## Abstract

Airflow obstruction and heterogeneities in airway constriction and ventilation distribution are well-described prominent features of asthma. However, the mechanistic link between these global and regional features has not been well defined. We speculate that peripheral airway resistance (*R*_{p}) may provide such a link. Structural and functional parameters are estimated from PET and HRCT images of asthmatic (AS) and nonasthmatic (NA) subjects measured at baseline (BASE) and post-methacholine challenge (POST). Conductances of 35 anatomically defined proximal airways are estimated from airway geometry obtained from high-resolution computed tomography (HRCT) images. Compliances of sublobar regions subtended by 19 most distal airways are estimated from changes in regional gas volume between two lung volumes. Specific ventilations (s*V̇*) of these sublobar regions are evaluated from ^{13}NN-washout PET scans. For each pathway connecting the trachea to sublobar region, values of *R*_{p} required to explain the *sV̇* distribution and global airflow obstruction are computed. Results show that *R*_{p} is highly heterogeneous within each subject, but has average values consistent with global values in the literature. The contribution of *R*_{p} to total pathway resistance (*R*_{T}) increased substantially for POST (*P* < 0.0001). The fraction *R*_{p}*/R*_{T} was higher in AS than NA at POST (*P* < 0.0001) but similar at BASE (range: 0.960–0.997, median: 0.990). For POST, *R*_{p}*/R*_{T} range was 0.979–0.999 (NA) and 0.981–0.995 (AS). This approach allows for estimations of peripheral airway resistance within anatomically defined sublobar regions in vivo human lungs and may be used to evaluate peripheral effects of therapy in a subject specific manner.

- distal lung
- peripheral airway
- structure and function
- modeling
- asthma

asthma is a chronic inflammatory disease of airways associated with excessive and heterogeneous airway constriction (3). Both pathological and physiological evidence suggests that small airways and lung parenchyma contribute to asthma pathogenesis (20). Detailed measurements of central airway structure illustrate a large degree of heterogeneous response to bronchoprovocation (4, 5, 18, 19). Severe heterogeneity in the distribution of ventilation and perfusion have also been observed in scans taken from positron emission tomography (PET) (23, 34, 35) and magnetic resonance imaging (6, 8, 31).

It is unclear how such regional heterogeneity relates to well studied global parameters of lung function such as those measured with FEV1 or specific airway conductance. Although it is possible that the missing link may be found in the behavior of small airways, limitations in imaging resolution have prevented accurate estimates of how much these airways affect regional heterogeneity and global lung function.

In this paper, we present an approach for assessing the resistance of peripheral airways that are too small to be accurately measured from HRCT images. Recognizing the variability among subjects, this approach characterizes respiratory structure and function in an individual specific manner. Using both computational modeling and imaging, we estimated regional resistance of small airways at sublobar level that were consistent with the regional distribution of ventilation, quantified from PET scans, the structure of large airways derived from high-resolution computed tomography (HRCT) images, and the global level of obstruction measured by spirometry (FEV1). Data were collected from subjects with and without asthma before and during MCh-induced bronchoconstriction. Our aim was to evaluate regional peripheral resistance that is consistent with the observed global lung obstruction and regional distribution of ventilation.

### Glossary

- BASE
- HRCT scan conducted under baseline conditions at the breathing mean lung volume
- C
_{e} - Effective compliance
- d
- Distance from the most ventral point in the lung of each sublobar region's geometric center normalized by the total lung height (from ventral to dorsal)
*f*- Breathing frequency
- Fgas
- Fractional gas content
- G
_{c} - Conductance of individual central airway
- G̃
_{c} - Pathway conductance of central airways
- G
_{p} - Pathway conductance of peripheral airways
- MLV
- Mean lung volume, measured during tidal breathing
- POST
- HRCT scan conducted following the administration of Methacholine at mean lung volume
- PostTLC
- HRCT scan conducted following the administration of Methacholine at total lung capacity
- P
_{pl} - Pleural pressure
*R*_{p}- Pathway resistance of peripheral airways
*R*_{T}- Total pathway resistance
- s
*V̇* - Specific ventilation (total ventilation per unit volume)
- s
*V̇*_{A} - Alveolar ventilation per unit volume
- TLC
- Total lung capacity
*V̇*_{A}- Alveolar ventilation
*V̇*_{D}- Dead space ventilation
*V*gas- Gas volume
*V*_{L}- Lung volume
*V*_{sub}- Gas volume of regions subtended by an airway
*V̇*_{T}- Total ventilation
*V*tis- Tissue volume
- Y
_{pw} - Total pathway admittance

