 2010. Nunn’s Applied Respiratory Physiology, Seventh Edition. Elsevier Ltd.
429
Otis A.B., Bembower W.C., 1949. Effect of gas density on resistance to respiratory gas
430 431
flow in man. Journal of Applied Physiology. 2, 300-306.
432
Otis A.B., Fenn W.O., Rahn H., 1950. Mechanics of Breathing in Man. Journal of
433
Applied Physiology. 2, 592-607.
434 435
Otis A.B., McKerrow C.B., Bartlett R.A., Mead J., McIlroy M.B., Selverstone N.J.,
436
Radford E.P., 1956. Mechanical factors in distribution of pulmonary ventilation. Journal
437 438 439 440 441 442 443 444 445
of Applied Physiology. 8, 427-443.
446 447 448 449 450 451 452 453 454 455 456 457 458
Swan, A.J., Clark, A.R., Tawhai, M.H., 2012. A computational model of the topographic distribution of ventilation in healthy human lungs. Journal of Theoretical Biology. 300, 222-231.
459 460 461
Wongviriyawong, C., Harris, R. S., Greenblatt, E., Winkler, T., and Venegas, J. G., 2013. Peripheral resistance: a link between global airflow obstruction and regional ventilation distribution. Journal of Applied Physiology, 114(4), 504-514.
462 463 464
Xi, J., Longest, P.W., and Martonen, T.B., 2008. Effects of the laryngeal jet on nanoand microparticle transport and deposition in an approximate model of the upper tracheobronchial airways. Journal of Applied Physiology, 104(6), 1761-1777.
Pedley, T.J., Schroter, R.C., Sudlow, M.F., 1970. Energy losses and pressure drop in models of human airways. Journal of Respiration Physiology. 9, 371-386. Rohrer, F., 1915. Der Strömungswiderstand in den menschlichen Atemwegen und der Einfluss der unregelmässigen Verzweigungen des Bronchialsystems auf den Atmungsverlauf in verschiedenen Lungenbezirken. Pflügers Arch ges Physiol, 162, 225–299.
Vilozni, D., Efrati, O., Barak, A., Yahav, Y., Augarten, A., Bentur, L., 2009. Forced inspiratory flow volume curve in healthy young children. Pediatric Pulmonology. 44(2), 105-111. White, F. M., 2010. Fluid Mechanics, 7th edition. Boston, USA: McGraw-Hill. Wilquem, F., Degrez. G., 1997. Numerical modeling of steady inspiratory airflow through a three-generation model of the human central airways. Journal of Biomechanical Engineering. 119(1), 59-65.
465
21
466 467 468
9. Tables
469
Table 1: Summary of child and adult subjects information and airway diameters of child and
470
adult replicas. Age
Subject
(year)
Number
4
Sex
Height
Weight
Diameter
Diameter
Diameter
Diameter
(cm)
(kg)
(stdev)
(stdev)
(stdev)
(stdev)
(Gen.0)
(Gen.1)
(Gen.2)
(Gen.3)
(mm)
(mm)
(mm)
(mm)
10c
F
99
16
7.15(0.04)
4.7(1.19)
3.48(0.27) 2.2(0.23)
14c
F
100
16
7.16(0.15)
6.47(2.77)
4.46(0.69) 2.23(0.76)
2c
M
117
22.9
7.05(0.09)
6.03(0.7)
6.00(1.4)
3c
M
112
20
7.99(0.51)
5.39(0.51)
4.93(0.49) 2.33(0.66)
9c
M
113
20
7.56(0.01)
6.44(0.87)
5.26(0.01) 3.38(1.05)
5c
F
112
18
7.99(0.42)
5.36(1.14)
5.35(1.41) 3.55(1.00)
6c
F
118
21.5
8.5(0.3)
6.75(1.32)
5.94(1.32) 2.9(1.47)
12c
F
124
24
7.41(0.46)
6.45(2.72)
3.28(0.26) 3.23(0.9)
7
13c
F
121
20
9.78(1.32)
7.64(0.09)
6.48(1.23) 3.58(0.01)
8
11c
M
124.5
24.5
10.49(0.95) 7.43(2.26)
6.23(1.42) 3.02(0.35)
50
7a
M
178
113
14.57(2.04) 12.34(1.08) 6.79(1.62) 3.6(0.15)
8a
F
155
99
12.4(0.01)
55
3a
F
159
68
14.94(2.67) 14(2.15)
9.45(1.65) 7.46(0.8)
62
4a
M
168
91
14.47(2.09) 13.63(2.5)
7.8(1.04)
80
5a
M
173
76
16.13(2.19) 14.27(2.3)
6.89(2.27) 4.47(1.87)
5
6
4.35(1.91)
10.83(2.68) 6.53(2.27) 5.18(0.23)
4.69(1.04)
471 472 473
Subject numbers ending with “c” indicate child subjects, while those ending with “a” indicate adult subjects.
474 475
Generation 0 (Gen.0) is the trachea, Gen.1 are the main bronchi, Gen.2 are the bronchi and Gen.3 are the segmental bronchi.
22
476 477
Stdev is the standard deviation of the range of diameters in each generation (Borojeni et al., 2014).
23
478 479
Table 2: Values of tracheal Reynolds number at different flow rate in different subjects
Subject 3a
Q (L/min) =2 ---
Q Q (L/min) (L/min) = = 10 5 -----
4U 0,trachea X Q Q (L/min) (L/min) = 30 = 15 1436 2872
Subject 4a
---
---
---
1483
2965
5931
8896
Subject 5a
---
---
---
1331
2661
5322
7984
Subject 7a
---
---
---
1473
2946
5893
8839
Subject 8a
---
---
---
1730
3460
6920
10380
Subject 9c
378
946
1892
2838
5676
11353
---
Subject 6c
337
842
1684
2526
5052
10104
---
Subject 5c
358
895
1790
2685
5371
10742
---
Subject 13c
292
731
1462
2193
4386
8773
---
Subject 2c
406
1014
2029
3043
6087
12174
---
Subject 3c
358
895
1790
2685
5370
10739
---
Subject 11c
273
682
1364
2046
4093
8185
---
Subject 10c
400
1000
2000
3000
6000
11999
---
Subject 12c
386
965
1930
2895
5790
11580
---
Subject 14c
400
999
1998
2997
5993
11987
---
Subject ID#
+ =
480 24
Q (L/min) = Q 60 (L/min) = 90 5744 8616
481
Table 3: Fraction of flow to the right lung. Data are presented as average (standard deviation).
