What Causes Uneven Aerosol Deposition in the Bronchoconstricted Lung? A Quantitative Imaging Study

Overview

In this section we define a general framework that may be used to isolate sources of variable deposition among lobes, or any other region of the lung periphery that is subtended by a unique airway. We begin by defining appropriate measures of deposition and ventilation that are not sensitive to differences in volume among lobes, and introduce the notions of retention fraction and aerosol concentration. We identify two factors that can influence the concentration along each airway and identify the net contribution of each factor to differences in aerosol deposition among lobes. We then collect all sources of variability into a single expression that describes lobar deposition in Equation (6). To better compare these sources of variability across different subjects, we account for differences in overall dosing, lung size, centrality of deposition, and restate the expression in nondimensional, normalized form in Equation (9). After several simplifications, we decouple the sources of variability in Equation (15).

Lobar measures of deposition and ventilation

Deposition beyond the subsegmental airways within a lobe can be characterized by a specific deposition sD_sl, where the subscript s indicates the subject studied and the subscript l indicates the lobe (or any other lung region). In this article, regions beyond the subsegmental airways are referred to as peripheral regions, while more proximal regions are referred to as central airways. sD_sl is defined as the aerosol deposition with a lobe's periphery normalized by the total volume of the lobe measured at the mean lung volume during the subjects' tidal breathing (MLV). The specific deposition allows for meaningful comparisons between lobes of different volumes.

The ventilation of a lobe or lung region is similarly characterized by its regional specific ventilation \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$s \dot{V}_{sl}$$ \end{document} , defined as the ventilation of the region per unit total volume at MLV to allow comparisons between these variables and among lobes or regions of different sizes.

Retention fraction and aerosol concentration

The retention fraction η _x of a location x along the airways is the fraction of the total aerosol that passes that location during inhalation that deposits. When ventilation data is acquired in conjunction with the deposition data, the concentration C_x of aerosol in the inhaled air can be evaluated at any location x along the airway tree. This is the average concentration along the cross section during inhalation of the aerosol, and is the ratio of aerosol to air volume that passed that location. The aerosol that passed the location is found by dividing the total deposition that deposited distal to that location D_x with the retention fraction of that location η _x (to account for exhaled aerosol that did not deposit). For a cumulative volume of air V_x passing location x during inhalation: \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*}C_x = \frac { D_x } { \eta_x V_x } \tag { { \rm Eq. } \ 1 } \end{align*} \end{document}

Note the units of the concentration will be aerosol per unit volume of air. The aerosol itself can be described in terms of mass, volume, or number of particles.

Individual airway branching factors and escape fractions

When the airflow and the aerosol entering into a bifurcation split differently between daughter branches, the aerosol concentration entering a daughter the k^th airway C_{airway k, ent.} is no longer equal to the aerosol concentration leaving the parent airway C_{parent k,exit}. We can therefore define a Branching Factor B_k for an airway k as: \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*}B_k = \frac { C_ { airway \ k , ent. } } { C_ { parent \ k , exit } } \tag { { \rm Eq. } \ 2 } \end{align*} \end{document}

Thus a value of B_k≠1 quantifies a change in aerosol concentration between mother and daughter airways (Fig. 1).

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FIG. 1.

Concentration changes at a bifurcation and along an airway k are characterized by two transitions: Bk captures the change in concentration at the inlet of the airway, and Ek captures the change in concentration along an airway.

A second factor affecting the concentrations is the aerosol deposition along airways; as aerosol deposits along an airway wall, the concentration of aerosol in suspension is reduced.⁽⁶⁾ The concentration of the aerosol leaving an airway, C_{airway k, exit}, and that entering it, C_{airway k, ent.} (Fig. 1), can be used to define an escape fraction, E_K, as the ratio of these concentrations: \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*}E_k = \frac { C_ { airway \ k , exit } } { C_ { airway \ k , ent. } } \tag { { \rm Eq. \ 3 } } \end{align*} \end{document}