## METHODS

This section is organized into three subsections: *Data Acquisition Study Protocol*, *Lumped-Parameter Network Model*, and *Derivations of Parameters in a Lumped Parameter Network Model*. The first subsection describes how the data from the HRCT-PET study was obtained, as well as the subject demographic data. The second subsection illustrates the flow of input and output parameters of the LPN and details the algorithm for estimating regional peripheral resistance serving as a link among a global parameter (FEV1), regional distribution of ventilation (from PET), and structural measurements of central airways (from HRCT). The last subsection covers how the obtained HRCT and PET data were used to derive functional and structural regional parameters that later became inputs to the LPN model.

### Data Acquisition Study Protocol

Images were obtained from 7 mild-to-moderate asthmatic and 8 healthy adult volunteers (demographics shown in Table 1). Subjects with mild-to-moderate asthma were selected according to the criteria of the National Institutes of Health Global Initiative for Asthma (23a) with forced exhaled volume within 1 s (FEV1) or forced vital capacity (FVC) ≥80% predicted, less than daily symptoms, and peak flow or FEV1 variability of ≤30%. We excluded subjects that were current smokers and those with >10 pack-years. Other exclusion criteria were the use of oral steroids, symptoms of upper and lower respiratory tract infection, or a history of hospitalizations for asthma in the past month, or history of cardiopulmonary disease other than asthma. No systemic or inhaled corticosteroids could be used within 1 wk prior to enrollment. The study protocol was approved by the Massachusetts General Hospital Institutional Review Board. All subjects gave their written informed consent.

Prior to the study date, all subjects underwent a methacholine (MCh) challenge to determine their PC20: the provocative concentration of inhaled MCh aerosol that caused a 20% reduction in FEV1. PC20 was determined based on the method published by Crapo et al. (7). The maximum dose given to asthmatics was 8 mg/ml and all nonasthmatic subjects were given 25 mg/ml, a dose that by protocol was less than their PC20. Spirometry during the initial screening of subjects was performed while subjects were in an upright position.

A Siemens Biograph 64 PET-CT tomography scanner was used in a helical mode to acquire 64 slices per rotation. The scanner setting was 0.6 mm collimation and a pitch of 1. The energy settings were 120 kV peak and 80 mA. Image reconstruction was done using the B31 kernel with a 0.75 mm slice thickness, 0.5 mm slice increment, and 0.25 mm overlap. With the subjects in the supine position, HRCT images were acquired during a short breath hold (∼12 s) under three consecutive conditions: *1*) at baseline (BASE) and imaged at mean lung volume (MLV), where MLV was defined as the average lung volume during stable tidal breathing; 2) following five deep breaths of MCh at his/her previously determined PC20 dose via a DeVilbiss nebulizer and Rosenthal dosimeter (model 646, DeVilbiss Healthcare, Somerset, PA) and imaged at the subject's spontaneous MLV (POST); and *3*) post MCh and imaged at total lung capacity (PostTLC). Dynamic PET emission scans of tracer gas ^{13}NN were acquired following the first and second CT scans. The kinetics of ^{13}NN were used to assess regional specific ventilation within the lung (36). After each PET scan, spirometry was done while the subject remained supine the PET-HRCT scanner. FEV1% predicted for each subject after each PET scan were presented in Table 2. Prior to each HRCT or PET scan, MLV was determined over the 30-s period of stable breathing. To guide the breath hold maneuver, the instantaneous lung volume and a line indicating MLV from an impedance plethysmograph (SomnoStarPT, SensorMedics, Yorba Linda, CA) were displayed to the subject by means of video goggles. Breathing frequency (*f*) was calculated for each subject at each condition as the dominant frequency of a Fast Fourier transform of the lung volume signal from the impedance plethysmography during the ^{13}NN washout period (23b).