482
N=5 for the adult replicas, and =10 for the child replicas.
483
Flow rate Katz (L/min) model
Adult Pedley model
Modified Pedley model -
Katz model
Child Pedley model
0.64(0.15)
0.63(0.15)
Modified Pedley model 0.63(0.15)
2
-
-
5
-
-
-
0.64(0.15)
0.63(0.15)
0.63(0.15)
10
-
-
-
0.64(0.15)
0.63(0.15)
0.63(0.15)
15
0.58(0.02)
0.58(0.03)
0.58(0.04)
0.63(0.15)
0.63(0.15)
0.63(0.15)
30
0.58(0.02)
0.58(0.03)
0.58(0.04)
0.62(0.13)
0.63(0.15)
0.62(0.15)
60
0.57(0.02)
0.58(0.03)
0.58(0.04)
0.61(0.12)
0.62(0.15)
0.62(0.15)
90
0.56(0.02)
0.58(0.04)
0.58(0.04)
484 485 486 487 488 489 490 491 492
25
-
-
-
493
9. Figure Captions
494 495 496
Figure 1: Experimental set-up used to measure pressure drop in adult and child CT-based
497
airway replicas.
498 499
Figure 2: Schematic of a simple branching airway network with numbering system used in the
500
analytical model.
501 502
Figure 3: Comparison of pressure drop data predicted by (a) the Katz et al. (2011) model, (b) the
503
Pedley et al. (1970) model, and (c) the modified Pedley model (van Ertbruggen et al., 2005) with
504
the measured data in adult replicas. The concordance correlation coefficient, ρc,between
505
modeled and experimental pressure drops is 0.84, 0.88, and 0.93 in panels (a), (b), and (c),
506
respectively.
507
508
Figure 4: (a) Comparison of pressure drop predicted by the Katz et al. (2010) model with the
509
measured data in child replicas. (b) Magnified view in the low range of pressure (0-100 Pa).
510
The concordance correlation coefficient, ρc, in panel (a) and (b) is 0.75 and 0.77, respectively.
511
512
Figure 5: (a) Comparison of pressure drop predicted by the Pedley et al. (1970) model with the
513
measured data in child replicas. (b) Magnified view in the low range of pressure (0-100 Pa). The
514
concordance correlation coefficient, ρc, in panel (a) and (b) is 0.61 and 0.84, respectively.
26
515
516
Figure 6: (a) Comparison of pressure drop predicted by the modified Pedley model (van
517
Ertbruggen et al., 2005) with the measured data in child replicas. (b) Magnified view in the low
518
range of pressure (0-100 Pa). The concordance correlation coefficient, ρc, in panel (a) and (b) is
519
0.43 and 0.65, respectively.
520
27
521 522 523 524 525
10. Figures 1
526 527 528 529
28
530 531 532 533
2
534 535 536 537 538 539 540 541 542 543 544 545 546 547 548 549 550 551 552 553 554 555 556 557 558
29
559 560
3 (a)
561 562 563
(b)
564 565 30
566
(c)
567
568 569 570 571 572 573 574 575 576 577 578 579 580 581 582 583 584 585 586 587 588 589 590 591
31
592 593
4 (a)
594 595
(b)
596 597 598 32
599 600 601 602 603
5 (a)
604 605
(b)
606 33
607 608 609 610 611 612
613 614
6 (a)
(b)
615
34
Validation of airway resistance Models for predicting pressure loss through anatomically realistic conducting airway Replicas of Adults and children Azadeh A.T. Borojeni, Michelle L. Noga, Andrew R. Martin, Warren H. Finlay
www.elsevier.com/locate/jbiomech
PII: DOI: Reference:
S0021-9290(15)00207-9 http://dx.doi.org/10.1016/j.jbiomech.2015.03.035 BM7117
To appear in:
Journal of Biomechanics
Accepted date: 27 March 2015 Cite this article as: Azadeh A.T. Borojeni, Michelle L. Noga, Andrew R. Martin, Warren H. Finlay, Validation of airway resistance Models for predicting pressure loss through anatomically realistic conducting airway Replicas of Adults and children, Journal of Biomechanics, http://dx.doi.org/10.1016/j.jbiomech.2015.03.035 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
1 2
Validation of Airway Resistance Models for Predicting Pressure Loss through
3
Anatomically Realistic Conducting Airway Replicas of Adults and Children
4
Azadeh A. T. Borojeni1,*, Michelle L. Noga2, Andrew R. Martin1’**, Warren H. Finlay1’**
5 6 7 8
9
10
11
1.
Department of Mechanical Engineering, University of Alberta, Edmonton, Alberta, Canada T6G 2G8
12
2.
Department of Radiology and Diagnostic Imaging, University of Alberta, Edmonton, Alberta, Canada
13 14
*
15
+1 780 492 2200
16
**
17
+1 780 492 4707; Fax: +1 780 492 2200
Corresponding author (during review and publication), Email: [email protected]; Tel: +1 780 492 2161; Fax:
Principal corresponding authors (post-publication), Email: [email protected], [email protected]; Tel:
18
19 20
Keywords: Airway resistance; conducting airways; pressure drop; pressure loss; ventilation
21
distribution
1
22
Abstract
23 24
This work describes in vitro measurement of the total pressure loss at varying flow rate through
25
anatomically realistic conducting airway replicas of ten children, 4 to 8 years old, and five adults.