Note that because the volume of air entering and exiting an airway is identical, the escape fraction can be determined without knowing the ventilation. By combining Equations (2) and (3), it can be shown that the product of B_k and E_k is the ratio of C_{airway k, exit} to C_{parent k, exit} : \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*}B_k E_k = \frac { C_ { airway \ k , exit } } { C_ { parent \ k , exit } } \tag { { \rm Eq. } \ 4 } \end{align*} \end{document}

Net branching factors and net escape fractions

The path between the carina and the periphery of each lobe includes a series of bifurcations and airways that partition and trap the aerosol. For a lobe of a subject we can calculate the ratio of average aerosol concentrations between that leaving the subsegmental airways of that lobe C_sl to that entering the tracheal carina C_carina,s. Thus, for a set of airways ψ_sl leading to a specific lobe of given subject the net escape fraction can be written in terms of the product of branching factor and escape fractions: \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} \frac { C_ { sl } } { C_ { carina , s } } = \mathop \prod \limits_ { j \in \psi_ { sl } } B_jE_j = \mathop \prod \limits_ { j \in \psi_ { sl } } B_j \mathop \prod \limits_ { k \in \psi_ { sl } } E_k = \Pi_ { B.sl } \Pi_ { E , sl } \tag { { \rm Eq. } \ 5 } \end{align*} \end{document}

Note that the product \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\Pi_{B.sl}$$ \end{document} is a net branching factor that characterizes the effect of the mismatch between air and aerosol division at bifurcations on the lobar aerosol concentration. Similarly, the product \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\Pi_{E , sl}$$ \end{document} is a net escape fraction that characterizes the effect of aerosol deposition within the airways along the pathway to the lobe. Note that to evaluate either of these metrics, the individual values of B_k and E_k of each airway along the pathway must first be evaluated.

Expressing specific deposition in terms of sources of variability

The lobar specific deposition of aerosol of subject s in lobe l is therefore: \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*}sD_{sl} = t_{N , s}s \dot{V}_{sl}C_{carina , s} \Pi_{B.sl} \Pi_{E , sl} \eta_{sl} \tag{{ \rm Eq.} \ 6}\end{align*} \end{document}

The term \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$t_{N , s}s \dot{V}_{sl}$$ \end{document} corresponds to the cumulative volume of air that reached the periphery of the lobe during nebulization (normalized by the total volume of the lobe). The term \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$C_{carina , s} \Pi_{A.sl} \Pi_{E , sl}$$ \end{document} corresponds to the concentration of aerosol entering the lobe periphery C_sl, explicitly expressed in terms of the two factors influencing it. The product of the normalized volume of air and its aerosol concentration yields the cumulative aerosol delivered to the lobar periphery (per unit volume of lobe), and when multiplied by the fraction that is not exhaled η _sl , we arrive at sD_sl. Note that since the nebulization time t_N,s and C_carina,s are constants for a given subject, the four remaining terms express the sources of variability of specific deposition among lobes for a given subject.

Rearranging into nondimensional form

A nondimensional form for Equation (6) can be used to compare data from subjects with different lung sizes who received different aerosol doses, different overall ventilation and deposition, and have different degrees of overall central deposition. It is also desirable to have the nondimensional variables vary about unity to simplify the analysis as is discussed below. Let us first describe the total aerosol passing a subject's carina D_0,s as the sum of the deposited aerosol past the carina D_T,s and the exhaled aerosol D_E,s. D_T,s itself can be expressed as the sum of the deposition in the periphery D_P,s and that in central airways D_C,s: \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*}D_{0 , s} = D_{T , s} + D_{E , s} , { \rm and} \ D_{T , s} = D_{P , s} + D_{C , s} \tag{{ \rm Eq.} \ 7}\end{align*} \end{document}

Using this we can expand the concentration at the carina using the definition of concentration given in Equation (1), rewritten in terms of the total deposition beyond the carina D_T,s and the total average inspiratory flow through the carina \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\dot{V}_{T , s}$$ \end{document} : \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*}C_ { carina , s } = \frac { D_ { carina , s } } { \eta_ { carina , s } V_ { carina , s } } = \frac { D_ { T , s } } { \eta_ { carina , s } \dot { V } _ { T , s } } \frac { 1 } { t_ { N , s } } = \frac { D_ { 0 , s } } { \dot { V } _ { T , s } } \frac { 1 } { t_ { N , s } } \tag { { \rm Eq. } \ 8 } \end{align*} \end{document}