### Lumped Parameter Network Analysis

For each subject and condition (BASE or POST*)*, the 35 anatomically defined central airways (0–6th generation) (Fig. 1*B*), each having an estimated conductance of G_{c} or resistance (G_{c}^{−1}), were interconnected in a network following the subject-specific airway tree structure (e.g., Fig. 1, *B* and *C*). Each of the 19 most distal resistances of the tree (colored in blue in Fig. 1*A*) was connected by a peripheral resistance (*R*_{p}), to an elastic element with effective compliance of C_{e}, and a common pleural pressure (P_{pl}) (Fig. 1*C*). Airflow (*V̇*) through each branch of the airway tree was assumed to be equal to the sum of flows of its daughter branches. Values of G_{c} and C_{e} were estimated from HRCT and PET images as detailed in the subsection *Derivations of Parameters for the Lumped Parameter Network Model*. To test the relevance of each lumped parameter element, we considered three possible network models. First, a model was composed only of elastic elements (C_{e}). Second, a model that had both central resistances (G_{c}^{−1}) and C_{e} and, finally, a model that considered pressure drops across all elements. Airway conductance of central airways along a pathway leading to a sublobar region, G̃_{c}, was defined as the ratio of airflow to that region over the cumulative sum of pressure drops across the airways leading to the region (Fig. 1*C* and appendix). For *Model 3*, a total pathway resistance (*R*_{T}) was defined as G̃_{c}^{−1}+*R*_{p}.

### Derivations of Parameters for the Lumped Parameter Network Model

Pulmonary Workstation 2.0 (PW2) software (VIDA Diagnostics, Iowa City, IA) was used to analyze and obtain airway dimensions of 35 defined central airways (Fig. 1*B*) for each condition. The following parameters were imported from PW2 into MATLAB (Mathworks, Natick, MA) for each airway: *1*) the average luminal area (*A*), *2*) airway length (*L*), and *3*) generation number (starting with 0 for the trachea). From the HRCT images, masks of the 19 sublobar regions, subtended by the 19 of the most distal of the identified airways, were defined (Fig. 1*B*) by PW2. For each region and imaging condition, the following primary parameters were gathered, *1*) *d*: relative distance of each sublobar region's geometric center from the most ventral point in the lung, normalized by the lung height (from ventral to dorsal); *2*) *V*tis: total parenchymal tissue volume defined as the volume of the sublobar region not occupied by air; *3*) Fgas: fraction of the region's volume occupied by gas.

These parameters were then used to derive other functional and structural parameters as follows.

#### Regional gas volume.

Initial analysis of the data showed that the regional tissue volume (*V*tis*)* for each subject, measured from PW2, was quite variable among the three imaging conditions. For example, the mean and standard deviation (Mean ± SD) of regional *V*tis_{i,BASE}/*V*tis_{i,POST} for all subjects was 1.01 ± 0.32, a much greater variability compared with that of measured for the whole lung (0.99 ± 0.04). This suggested that the variability in regional *V*tis_{i} was likely due to inconsistent sublobar segmentation from scan to scan for a given subject—an observation that was consistent with visual inspection. Given that parenchymal segmentation appeared to be most anatomically accurate at TLC and that *V*tis of the lung did not significantly change with imaging condition, for each sublobar region *V*tis was assumed to be equal to that measured at TLC. Moreover, since regional *F*gas is *V*tis normalized by the sublobar region's total volume, it should be less sensitive to inaccuracy in sublobar segmentation than *V*gas. Hence, *V*gas of each sublobar region *i* was estimated from its regional Fgas (Fgas_{i}) and *V*tis_{i} at TLC as,

*Subtended gas volume (V*_{sub}) by each central airway was computed as the sum of gas volumes of all sublobar regions subtended by that airway.

#### Alveolar ventilation per unit volume (sV̇_{A}).

Within each sublobar region, the kinetics of ^{13}NN concentration during washout, measured with PET, was evaluated assuming a two-compartment model (36) to estimate alveolar ventilation per unit of gas volume. Sublobar s*V̇*_{A} was computed as the blood flow weighted s*V̇*_{A} of the two compartments.