26
Experimental results were compared with analytical predictions made using published airway
27
resistance models. For the adult replicas, the model proposed by van Ertbruggen et al. (J. Appl.
28
Physiol. 98:970-980,2005) most accurately predicted central conducting airway resistance for
29
inspiratory flow rates ranging from 15 to 90 L/min. Models proposed by Pedley et al. (J. Respir.
30
Physiol. 9:371-386,1970) and by Katz et al. (J. Biomechanics 44:1137-1143,2011) also provided
31
reasonable estimates, but with a tendency to over predict measured pressure loss for both models.
32
For child replicas, the Pedley and Katz models both provided good estimation of measured
33
pressure loss at flow rates representative of resting tidal breathing, but under predicted measured
34
values at high inspiratory flow rate (60 L/min). The van Ertbruggen model, developed based on
35
flow simulations performed in an adult airway model, tended to under predict measured pressure
36
loss through the child replicas across the range of flow rates studied (2 to 60 L/min). These
37
results are intended to provide guidance for selection of analytical pressure loss models for use in
38
predicting airway resistance and ventilation distribution in adults and children.
39 40 41
2
42
1. Introduction
43
Airway resistance is an important parameter used in describing the mechanics of breathing. The
44
relationship between pressure loss and flow rate through the conducting airways influences work
45
of breathing, and is widely used for diagnostic purposes in detecting and classifying respiratory
46
diseases. In addition, variation in airway resistance between individual airways or groups of
47
airways along different paths through the lung plays a role in determining the distribution of
48
ventilation between different lung regions. Ventilation distribution in turn influences regional
49
gas transport, mixing, and uptake, as well as transport and deposition of inhaled aerosols.
50
Analytical models of airway resistance have been employed recently, for example, in modeling
51
topographic distribution of ventilation in healthy lungs (Swan et al., 2012), in estimating
52
contributions of peripheral airway resistance to ventilation distribution measured by PET
53
imaging (Wongviriyawong et al., 2013), in predicting distribution of pressure through the airway
54
tree during xenon anesthesia (Katz et al., 2012), and in optimizing algorithms used to predict
55
alveolar pressure during mechanical ventilation (Damanhuri et al., 2014).
56 57
Study of pressure loss over the conducting airways dates to the foundational work of Rohrer
58
(1915). For a given flow rate, the difference in pressure between the airway opening (nose or
59
mouth) and the alveoli was expressed as a summation of linear and quadratic terms, each term
60
weighted by a dimensional constant.
61
depending on airway dimensions and geometry. The utility of such an approach was clearly
62
demonstrated by Otis and colleagues, who used the Rohrer airway resistance model in
63
combination with experimental measurements to describe work of breathing (Otis et al., 1950),
64
ventilation distribution (Otis et al., 1956), and effects of breathing different gas mixtures on
Values of these constants varied between subjects,
3
65
airway resistance (Otis et al., 1949). These studies provided early insight into the functional
66
relationship between pressure loss, flow rate, gas properties, and airway geometry. However,
67
significant further understanding was gained through fundamental experiments and fluid
68
dynamics analysis performed by Pedley and colleagues (1970). These authors presented an
69
analytical model of pressure loss through conducting airways of the human bronchial tree based
70
on measurements performed in idealized, symmetric airway bifurcations (Pedley et al., 1970).
71
Pressure loss was found to exceed that predicted by assuming fully developed Hagen-Poiseuille
72
flow, and a correction factor was proposed based on airway length and diameter, and the gas
73
Reynolds number (Pedley et al., 1970).
74 75
More recently, computational fluid dynamics (CFD) simulations have been performed to
76
investigate pressure loss through idealized (e.g. Wilquem and Degrez, 1997; Comer et al., 2001;
77
Kang et al., 2011; Katz et al., 2011) and anatomically realistic (e.g. van Ertbruggen et al., 2005;
78
Gemci et al., 2008; Ismail et al., 2013) adult airway bifurcations. Previous authors have noted
79
that, for adult airways, the Pedley model overestimates simulated pressure losses (Comer et al.,
80
2001; van Ertbruggen et al., 2005; Katz et al., 2011; Ismail et al., 2013).
81
alternative analytical models have been proposed (van Ertbruggen et al., 2005; Katz et al., 2011).
82
In contrast, considerably less work has been done to model airway pressure losses for the
83
conducting airways of children. The accuracy of using analytical models developed based on
84
measurements or simulations performed in adult airway models in predicting pressure loss in
85
children’s airways is presently unknown.
86
4
Accordingly,
87
In this paper, we describe in vitro measurements of the total pressure loss at varying flow rate
88
through anatomically realistic conducting airway replicas of adults and children. The in vitro
89
approach employed presents a unique opportunity to reduce the pressure-flow relationship
90
through the three-dimensional, multi-branching, conducting airway geometry to the classic fluid
91
mechanics problem of flow through a piping network comprising nonlinear resistances (White
92
2010). Accordingly, analytical predictions of the total pressure loss through the airway replicas
93
made using airway resistance models proposed by Pedley et al. (1970), van Ertbruggen et al.
94
(2005), and Katz et al. (2011) are compared to measured experimental values. Our results are
95
intended to provide guidance for selection of analytical models for use in predicting airway
96
resistance in adults and children.
97 98
2.
Methods
99 100
2.1 Airway pressure drop measurement
101 102
Fabrication of realistic adult and child conducting airway replicas has been explained in detail in
103
previous work (Borojeni et al., 2014). In brief, conducting airways were identified using Mimics
104
software (MIMICS 3D, Materialise, Ann Arbor, MI, USA) as regions of interest in CT-images of
105
5 adults and 10 children. Subject information and average airway dimensions are shown in
106
Table 1. A rapid prototyping technique (Invision® SR 3D printer, 3D systems, Rock Hill, SC,
107
USA) was used to fabricate the conducting airway replicas from acrylonitrile butadiene styrene
108
(ABS) plastic (P430, Stratasys, Eden Prairie, MN).