After substituting this relationship into Equation (6), a dimensionless form can be obtained by rearranging the terms and dividing by the total lung volume V_L,s, and by D_P,s/D_T,s: \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*}sD^{*}_{sl} = s \dot{V}^{*}_{sl} \Pi^{*}_{B.sl} \Pi^{*}_{E , sl} \eta^{*}_{sl} \tag{{ \rm Eq.} \ 9}\end{align*} \end{document}

Here the dimensionless specific deposition and ventilation are: \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} sD^ {*}_{sl} = \frac {{sD}_ {sl}} { \overline { sD_ {s}}} = \frac {sD_ {sl}} {{D}_ {P , s} / V_ {L , s}} ; \quad s \dot {V} ^ {*} _ {sl} = \frac {s \dot {V} _ {sl}} {\overline {s \dot {V} _ {s}}} = \frac {s \dot {V} _ {sl}} {\dot {V} _ {T , s} / V_ {L , s}} \tag { { \rm Eq.} \ 10 } \end{align*} \end{document}

These are the specific deposition of each lobe normalized by the specific deposition in the entire periphery of each subject \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\overline{sD_s}$$ \end{document} , and the specific ventilation of each lobe normalized by the specific ventilation of the entire lung for each subject \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\overline{s \dot{V}_s}$$ \end{document} . Lobar values for these numbers for each subject distribute about unity. The already dimensionless product \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\Pi_{B.sl}$$ \end{document} is normalized by the ventilation weighted lung average for each subject \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\overline{ \Pi_{B , s}}$$ \end{document} . Thus, if \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$f_{ \dot{V} , sl}$$ \end{document} is the fraction of a subject's ventilation that is delivered to a lobe, we have that: \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} \Pi^ {*} _ {B.sl} = \frac {\Pi_ {B.sl}} {\overline {\Pi_ {B , s.}}} = \frac {\Pi_ {B.sl}} {\mathop \sum \limits_ { j } f_ { \dot { V } , sj } \Pi_ { B.sj } } = \Pi_ { B.sl } \tag {{\rm Eq.} \ 11} \end{align*} \end{document}

Note that the ventilation weighted average of \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\Pi_{B.sl}$$ \end{document} is unity because the branching factors do not provide a mechanism for deposition and therefore conserve the total amount of aerosol. Similarly, the already dimensionless \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\Pi_{E , sl}$$ \end{document} has been normalized by the average escape fraction of the of the airway tree \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\overline{ \Pi_{E , s}}$$ \end{document} for each subject: \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} \Pi^ {*}_ {E.sl} = \frac {\Pi_ {E.sl}} {\overline {\Pi_ {E , sl}}} = \frac {\Pi_ {E.sl}} {\frac {{D_ {P , s}} + {D_ {E , s}}} {{D_ {O , s}}}} \tag {{\rm Eq.} \ 12} \end{align*} \end{document}

Finally, the retention fraction of each lobe has been normalized by the retention of the entire periphery of each subject \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\overline{ \eta_{s}}$$ \end{document} : \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*}\eta^ { * } _ {sl} = \frac { \eta_ {sl}} {\overline { \eta_ { s } } } = \frac {\eta_ { sl }} {\frac {{ D_ {P , s}}} {{ D_ { P , s } + D_ { E , s }}}} \tag { {\rm Eq. } \ 13 } \end{align*} \end{document}

Decoupling sources of variability

Taking the log of Equation (9) transforms the product of effects into a sum of effects. Given that all the variables have been defined to vary about unity, we can further approximate the logarithm with the first order Taylor expansion of the logarithm about unity (log(x)≈x-1)), Equation (9) can then be approximated as: \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*}sD^{*}_{sl} \approx sV^{*}_{sl} + \Pi^{*}_{B , sl} + \Pi^{*}_{E , sl} + \eta^{*}_{sl} - 3 \tag{{\rm Eq.} \ 14}\end{align*} \end{document}