*Total ventilation (V̇*_{T,i}*)* of a sublobar region, *i*, was defined as the sum of its alveolar (*V̇*_{A,i}) and dead space (*V̇*_{D,i}) ventilations. Because specific ventilation s*V̇*_{A} was the total ventilation per gas volume within each subtended region, *V̇*_{A,i} was the product of s*V̇*_{A,i} and *V*gas_{i}. *V̇*_{D,i} was assumed to be 1/3 of *V̇*_{T,i}. We discussed the validity of this assumption in *Model Assumptions* section in the discussion. An average airflow through each airway was computed from the sum of *V̇*_{T,i} to its subtended regions. Global lung specific ventilation, s*V̇*^{*}, was defined as the mean-normalized s*V̇* of all sublobar regions weighted by their corresponding *V*gas_{i}.

#### Regional effective compliance.

Given that the model assumes a constant *P*_{pl} throughout the pleural space, and thus equal transpulmonary pressure (*P*_{tp}) for all regions, the effective compliance of a sublobar region (C_{e,i}) is estimated as the change of its regional gas volume (Δ*V*_{i}) between MLV and TLC divided by the change in transpulmonary pressure (Δ*P*_{tp}).
*P*_{tp} was found from the relationship below,

Hence, for both BASE and POST, we computed C_{e,i} as,
*V*gas_{i} and *V*gas_{L} were the gas volumes in the sublobar region and the whole lung, respectively; the subscript MLV and TLC denoted the lung volume state at which *V*gas was calculated; sC was the specific compliance of the lung and assumed to be 0.1 l/cmH_{2}O per 1 liter of FRC (17, 27). This method was equivalent to those described in the literature by Fuld et al. (9). Each subject's FRC was estimated from their TLC and body mass index using the empirical table by Jones et al. (13). Figure 2*B* shows a plot *V*gas_{i}*/V*gas_{i,TLC} vs. whole lung *V*gas_{L}*/V*gas_{L,TLC} for all sublobar regions of a subject at BASE (in black) and POST (in red). In this plot, the data from a sublobar region, whose specific compliance is equal to that of the lung, would follow the identity line. Also, the variation of the sublobar effective compliance is reflected by differences in the slope.

*Conductance of individual airways (G*_{c}*)* was defined as the ratio of airflow to pressure loss along each of the 35 anatomically defined airways. The pressure loss along each airway was computed as the sum of major and minor losses based on the work by Katz et al. (16) that accounted for the local characteristics of flow in the human airway tree. Major and minor losses were estimated based on a local Reynolds number, average airway diameter, length, average flow velocity, and empirical table of minor loss coefficients. That table provides constants for each generation number based on computational modeling and experimental results of flow through a bifurcating structure. This method was applied to all airways except those whose cross sections, by inspection, did not appear perpendicular to their centerlines due to the segmentation error in PW2. On average, this amounted to one, but no more than two, airways per subject. Conductance of erroneously segmented airways was estimated based on the observed correlation between log(G_{c}) and log(*V*_{sub,PostTLC}) for all airways measured within each individual (Fig. 3). For BASE or POST condition, linear regression was applied to the plot of log (G_{c}) vs. log(*V*_{sub,PostTLC}). The regression parameters for each subject were used to estimate G_{c} of those airways based on their measured subtended volume, *V*_{sub,PostTLC}.

#### Peripheral resistance (R_{p}).

The plot of model-predicted (s*V̇*_{pred,i}^{*}) vs. measured mean-normalized (s*V̇*_{i}^{*}) specific ventilations of sublobar regions shows deviations from the identity line of the data for *Model* 1 and *Model 2* (Fig. 4, *A* and *B*) and implies that regional peripheral resistance (where *R*_{p} >0) needs to be included to the network to account for the distribution of s*V̇*_{i} measured from PET scans. To estimate *R*_{p} for each pathway, we used the following algorithm. First, assuming a known oscillatory driving pressure and in-phase flow throughout the tree, we obtained approximate values of sublobar *R*_{p} values such that the magnitude criteria in the following equation was satisfied:
*k* is an index defining specific airways along a pathway, Wi, and *P*_{pl} is the pleural pressure.

Values of estimated *R*_{p} values entered into *Model 3* allow airflow to be calculated for each sublobar region *i* (including both phase and magnitude). These airflows were used to calculate s*V̇*_{pred,i} that were then compared against the corresponding PET measured s*V̇*_{i}. This yielded a prediction error of less than 1% for the initial estimate and was rapidly reduced by iterative estimation of *R*_{p}. For each scan in each subject, *P*_{pl} was selected such that specific airway conductance of the entire respiratory system (sG_{aw}) would be equal to that predicted from the published correlation between FEV1 and sG_{aw} by Van Noord et al. (*R* = 0.81) (33).