109
5
110
In the present work, pressure drop across the airway replicas was measured in vitro using a
111
digital pressure gauge (HHP-103, Omega Engineering, Inc., Stamford, CT) with low range (0-
112
498 Pa), high range (0-2590 Pa) and accuracy of ±0.2% full scale. The upstream pressure tap
113
was open to atmosphere and the downstream pressure tap was positioned in the tubing that
114
connected the outlet of an artificial thoracic chamber to a vacuum pump (Figure 1). Preliminary
115
measurements were made with the chamber open to characterize the pressure drop across only
116
the chamber outlet at flows ranging from 2 to 90 L/min. Constant flow rates were then drawn
117
through the chamber with the replicas in place by operating a vacuum pump, with flow rates
118
chosen in the range of 15 to 90 L/min for adult replicas and 2 to 60 L/min for child replicas. The
119
desired flow rate was set using a needle valve and monitored by a digital mass flowmeter (TSI
120
Model 4000 series, TSI Inc., Shoreview, MN, USA) downstream of the thoracic cylindrical
121
chamber. The pressure drop across the airway replicas is reported below as the difference
122
between the pressure drop measured with and without the airway replicas in place. Each
123
experimental measurement was repeated three times.
124 125
2.2 Analytical pressure loss model
126 127
Measured values of pressure drop across the airway replicas were compared with predicted
128
values made using three previously published models of airway resistance (Pedley et al., 1970;
129
Katz et al., 2011; van Ertbruggen et al., 2005). For each replica, subject-specific geometry of the
130
conducting airways was required as input in the analytical models of flow distribution and
131
resistance.
132
6
133 134
2.2.1 Determination of Airway Dimensions
135
Diameters and lengths for all airway segments in each replica were measured from segmented
136
CT images using a - post processing software package (MAGICS, Materialise, MI, USA). The
137
length of a segment was defined as the centerline path length from one bifurcation to the next.
138
The diameter was defined as the average of equivalent cross-sectional area diameters evaluated
139
every 2 mm along the length of the segment.
140 141
2.2.2 General Solution Procedure
142 143
For convenience of calculation, a hydraulic resistance, R, is defined for a segment as R =
144
∆h
(1)
145
146
where Q is the flow rate through the airway and ∆h is a form of the head loss, expressed in
147
dimensions of energy per unit mass as the ratio between pressure drop across the airway (∆p) and
148
fluid density (ρ), i.e. Δh=Δp/ρ .
149
Expressions for the head loss as a function of airway geometry, fluid properties and flow rate
150
result from the application of specific airway resistance models. These will be provided in the
151
following section. At present, Equation 1 is used to outline the solution procedure in general.
152
For example, for the simple network of segments shown in Fig. 2 we wish to determine the
153
fraction of the total flow entering airway 0 that flows through each branch, 1a and 1b. To do so,
154
the resistances of branches 1a and 1b must be taken into account, as must be the resistances of all 7
155
subsequent segments downstream. The equivalent hydraulic resistance of either branch can be
156
determined as follows (White, 2010): eq, 1a = +
/
1
/
[ 2aa + 2ab ] (2)
157
158
eq, 1b = +
/ [ 2ba
1
/
+ 2bb ] (3)
159
160
In the tracheobronchial tree, the distal airways shown in Figure 2 will proceed to bifurcate into
161
further downstream airway generations. It is noted that hydraulic resistances of these
162
downstream segments (i.e. R2aa, R2ab, R2ba, or R2bb) may themselves be expressed as equivalent
163
hydraulic resistance defined using these same principles.
164 165
The head loss through any branch can then be expressed as
166 167
∆hn = R eq,n Q2n (4)
168
169
where the equivalent hydraulic resistance Req,n of that branch is formulated by generalization of
170
equations 2 and 3 and is itself a function of Qn.
171
8
172
Applying this procedure to a branching airway network results in a system of nonlinear
173
equations. For a multiple-path network with n paths, a system of n equations is obtained. This
174
system of equations can be solved iteratively provided that the head loss is the same across each
175
path. This condition is met in the experiments described above given that all terminal branches
176
supply the artificial thorax chamber. The head loss calculation for each path included a minor
177
loss term accounting for dissipation of kinetic energy due to discharge of flow from terminal
178
branches into the artificial thorax. This term was in all cases small compared with the total head
179
loss.
180 181
The iterative solution procedure starts with an initial guess for the flow rate in each branch. This
182
was made assuming evenly distributed flow between daughter branches at all bifurcations. The
183
flow rates through each branch at subsequent iterations may then be calculated according to:
184
a
=
/
( eq,b )# (Qinlet ) /
/
[( eq,a )# + ( eq,b )# ] (5)
185
186
where i+1 and i denote the current and previous iteration, respectively, and subscripts a and b
187
indicate daughter airways. Equivalent hydraulic resistances are flow rate dependent and thus
188
update on each iteration.
189 190
After each iteration, flow rates along each path were compared with corresponding values from
191
the previous iteration. The root mean square (RMS) difference in flow rates between successive
192
iterations was used to determine convergence. The solution was deemed converged when this 9
193
RMS difference fell below 0.006 L/min. The sensitivity of the total head loss to this criterion
194
was trivial for values less than 0.006 L/min. For the converged solution, the total head loss
195
across the airway tree was then calculated using the branch flow rates and resistances.
196 197 198
2.2.3
Airway Resistance Models
199
2.2.3.1 Pedley Model
200
Pedley and colleagues (1970) determined the pressure drop through idealized symmetric models
201
of bronchial airways based on measurement of changes to inspiratory flow profiles. They
202
derived an expression for the ratio relating the actual pressure drop through an airway section to
203
that which would be predicted assuming fully-developed Hagen-Poiseuille flow. This ratio was
204
given by:
205 /
1.85
, $= * + . 4√2
(6)
206
207 208
where D and L are airway diameter and length, and Re is the flow Reynolds number:
209
+ =
4 0,1 (7)
210
211
where 1 is the kinematic viscosity of the fluid (air). 10
212 213
The pressure drop is then calculated as: ∆2pedley = $∆2Hagen-Poiseuille (8)
214
215
216
where, ∆2Hagen-Poiseuille =
128µ- 0,? (9)
217
218
where µ is the fluid dynamic viscosity.