In the specific case where these four effects are independent (i.e., there is weak covariance among the terms) we arrive at a decomposition of the variance in specific deposition in terms of the variance of the four factors that independently influence it. For the set S of all lobes among all subjects: \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*}\mathop{ \rm var} \limits_{j \in S} \left( sD^{*}_{j} \right) \approx \mathop{ \rm var} \limits_{j \in S} \left( sV^{*}_{j} \right) + \mathop{ \rm var} \limits_{j \in S} \left( \Pi^{*}_{B , j} \right) + \mathop{ \rm var} \limits_{j \in S} \left( \Pi^{*}_{E , j} \right) + \mathop{ \rm var} \limits_{j \in S} \left(\eta^{*}_{j} \right) \tag{{\rm Eq.} \ 15}\end{align*} \end{document}

This final equation allows us to estimate the fraction of the variance that comes from each of the four factors. This equation states that the variability in relative specific deposition is approximately the sum of the variability in relative specific ventilation, the normalized net branching factor, the normalized net escape fraction and the normalized retention fraction.

Overview

This section describes the imaging technique, and the method to extract the abovementioned parameters in a group of bronchoconstricted asthmatic subjects from PET-CT images of ventilation and aerosol deposition. The imaging protocol timeline is shown in Figure 2.

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FIG. 2.

The imaging protocol sequence. Color images available online at www.liebertpub.com/jamp

Imaging protocol

The imaging protocol was completed at Massachusetts General Hospital with IRB approval (Application No. 2007P000493). Twelve asthmatic subjects were imaged with PET-CT (Biograph 64; Siemens AG). The subjects were young (average 20.1 years), predominantly female (3 male and 9 female) and not obese (BMI's less than 32 Kg/m²). All subjects had mild intermittent or mild persistent asthma as defined by the NIH Global Initiative for Asthma⁽¹²⁾ with FEV₁ and FVC≥80% predicted, fewer than daily symptoms, and peak flow or FEV₁ variability of less than 30%. All subjects demonstrated reversible obstruction with inhaled albuterol (≥12% on previous spirometry).

During an initial screening visit the concentration of methacholine (MCh) required to cause a 20% drop in the subjects FEV₁ (PC₂₀) was estimated. On a second visit that same concentration was administered with five deep breaths of methacholine aerosolized by a DeVilbiss nebulizer and Rosenthal dosimeter (model 646, DeVilbiss Healthcare, Somerset, PA) with the subject in the supine position in the PET-CT camera. Three HRCT images were obtained during breath hold: one at total lung capacity (TLC) before bronchoconstriction, a second at mean lung volume (MLV) approximately 5 minutes after bronchoconstriction, and a final HRCT at TLC approximately 30 minutes after the MCh challenge.

To monitor lung volume and guide its value during imaging, a real time trace of instantaneous volume was obtained with an inductance plethysmograph (RIP) (SomnoStar PT, SensorMedics Corp, Yorba Linda, CA) and presented to the subject through video goggles. When imaging the lung at MLV with HRCT, the subject was instructed to stop breathing and hold his/her breath when the trace reached a line defining their MLV (previously determined during a 30 sec window of spontaneous breathing before the scan).

After the MLV HRCT image (16±5 min after the MCh challenge), an aerosol of ¹³NH3-labeled isotonic saline was generated with a vibrating mesh nebulizer (Aeroneb Solo, Aerogen, Galway, Ireland) and delivered via an Idehaler holding chamber (Aerodrug, Cedex, France) into the mouthpiece through which the subject was breathing while wearing a nose clip. The aerosol reaching the mouthpiece was previously characterized by laser diffraction as having an approximately lognormal distribution with a median aerosol diameter of 4.9 μm and a geometric standard deviation of 1.8 (4.9 μm VMD with a GSD of 1.8). This indicates that approximately 95% of the aerosol volume was between 1.5 and 15 μm.⁽¹³⁾ While the subjects were supine in the PET-CT camera, intrapulmonary aerosol deposition was imaged with dynamic PET during the 2-min period of inhalation and the following 8 min of spontaneous breathing.