#### Specific pathway conductance.

The peripheral airway component of pathway specific conductance (sG_{p,i}) and central airway component of pathway specific conductance (sG̃_{c,i}) were defined as *R*_{p,i}^{−1}*/V*_{gas,i} and G̃_{c,i}*/V*_{gas,i}, respectively.

#### Statistical analysis.

All reported measurements were the mean ± standard error (range: minimum − maximum). Statistical analyses were performed using SAS 9.2 (SAS Institute, Cary, NC). One-way ANOVA was used to determine the effect of MCh challenge on C_{e}. When the normality test failed, the Wilcoxon rank sum test was used instead of the one-way ANOVA. Two-way ANOVA with repeated measurements was used to determine the effect of MCh challenge and the effect of asthma. If the normality test failed, two-way ANOVA was reapplied to the log-transformed data. For all statistical analyses, *P* < 0.05 was considered significant.

## RESULTS

#### Fractional gas content.

Fgas within each subject increased after MCh challenge (*P* < 0.0001), reflecting an increase in total lung volume (*P* < 0.001). Lung volume increased after the challenge on average by 24.8 ± 7.0% in NA and 26 ± 8.8% in AS subjects, but were not significantly different between the two groups. Fgas measured at MLV had a vertical dependency that was reduced when lung volume increased after MCh challenge (*P* < 0.05). Fgas at PostTLC had a small but significant vertical dependency (*P* < 0.001) as illustrated in Fig. 2*A*. During inflation from MLV to TLC, sublobar gas volume (*V*gas_{i}) increased differently among regions, with some expanding more and others expanding less than the average expansion of the lung (Fig. 2*B*). Sublobar effective compliance (C_{e,i}) derived from the regional changes in *V*gas between MLV and TLC, did not significantly change after MCh challenge in either the NA or AS group (Wilcoxon rank sum test).

#### Conductance of individual central airway.

Log(G_{c}) varied linearly with log(*V*_{sub}) at TLC in both NA and AS subjects at BASE and POST (Fig. 3). Linear regression parameters for that relationship had an average correlation coefficient of 0.85 (SE of 0.01) and average regression slope of 1.12 (SE of 0.05). Two-way ANOVA with repeated measurements performed on these correlation coefficients and slopes showed no significant effect of MCh challenge or the presence of asthma.

#### Effect of regional compliance and central airway conductance on ventilation distribution.

Except for most NA subjects at baseline, plots of model-predicted (s*V̇*_{pred,i}^{*}) vs. measured mean-normalized (s*V̇*_{i}^{*}) specific ventilations of sublobar regions showed deviations from the identity line of the data for *Model 1* and *Model 2* (Fig. 4, *A* and *B*). This implies that regional peripheral resistance where *R*_{p} > 0 need to be included to the network to account for the distribution of s*V̇*_{i} measured from PET scans.

#### Central component of pathway specific conductance.

The data taken from all pathways in all subjects of each group show that sG̃_{c,i} was highly variable as seen in the broad cumulative distributions both at BASE and POST (Fig. 5*A*). In addition, MCh challenge caused a left shift in these distributions in both groups, reflecting a systematic reduction in median of sG̃_{c,i} from 3.16 cmH_{2}O/s to 2.41 cmH_{2}O/s (Fig. 5*A*). Analysis by individual subject shows a significant, but small, reduction of the individual geometric mean (μ_{g}) of sG̃_{c,i} after MCh challenge for both groups (*P* < 0.0001; average reduction of 27.7% ± 0.04% SE; Fig. 6*A*). The heterogeneity of sG̃_{c,i} within subjects measured by the geometric variance (σ_{g}^{2}), was not different between NA and AS subjects. Although there was an increase in σ_{g}^{2} of sG̃_{c,i} after MCh challenge in NA subjects, the increase did not reach significance (Fig. 6*C*). In the AS group, the response of σ_{g}^{2} (sG̃_{c,i}) to MCh challenge was highly variable between subjects.