219
Accordingly, the expression for hydraulic resistance, R in a given airway branch for the Pedley
220
model is as follows:
pedley =
128 ∙ 1.85 - A.B A.B * . + √20 ,? , (10)
221
222
where ρ is fluid density.
223
Note that in analyzing the realistic airway replicas, the value of Z was less than 1 at some distal
224
airways where airway diameter was considerably smaller than the airway length and Reynolds
225
number was small. This is not a physically realistic result as the dissipation of energy in realistic
11
226
airways cannot be less than dissipation of energy in the ideal Hagen-Poiseuille flow. Therefore,
227
the value of $ was replaced by 1 in these cases.
228
229
2.2.3.2 Modified Pedley Model (van Ertbruggen model)
230
van Ertbruggen and colleagues ( 2005) used CFD to simulate flow through an anatomically-
231
based model of the airways extending from the trachea to the segmental bronchi. This model
232
combined morphometric data data from Horsfield et al. (1971) with orientation of branching
233
planes based on bronchoscope and CT images. Simulated pressure drop through each airway
234
generation in the model was compared by van Ertbruggen and colleagues ((2005) to values
235
predicted using the Pedley model described above. Pressure drop predicted using the Pedley
236
model was consistently higher than simulated values. The authors therefore proposed a modified
237
form of equation 8, as follows:
∆2ModiDied-Pedley
/ 32µ-G
, = E * + . -
,
238
(11)
239
where values of γ for generations 0-3 are 0.162, 0.239, 0.244 and 0.295 respectively (van
240
Ertbruggen et al., 2005).
241
Using these values for γ, the expression for hydraulic resistance, R, in a given airway branch for
242
the modified Pedley model is: modiDied Pedley =
512 - A.B E ( ) 0 ,? , + A.B
12
(12)
243
244
2.2.3.3 Katz Model
245
Katz and colleagues (2011) presented a model for calculating pressure drop through the
246
conducting airways based on concepts of major and minor head losses.
247
The expression for hydraulic resistance, R, in a given airway branch for the Katz model is:
248
Katz =
8
0 ,? L
[M
+ N] , (13)
249
250
where M is the friction factor, assumed to be 64/Re for Re < 2000 (laminar flow). Above this
251
value, flow is assumed turbulent and the friction factor is calculated using the Blasius correlation
252
for smooth tubes: M=
0.316 + A.B
253
(14)
254
The minor loss coefficient (N) in Equation 13, is calculated using the following correlation based
255
on CFD simulations of flow through idealized symmetric airway bifurcations (Katz et al., 2011): N=
Q +S(log +) + ,(log +) + T + R (15)
256
13
257
Values for the constants A through E in Equation 15 are provided for each tracheal-bronchial
258
lung generation by Katz and colleagues (2011).
259
For each of the three airway resistance models, prediction of measured pressure drop through the
260
adult and child airway replicas was assessed using Lin’s concordance correlation coefficient, ρc
261
(Lin 1989). In the present work, reported values of the concordance correlation coefficient
262
indicate goodness of fit of plotted predicted vs. measured data to the identity line, with higher
263
values indicating better fit and ρc = 1.0 indicating a perfect fit.
264
265
3. Results
266 267
Tracheal Reynolds numbers for adult and child subjects are presented in Table 2. At an
268
equivalent volumetric flow rate, Reynolds numbers were higher in child than in adult replicas,
269
owing to the smaller tracheal diameters (and thus higher flow speeds) of the child replicas.
270
Comparison of measured versus calculated pressure drop through the adult replicas is made in
271
Figures 3 for the three airway resistance models employed. These comparisons indicate there is,
272
in general, reasonable agreement between measured and predicted values, for all three resistance
273
models. It can be further observed that the experimental data are in best agreement with the
274
predicted values made using the modified Pedley model.
275 276
Comparison of measured versus calculated pressure drop through the child replicas is
277
demonstrated in Figures 4-6. Here the Katz model and original Pedley model displayed better
14
278
agreement with measured values than did the modified Pedley model, though all three models
279
had a tendency to under predict measured data for the highest measured values of pressure drop,
280
corresponding to higher flow rates through the replicas.
281 282
In calculating the pressure drop across the replicas, the distribution of flow through the
283
branching airways is also calculated. Calculated values of the fraction of flow through the right
284
lung are presented in Table 3. For both adults and child replicas, a greater fraction of the
285
inhalation flow passes through airway of the right lung. The standard deviation between replicas
286
was higher in the child group than in the adult group (0.15 versus 0.03 on average, respectively).
287 288 289
4. Discussion
290 291
In the present work, we measured pressure loss at varying flow rate through conducting airway
292
replicas of adults and children. Measurements were compared with values predicted using
293
analytical airway resistance models previously developed based on measurements or simulations
294
performed in adult conducting airway geometries. For the present five adult airways, the airway
295
resistance model proposed by Pedley and colleagues (1970) tended to over-predict measured
296
pressure loss. This is consistent with the results of previous CFD studies comparing predictions
297
of conducting airway pressure loss made using the Pedley model to simulated values (Comer et
298
al., 2001; van Ertbruggen et al., 2005; Katz et al., 2011; Ismail et al., 2013). Of the three
299
resistance models evaluated, the modified Pedley model proposed by van Ertbruggen et al.