After the end of the deposition image collection (30±5 min post MCh) regional specific ventilation was assessed during a V/Q scan from the washout of a bolus of ¹³NN gas as described in detail elsewhere.⁽¹⁴⁾ Briefly, during a V/Q scan nitrogen (¹³N-N) in saline solution was injected intravenously as a bolus during apnea. The poorly soluble radioactive nitrogen gas diffuses from the plasma to the alveolar airspace resulting in an intrapulmonary tracer distribution proportional to that of regional pulmonary blood flow. Following a 20 sec breath hold, the subject resumes breathing and the washout of the tracer is used to determine regional specific ventilation.

It was important to conduct the ventilation measurement as close as possible to that of deposition to minimize changes in the degree of constriction. The close proximity (15±2 min) between the deposition and ventilation images meant that there was some residual labeled aerosol that was registered as ¹³NN gas in the washout image. This residual activity was very small compared to the activity coming from the ¹³NN (<4%) and similar among lobes, and thus weakly affects \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$s \dot{V}^{*}_{sl}$$ \end{document} .

In 9 of the 12 subjects with ventilation images, we were also able to estimate the breathing frequency during aerosol inhalation from the RIP signal; the data from the first three subjects were recorded on an older laptop, and the signals could not be recovered.

Evaluating deposition and ventilation from the images

The PET images were analyzed to quantify the anatomical location of the aerosol deposition and the alveolar ventilation. The local tracer from the deposition images was assigned to anatomical regions (ARs) using the Grayscale method previously developed.⁽³⁾ This technique allowed quantification of aerosol deposition within anatomically defined portions of the central airway tree (CA) and on distal regions fed by them, after accounting for blurring of the PET image due to breathing motion, limited spatial resolution of PET, and PET-CT co-registration uncertainties. Using Apollo^™ software (Vida Diagnostics, Mountain View, CA) the lung was segmented into 14 ARs, including 5 peripheral lobar regions and 9 segments of the airway tree (Fig. 3). The lung periphery included the five lobes; left upper lobe (LUL), left lower lobe (LLL), right upper lobe (RUL), right middle lobe (RML), and right lower lobe (RLL). Central airways included the bronchus intermedius (BINT), the right and left main bronchus (RMB and LMB) and the trachea (TRC), as well as five lobar central airway trees feeding each lobe that included the lobar, segmental, and subsegmental airways.

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FIG. 3.

Rendering of the 14 anatomically consistent regions (ARs) used in this study. The lobar central airways are color matched to the lobes that they feed. The 16 cm PET field of view and the typical placement of the PET image are shown with the rectangle. The PET field of view typically covers 83±8% of the lung volume, with 4±3% missing from the apex, and 13±7% missing from the base. Color images available online at www.liebertpub.com/jamp

The distribution of deposition throughout the lung and airways was described as the relative specific deposition \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$s{D}^{*}_{sl}$$ \end{document} . Note that the radionuclide concentration of the aerosol must be known to evaluate \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$s{D}_{sl}$$ \end{document} from the activity within the image. However, once normalized by the average specific deposition \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\overline{s{D}^{s}}$$ \end{document} (as in \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$s{D}^{*}_{sl}$$ \end{document} ), the relative depositions can be determined directly from the relative activities in the image.