#### Peripheral component of pathway specific conductance.

The broad cumulative distributions of sG_{p,i}, pooled for all pathways in all subjects of each group, illustrates a large variability in peripheral pathway conductance in both groups at BASE and POST (Fig. 5*B*). MCh challenge also caused a left shift in these distributions of sG_{p,i} in both groups, reflecting a substantial reduction in the median of sG_{p,i} from 0.04 cmH_{2}O/s to 0.01 cmH_{2}O/s (Fig. 5*B*). Analysis by individual showed that MCh challenge caused a significant and substantial reduction in μ_{g} of sG_{p,i} in both groups (*P* < 0.0001; average reduction of 69.7 ± 0.04% SE), but μ_{g} of sG_{p,i} was not different between groups (Fig. 6*B*). The geometric variance (σ_{g}^{2}) of sG_{p,i} was not different between NA and AS subjects and was not affected by MCh challenge in a systematic manner (Fig. 6*D*). The peripheral component of specific conductance was more than two orders of magnitude smaller than the central component in both groups for both conditions (Fig. 6, *A* and *B*). This large contribution of peripheral airways to mechanical obstruction can best be presented in terms of the ratio between peripheral resistance (*R*_{p,i}) and total resistance (*R*_{T,i}) of each pathway. The right shift in the cumulative distributions of *R*_{p,i}/*R*_{T,i}, pooled for all pathways in all subjects of each group, shows that increased significantly after MCh challenge (*P* < 0.0001) in both groups and with higher *R*_{p,i}/*R*_{T,i} during bronchoconstriction in AS compared with NA subjects (*P* < 0.0001; Fig. 7). Baseline values of *R*_{p,i}/R_{T,i} ranged from 0.960 to 0.997 (median of 0.990) in both groups.

For POST, values of *R*_{p,i}*/R*_{T,i} ranged from 0.979 to 0.999 (median of 0.994) in NA subjects and 0.981 to 0.995 (median of 0.996) in AS subjects.

## DISCUSSION

This paper presents an image-based modeling approach that links global and regional parameters of respiratory structure and function. With this approach, a peripheral airway resistance (*R*_{p}) was estimated to provide a coherent link between heterogeneities in ventilation and well studied global parameters of lung function, i.e., FEV1 and specific airway conductance (sG_{aw}). Applying this approach to imaging data from asthmatic (AS) and nonasthmatic (NA) subjects, we found that the relative contribution of such *R*_{p} was elevated in AS compared with NA subjects at baseline and substantially increased during bronchoconstriction (BC) in both groups. Moreover, *R*_{p} was highly heterogeneous within and among subjects. *R*_{p} represented a large fraction (>96%) of the total resistance (*R*_{T}) of any pathway at BASE and increased to account for >98% of *R*_{T} during BC (with a larger increase in *R*_{p}/*R*_{T} in AS compared with NA subjects). These results demonstrate that airway constriction during induced bronchoconstriction occurs mostly in peripheral airways (<2 mm in diameter).

We also found that only in NA at baseline, plots of model-predicted (s*V̇*_{pred,i}^{*}) vs. measured mean-normalized (s*V̇*_{i}^{*}) specific ventilations of sublobar regions were close to the identity line for *Models 1* and *2*. This implies that only in NA with nonconstricted airways, the distribution of ventilation could be predicted based on the elastic properties of the parenchyma. During bronchoconstriction in NA and in AS in both conditions, *R*_{p} > 0 needs to be included to the network to account for the distribution of s*V̇*_{i} measured from PET scans. A corollary of this finding is that the distribution of ventilation in constricted lungs cannot be predicted based volumetric changes of the parenchyma measured from static CT images. This may be of relevance in models attempting to predict the aerosol distribution within the diseased lung. Currently, most models have assumed that the distribution of airflow along the bronchial tree is that predicted by the elastic properties of the lung parenchyma (22).

We will discuss advantages of our approach over existing modeling approaches in two categories.

#### Models of regional lung mechanics.

Distributed inverse models applied to frequency response data have been widely used to investigate mechanical responses during bronchoconstriction (14, 15, 28, 40). Parameters such as resistance and elastance from the distributed model are estimated from whole organ measurements. They may provide measurements of functional heterogeneity within the lung but not for anatomically defined regions. In contrast, the modeling approach proposed in this paper provides estimates of regional mechanics within anatomically defined regions of interest. Moreover, the proposed approach not only allows for the quantification of functional impairment but also links the source of global functional impairment to anatomical structures.