300
(2005) demonstrated closest agreement with our measured adult data. These authors modified
301
the original Pedley model by proposing alternative coefficients for predicting the pressure loss 15
302
through each proximal lung generation. The modified coefficients were based on pressure losses
303
determined by CFD simulation of laminar flow through a single, representative, anatomically-
304
based adult airway geometry. The good agreement of predicted pressure losses made using their
305
model with our data in five different adult airway replicas, at tracheal Reynolds numbers
306
exceeding 10,000 in the highest case (Table 2), was therefore not expected a priori. However,
307
van Ertbruggen et al. (2005) argue that below Reynolds number averaging 8,000 the flow regime
308
through conducting airways can be considered laminar or ‘lightly’ transitional, but not turbulent.
309
Again with reference to Table 2, for a strong majority of cases we studied tracheal Reynolds
310
numbers in the adult replicas that fell below 8,000.
311
angle (and branching asymmetry) are not explicitly accounted for in the modified Pedley model,
312
and surely existed between our five adult replicas, Kang et al. (2011) found only a weak
313
influence of branching angle on simulated pressure loss through a model airway bifurcation.
Furthermore, while variation in branching
314 315
For the child airway replicas, predicted and measured pressure losses were in closer agreement
316
for the Katz model and the original Pedley model than for the modified Pedley model. Poorer
317
agreement of the modified Pedley model with the child versus adult data may be due to
318
differences in central conducting airway geometry between children and the single adult
319
geometry used in creating this model, that are not captured explicitly in Equation 12 (recalling
320
that model coefficients were fit to simulation data obtained in a representative single adult
321
airway geometry).
322 323
All three models tended to under predict measured pressure loss at the highest flow rate studied
324
in children (60 L/min), where tracheal Reynolds numbers averaged over 10,000. Indeed, for
16
325
replicas from the youngest children studied (ages 4 and 5), the pressure loss at 60 L/min was
326
sufficiently high that such an inspiratory flow rate is likely to be achieved in vivo by the
327
corresponding child subjects only near maximum inspiratory effort (Vilozni et al., 2009). At
328
lower flow rates, corresponding to lower pressure loss, the original Pedley model was the most
329
accurate of the three models studied. Pedley et al. (1970) based their model on measurements
330
made in idealized symmetric models of branching airways, where Reynolds number ranged from
331
~100 to ~1,000. Tracheal Reynolds numbers in our child replicas were on average within this
332
range for the two lowest flow rates studied (2 and 5 L/min).
333
relatively close agreement between our measured data and predictions made using the Pedley
334
model for low pressure loss (where flow rate and Reynolds number were also low) and lack of
335
such agreement for higher pressure loss.
This very likely explains the
336 337
In calculating the total pressure loss through the airway replicas using each of the three airway
338
resistance models, the distribution of flow through the replicas is also obtained. The fraction of
339
flow directed to the right lung side of the replicas was found to average 0.58 in the adult replicas,
340
with little variation with flow rate or between replicas.
341
distribution of ventilation to the right lung observed in vivo (Lumb 2010), especially given that
342
for the intact, healthy, in vivo lung, ventilation distribution is determined more by downstream
343
tissue compliance than by airway resistance (Swan et al., 2012). Ostensibly, the dimensions of
344
the central conducting airways act to reinforce, rather than oppose, this ventilation distribution.
345
For the child replicas, the fraction of flow directed to the right lung side of the replicas
346
(averaging 0.63) did not vary significantly from that for the adult replicas. However, variation
347
between replicas was considerably greater for child than for adult replicas. This variation is
17
This value is consistent with the
348
potentially attributable to a greater variation in airway geometry for the child replicas, given that
349
these geometries were obtained from images of children aged 4-8 years, and therefore at
350
different stages of physical development.
351 352
A limitation in the present work is the absence of any upper, extrathoracic airway upstream from
353
our conducting airway replicas. In this regard, our replicas are similar to the original single and
354
multiple bifurcation geometries used to develop the three analytical resistance models studied
355
herein. None of these models was based on measurements or simulations done in airway
356
geometries that included the upper airway.
357
performed by Xi et al. (2008), the upper airway most significantly influences flow features
358
within the trachea, whereas flow in subsequent downstream airway branches is much less
359
affected. The accuracy of the resistance models in predicting in vivo pressure loss through the
360
trachea, directly downstream from the upper airway, was not assessed in the present study.
As observed in numerical flow simulations
361 362
5. Conclusion
363
Pressure loss was experimentally measured at varying flow rate through anatomically realistic
364
conducting airway replicas of adults and children. Measured values were compared with values
365
predicted using published analytical airway resistance models. The modified Pedley model
366
proposed by van Ertbruggen et al. (2005) provided close agreement with measured central
367
conducting airway resistance in adult replicas across a wide range of inspiratory flow rates, from
368
15 to 90 L/min. In contrast, in child airway replicas, the Pedley et al. (1970) model most closely
369
matched the experimental data.
370
18
371 372
6. Acknowledgements
373
Andrew Grosvenor and Jamie Schmitt from the Institute for Reconstructive Sciences in Medicine
374
(iRSM) are acknowledged for their technical help in fabricating the replicas. Greg Wandzilak
375
from PET/CT center, University of Alberta, Stollery Children’s Hospital is gratefully
376
acknowledged for providing CT scans. Travis MacLean and John Chen are appreciated for their
377
help in measuring pressure drop in the child replicas.
378 379
7. Conflict of Interest
380
The authors report no conflict of interest.
381
19
382 383 384 385 386 387 388 389 390 391 392 393 394 395
8. References
396 397 398 399 400 401 402
van Ertbruggen, C., Hirsch, C., Paiva, M., 2005. Anatomically based three-dimensional model of airways to simulate ow and particle transport using computational fluid ynamics. Journal of Applied Physiology. 98, 970-980.
403 404 405 406 407 408 409 410 411 412
Horsfield, K., Dart, G., Olson, D.E., Filley, G.F., Cumming, G., 1971. Models of the human bronchial tree. Journal of Applied Physiology. 31, 207–217.
413 414 415 416 417 418 419 420 421 422 423 424 425
Katz, I.M., Martin, A.R., Muller, P.-A., Terzibachi, K., Feng, C.-H., Caillibotte, G., Sandeau, J., Texereau, J., 2011. The ventilation distribution of helium-oxygen mixtures and the role of inertial losses in the presence of heterogeneous airway obstructions. Journal of Biomechanics. 44, 1137-1143.