The relative distribution of ventilation during aerosol inhalation was assumed to be equal to that measured during the subsequent ¹³NN washout during the V/Q scan. Voxel by voxel ventilation was estimated from the best model chosen by the Akaike Information Criterion⁽¹⁴⁾ among a two-compartment model, a one compartment model, a partial gas trapping model, and a full gas trapping model. Net alveolar ventilation for each lobe was estimated as the sum of voxel products of compartment's turnover rate times its gas volume. The gas volume in each compartment within a voxel was estimated by the gas content of the voxel (measured from the CT scan at MLV) multiplied by the fraction of perfusion of the voxel that was delivered to that compartment. The voxel by voxel estimates were then distributed among lobes based on how each lobe contributed to the activity in that voxel.⁽³⁾ The total ventilation of the lobe is divided by the lobe volume to identify the specific ventilation \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$s \dot{V}_{sl}$$ \end{document} . Dividing by the average specific ventilation among all lobes in the subject yielded \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$s \dot{V}^{*}_{sl}$$ \end{document} . Note that although this is a measure of relative alveolar specific ventilation, if the deadspace ventilation distributes in a similar way as the fresh gas,⁽¹⁵⁾ this is also a measure of relative total specific ventilation.

Estimating specific inflation from 2 HRCTs

An alternative estimate of lobar ventilation distribution, similar to one used in previous studies,^(15,16) uses the change in lobar volume between two static HRCT scans from MLV to TLC as a proxy for the ventilation during tidal breathing. The specific inflation \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$sI^{*}_{sl}$$ \end{document} was therefore: \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*}sI_ { sl } = \frac { V_ { TLC , sl } - V_ { MLV , sl } K_ { sl } } { V_ { MLV , sl } K_ { sl } } \tag { {\rm Eq. } \ 16 } \end{align*} \end{document}

where the V terms are the different total lung volumes post constriction and K_sl is a number close to 1 that corrects for lobar segmentation differences of the lobe between the two images. Under the assumption that differences in the tissue volume of a perfectly segmented lobe should not differ between the two images, and that changes in blood volume can be estimated from the entire lung, K_sl can be expressed in terms of the blood and tissue volumes V_Tissue of the lobes and lungs: \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*}K_ { sl } = \left[ \frac { V_ { Tissue , TLC , sl } } { V_ { Tissue , MLV , sl } } \right] * \left[ \frac { V_ { Tissue , MLV , Total } } { V_ { Tissue , TLC , Total } } \right] \tag { {\rm Eq. } \ 17 } \end{align*} \end{document}

A dimensionless form for \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$sI^{*}_{sl}$$ \end{document} is obtained by normalizing by the average lung specific inflation \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\overline{sI_s}$$ \end{document} , or the total lung inflation \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$I_{T , s}$$ \end{document} divided by the MLV lung volume: \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*}sI^ { * } _ { sl } = { \frac { sI_ { sl } } {\overline { sI_s } } } = \frac { sI_ { sl } } { I_ { T , s } / V_ { L , s } } \tag { {\rm Eq. } \ 18 } \end{align*} \end{document}

Estimating lobar retention fractions

Lobar retention fractions could not be directly estimated from our data, nor were there values in the literature that could be directly used. However, based on global lung bolus retention data presented by Kim et al.,⁽⁸⁾ we could estimate a global retention fraction of the periphery \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\overline{ \eta_s}$$ \end{document} as a function of the settling distance d_s (see details in Appendix I). A general relationship could be derived that explained 97% of the variance in the measured peripheral retention fraction among nine experiments with varying particle sizes and flow rates (Fig. 4). Given the settling distance d_s that a particle of a given diameter would descend at the Stokes settling velocity v_s during the average residence time in the periphery t_s, we found that the retention fraction could be described by the equation: \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*}\overline { \eta_ { fit } } = 1 - e^ { - \frac { d_s } { 371 \mu m } } , \quad { \rm where } \quad d_s = v_s t_s \tag { {\rm Eq. } \ 19 } \end{align*} \end{document}

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FIG. 4.

The global retention fraction estimated from the sequential bolus data presented in Kim et al.⁽⁸⁾ versus the settling distance.

Evaluating \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\overline{ \eta}_s$$ \end{document} for the polydisperse aerosols used in our study showed that we can expect those subjects breathing at the highest frequencies during nebulization f_N should show approximately 20%–60% lower average retention. The value of \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\overline{ \eta}_s$$ \end{document} was highly sensitive to proximal filtering effects; when larger aerosols are selectively filtered in the upper and central airways, the retention fraction of the periphery is significantly reduced (Fig. 5).