Borojeni, A.A., Noga, M.L., Vehring, R., Finlay, W.H., 2014. Measurements of total aerosol deposition in intrathoracic conducting airway replicas of children. Journal of Aerosol Science. 73, 39-47. Comer, J.K., Kleinstreuer C, Zhang Z., 2001. Flow structures and particle deposition patterns in double bifurcation airway models. 1. Airflow fields. Journal of Fluid Mechanics. 435, 25–54. Damanhuri, N.S., Docherty, P.D., Chiew, Y.S., van Drunen, E.J., Desaive, T., and Chase, J.G., 2014. A Patient-specific airway branching model for mechanically ventilated patients. Computational and Mathematical Methods in Medicine, Article ID 645732.
Gemci, T., Ponyavin, V., Chen, Y., Chen, H., Collins, R., 2008. Computational model of airflow in upper 17 generations of human respiratory tract. Journal of Biomechanics. 41(9), 2047-2054.
Ismail, M., Comerford, A., Wall, W.A., 2013. Coupled and reduced dimensional modeling of respiratory mechanics during spontaneous breathing. International Journal For Numerical Methods In Biomedical Engineering. 29, 1285-1305. Kang, M-Y, Hwang, J., and Lee, J-W., 2011. Effect of geometric variations on pressure loss for a model bifurcation of the human lung airways. Journal of Biomechanics. 44, 1196–1199.
Katz, I. M., Martin, A. R., Feng, C. H., Majoral, C., Caillibotte, G., Marx, T., Bazin, J.-E., and Daviet, C. 2012. Airway pressure distribution during xenon anesthesia: the insufflation phase at constant flow (volume controlled mode). Applied Cardiopulmonary Pathophysiology, 16, 5-16. Lin, IL. 1989. A concordance correlation coefficient to evaluate reproducibility. Biometrics, 45, 255-268.
20
426 427 428
Lumb, A.B., 2010. Nunn’s Applied Respiratory Physiology, Seventh Edition. Elsevier Ltd.
429
Otis A.B., Bembower W.C., 1949. Effect of gas density on resistance to respiratory gas
430 431
flow in man. Journal of Applied Physiology. 2, 300-306.
432
Otis A.B., Fenn W.O., Rahn H., 1950. Mechanics of Breathing in Man. Journal of
433
Applied Physiology. 2, 592-607.
434 435
Otis A.B., McKerrow C.B., Bartlett R.A., Mead J., McIlroy M.B., Selverstone N.J.,
436
Radford E.P., 1956. Mechanical factors in distribution of pulmonary ventilation. Journal
437 438 439 440 441 442 443 444 445
of Applied Physiology. 8, 427-443.
446 447 448 449 450 451 452 453 454 455 456 457 458
Swan, A.J., Clark, A.R., Tawhai, M.H., 2012. A computational model of the topographic distribution of ventilation in healthy human lungs. Journal of Theoretical Biology. 300, 222-231.
459 460 461
Wongviriyawong, C., Harris, R. S., Greenblatt, E., Winkler, T., and Venegas, J. G., 2013. Peripheral resistance: a link between global airflow obstruction and regional ventilation distribution. Journal of Applied Physiology, 114(4), 504-514.
462 463 464
Xi, J., Longest, P.W., and Martonen, T.B., 2008. Effects of the laryngeal jet on nanoand microparticle transport and deposition in an approximate model of the upper tracheobronchial airways. Journal of Applied Physiology, 104(6), 1761-1777.
Pedley, T.J., Schroter, R.C., Sudlow, M.F., 1970. Energy losses and pressure drop in models of human airways. Journal of Respiration Physiology. 9, 371-386. Rohrer, F., 1915. Der Strömungswiderstand in den menschlichen Atemwegen und der Einfluss der unregelmässigen Verzweigungen des Bronchialsystems auf den Atmungsverlauf in verschiedenen Lungenbezirken. Pflügers Arch ges Physiol, 162, 225–299.
Vilozni, D., Efrati, O., Barak, A., Yahav, Y., Augarten, A., Bentur, L., 2009. Forced inspiratory flow volume curve in healthy young children. Pediatric Pulmonology. 44(2), 105-111. White, F. M., 2010. Fluid Mechanics, 7th edition. Boston, USA: McGraw-Hill. Wilquem, F., Degrez. G., 1997. Numerical modeling of steady inspiratory airflow through a three-generation model of the human central airways. Journal of Biomechanical Engineering. 119(1), 59-65.
465
21
466 467 468
9. Tables
469
Table 1: Summary of child and adult subjects information and airway diameters of child and
470
adult replicas. Age
Subject
(year)
Number
4
Sex
Height
Weight
Diameter
Diameter
Diameter
Diameter
(cm)
(kg)
(stdev)
(stdev)
(stdev)
(stdev)
(Gen.0)
(Gen.1)
(Gen.2)
(Gen.3)
(mm)
(mm)
(mm)
(mm)
10c
F
99
16
7.15(0.04)
4.7(1.19)
3.48(0.27) 2.2(0.23)
14c
F
100
16
7.16(0.15)
6.47(2.77)
4.46(0.69) 2.23(0.76)
2c
M
117
22.9
7.05(0.09)
6.03(0.7)
6.00(1.4)
3c
M
112
20
7.99(0.51)
5.39(0.51)
4.93(0.49) 2.33(0.66)
9c
M
113
20
7.56(0.01)
6.44(0.87)
5.26(0.01) 3.38(1.05)
5c
F
112
18
7.99(0.42)
5.36(1.14)
5.35(1.41) 3.55(1.00)
6c
F
118
21.5
8.5(0.3)
6.75(1.32)
5.94(1.32) 2.9(1.47)
12c
F
124
24
7.41(0.46)
6.45(2.72)
3.28(0.26) 3.23(0.9)
7
13c
F
121
20
9.78(1.32)
7.64(0.09)
6.48(1.23) 3.58(0.01)
8
11c
M
124.5
24.5
10.49(0.95) 7.43(2.26)
6.23(1.42) 3.02(0.35)
50
7a
M
178
113
14.57(2.04) 12.34(1.08) 6.79(1.62) 3.6(0.15)
8a
F
155
99
12.4(0.01)
55
3a
F
159
68
14.94(2.67) 14(2.15)
9.45(1.65) 7.46(0.8)
62
4a
M
168
91
14.47(2.09) 13.63(2.5)
7.8(1.04)
80
5a
M
173
76
16.13(2.19) 14.27(2.3)
6.89(2.27) 4.47(1.87)
5
6
4.35(1.91)
10.83(2.68) 6.53(2.27) 5.18(0.23)
4.69(1.04)
471 472 473
Subject numbers ending with “c” indicate child subjects, while those ending with “a” indicate adult subjects.