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FIG. 5.

Estimates of the global retention in the lobar periphery for the polydisperse aerosols used in this study. The different lines describe different extents of proximal filtering of the larger sized aerosols; ‘No Filter’ assumes that the aerosol distribution entering the periphery is identical to the distribution at the mouth, while the filtered estimates assume that all particles above the filter size have already been deposited in the upper airways.

While this population averaged and lung averaged data can provide a useful estimate of the degree of average retention among all lobes, lobar retention fractions may vary substantially between subjects and lobes. For example, expanded lobes may have reduced η _sl due to both the longer sedimentation distances and lower likelihood of impaction in the distended airways. The expansion of a lobe F_VOL,sl can be defined as the total volume of gas V_gas per unit of tissue and blood volume V_Tissue in a lobe at MLV: \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*}F_ { VOL , sl } = \frac { V_ { Gas , MLV , sl } } { V_ { Tissue , MLV , sl } } \tag { {\rm Eq. } \ 20 } \end{align*} \end{document}

Note that F_VOL,sl is a measure of the degree of regional expansion during a breath hold at MLV and is not directly related to regional tidal volume. Assessment of both f_N and F_VOL,sl may provide insight into retention fraction effects.

Apparent net branching factor and escape fraction

The net branching factor \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\Pi^{*}_{B , sl}$$ \end{document} and net escape fraction \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\Pi^{*}_{E , sl}$$ \end{document} are both functions of changes in aerosol concentration along the airway tree and are thus intricately coupled with the lobar retention fractions. If these parameters are evaluated under the assumption of complete deposition in all lobes, an apparent net branching factor \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\hat{ \Pi}^{*}_{B , sl}$$ \end{document} and apparent net escape fraction \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\hat{ \Pi}^{*}_{E , sl}$$ \end{document} are obtained. Appendix II demonstrates that if \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\eta_{carina , sl}$$ \end{document} is defined as the retention fraction of a specific lobar pathway at the carina, then the apparent branching factor and apparent escape fraction have the following relationship with the actual values: \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} \frac { \hat { \Pi } ^ { * } _ { B , sl } } { \Pi^ { * } _ { B , sl } } = \frac { \eta_ { carina , sl } } { \eta_ { carina , s } } = h^ { * } _ { sl } \quad { \rm and } \quad \frac { \hat { \Pi } ^ { * } _ { E , sl } } { \Pi^ { * } _ { E , sl } } = \frac { \eta_ { sl } / \eta_ { carina , sl } } {\overline { \eta } _s / \eta_ { carina , s } } = g^ { * } _ { sl } \tag { {\rm Eq } \ 21 } \end{align*} \end{document}

Note that \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$g^{*}_{sl}$$ \end{document} and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$h^{*}_{sl}$$ \end{document} are both dimensionless terms that both distribute around 1 and combine to form the lobar retention fraction \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\eta^{*}_{sl}$$ \end{document} : \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*}\eta^{*}_{sl} = g^{*}_{sl}h^{*}_{sl} \tag{{\rm Eq.} \ 22}\end{align*} \end{document}

In this article we will present the apparent parameters as a proxy for the actual factors, noting that they differ from the actual factors by the \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$g^{*}_{sl}$$ \end{document} and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$h^{*}_{sl}$$ \end{document} factors defined in Equation (21).

The apparent branching factors can be evaluated from PET-CT deposition and ventilation images for the first four bifurcations in the airway tree (Fig. 6) by setting η_x as unity throughout the airway tree. The apparent escape fractions can be similarly calculated up to the lobar central airways. Due to the limited spatial resolution of PET, proximal airways within each lobe, up to the subsegmental airways, had to be lumped into a single compartment with an average lobar escape fraction.

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FIG. 6.

The four bifurcations in which the changes in concentration between the parent and daughter can be evaluated using PET-CT. These bifurcations are at the terminal ends of 1) the trachea, 2) the left main bronchus, 3) the right main bronchus, and 4) the bronchus intermedius. Color images available online at www.liebertpub.com/jamp