474 475
Generation 0 (Gen.0) is the trachea, Gen.1 are the main bronchi, Gen.2 are the bronchi and Gen.3 are the segmental bronchi.
22
476 477
Stdev is the standard deviation of the range of diameters in each generation (Borojeni et al., 2014).
23
478 479
Table 2: Values of tracheal Reynolds number at different flow rate in different subjects
Subject 3a
Q (L/min) =2 ---
Q Q (L/min) (L/min) = = 10 5 -----
4U 0,trachea X Q Q (L/min) (L/min) = 30 = 15 1436 2872
Subject 4a
---
---
---
1483
2965
5931
8896
Subject 5a
---
---
---
1331
2661
5322
7984
Subject 7a
---
---
---
1473
2946
5893
8839
Subject 8a
---
---
---
1730
3460
6920
10380
Subject 9c
378
946
1892
2838
5676
11353
---
Subject 6c
337
842
1684
2526
5052
10104
---
Subject 5c
358
895
1790
2685
5371
10742
---
Subject 13c
292
731
1462
2193
4386
8773
---
Subject 2c
406
1014
2029
3043
6087
12174
---
Subject 3c
358
895
1790
2685
5370
10739
---
Subject 11c
273
682
1364
2046
4093
8185
---
Subject 10c
400
1000
2000
3000
6000
11999
---
Subject 12c
386
965
1930
2895
5790
11580
---
Subject 14c
400
999
1998
2997
5993
11987
---
Subject ID#
+ =
480 24
Q (L/min) = Q 60 (L/min) = 90 5744 8616
481
Table 3: Fraction of flow to the right lung. Data are presented as average (standard deviation).
482
N=5 for the adult replicas, and =10 for the child replicas.
483
Flow rate Katz (L/min) model
Adult Pedley model
Modified Pedley model -
Katz model
Child Pedley model
0.64(0.15)
0.63(0.15)
Modified Pedley model 0.63(0.15)
2
-
-
5
-
-
-
0.64(0.15)
0.63(0.15)
0.63(0.15)
10
-
-
-
0.64(0.15)
0.63(0.15)
0.63(0.15)
15
0.58(0.02)
0.58(0.03)
0.58(0.04)
0.63(0.15)
0.63(0.15)
0.63(0.15)
30
0.58(0.02)
0.58(0.03)
0.58(0.04)
0.62(0.13)
0.63(0.15)
0.62(0.15)
60
0.57(0.02)
0.58(0.03)
0.58(0.04)
0.61(0.12)
0.62(0.15)
0.62(0.15)
90
0.56(0.02)
0.58(0.04)
0.58(0.04)
484 485 486 487 488 489 490 491 492
25
-
-
-
493
9. Figure Captions
494 495 496
Figure 1: Experimental set-up used to measure pressure drop in adult and child CT-based
497
airway replicas.
498 499
Figure 2: Schematic of a simple branching airway network with numbering system used in the
500
analytical model.
501 502
Figure 3: Comparison of pressure drop data predicted by (a) the Katz et al. (2011) model, (b) the
503
Pedley et al. (1970) model, and (c) the modified Pedley model (van Ertbruggen et al., 2005) with
504
the measured data in adult replicas. The concordance correlation coefficient, ρc,between
505
modeled and experimental pressure drops is 0.84, 0.88, and 0.93 in panels (a), (b), and (c),
506
respectively.
507
508
Figure 4: (a) Comparison of pressure drop predicted by the Katz et al. (2010) model with the
509
measured data in child replicas. (b) Magnified view in the low range of pressure (0-100 Pa).
510
The concordance correlation coefficient, ρc, in panel (a) and (b) is 0.75 and 0.77, respectively.
511
512
Figure 5: (a) Comparison of pressure drop predicted by the Pedley et al. (1970) model with the
513
measured data in child replicas. (b) Magnified view in the low range of pressure (0-100 Pa). The
514
concordance correlation coefficient, ρc, in panel (a) and (b) is 0.61 and 0.84, respectively.
26
515
516
Figure 6: (a) Comparison of pressure drop predicted by the modified Pedley model (van
517
Ertbruggen et al., 2005) with the measured data in child replicas. (b) Magnified view in the low
518
range of pressure (0-100 Pa). The concordance correlation coefficient, ρc, in panel (a) and (b) is
519
0.43 and 0.65, respectively.
520
27
521 522 523 524 525
10. Figures 1
526 527 528 529
28
530 531 532 533
2
534 535 536 537 538 539 540 541 542 543 544 545 546 547 548 549 550 551 552 553 554 555 556 557 558
29
559 560
3 (a)
561 562 563
(b)
564 565 30
566
(c)
567
568 569 570 571 572 573 574 575 576 577 578 579 580 581 582 583 584 585 586 587 588 589 590 591
31
592 593
4 (a)
594 595
(b)
596 597 598 32
599 600 601 602 603
5 (a)
604 605
(b)
606 33
607 608 609 610 611 612
613 614
6 (a)
(b)
615
34