#### Morphometric models.

Morphometric models of a human lung such as those developed by Weibel et al. (38) and Horsfield et al. (11) have been used to estimate the contribution of airway generations to the total pressure losses during breathing (24). In Weibel and Horsfield models, assumptions about the airway tree were made to generalize its morphometry within a population, disregarding branching asymmetry and differences in airway morphometry among individuals, which may influence the regional distribution of ventilation. Rather than a population-based model, the described approach allowed us to build a subject-specific model for the lung based on each individual's anatomical information.

Another morphometric model recently developed by Tawhai et al. (29) used lobar geometry obtained from the segmentation of subject-specific CT scan as an input to an algorithm to grow distal generations of the airway tree. This model uses a volume-filling algorithm to generate a tree structure that closely resembles that of the human lung. Although the model creates an airway tree using subject-specific constraints, assumptions had to be made about the geometry of specific airways, which may differ in vivo from baseline to during bronchoconstriction. Nevertheless, the Tawhai model could be combined with our approach to investigate functional changes at smaller length scales within sublobar regions.

Tgavalekos and colleagues (30) combined data from PET imaging, oscillatory ventilation mechanics, and the Tawhai model to identify potential patterns of airway constriction that could explain the observed size and location of ventilation defects during bronchoconstriction. Such statistical modeling tested assumptions about the distribution of constriction at different levels of the tree to match global oscillatory mechanics but were not necessarily consistent with quantitative measurements of regional ventilation. Our modeling approach without making an assumption about constriction patterns could estimate values of peripheral resistance for airways with diameter <2 mm within anatomically defined sublobar regions.

### Model Assumptions

Assumptions about unmeasured variables had to be made in our model to derive estimates of peripheral resistance. Here, we discuss the potential implications that these assumptions may have on our results and, when possible, how they can be avoided in the future.

*Breathing flow waveform* was assumed to be sinusoidal at a frequency equal to the dominant frequency evaluated from Fourier analysis of the lung volume signal recorded from impedance plethysmography during PET scan used to measure regional ventilation distribution. Therefore, this estimated flow waveform is a first-order approximation of the actual flow, and it should not result in substantial differences in estimates of peripheral resistance. Deviations from the actual flow waveform can be represented by additional sinusoids of lower amplitude at frequencies other than the dominant frequency, which should result in a small effect on the estimated pressure losses.

*Regional lung compliance* was estimated from total lung compliance and regional changes in sublobar gas volumes derived from HRCT images. Total lung compliance was not measured, but was estimated for each subject based on their lung volume assuming a population-based specific compliance (sC), FRC, that was the product of TLC in the supine position and a subject-specific FRC-to-TLC ratio. This method allowed us to arrive at a first-order approximation of total lung compliance (C_{L}) for each subject in a supine position, a value that can be measured in future studies.

#### Total lung compliance.

Although C_{L} depends on the size of the individual and the lung volume at which it is measured, sC should be insensitive to these effects, including the change in lung volume between a supine and upright position. However, airway closure could affect C_{L} because it is calculated using the full gas volume of the region or the lung, but not the ventilating volume. Thus, in such case, C_{L} could have been overestimated because the volume multiplying sC is overestimated. This could result in an overestimation of the peripheral resistance to compensate for the reduced compliance.

#### Dead space ventilation.

To compute total flow along airways from alveolar ventilation per unit of gas volume (s*V̇*_{A}), we need an estimate of dead space ventilation (*V̇*_{D}). Hardman et al. (10) showed in a modeling analysis that maximum *V*D*/V*T under normal physiological conditions was ∼1/3 (10). Therefore, we assumed *V*D*/V*T ratio to be 1/3 and equal for all sublobar regions. In reality, regional dead space ventilation is difficult to estimate and may differ between regions depending on several factors. For example, intraregional gas mixing, anatomical volumes of airway feeding sublobar regions, characteristics of regional airflow, or incomplete gas mixing between inhaled and alveolar gas can affect the efficiency of gas exchange and thus our estimates of regional airflow from PET scans. Although studies by Wellman et al. (38) and Simon et al. (25) demonstrated a high degree of correlation between s*V̇*_{A} and specific alveolar volume change during breathing in animals, the effect of this assumption on our estimates has to be further investigated in humans.

#### Correlation between FEV1 and sG_{aw}.

When either the pleural pressure (P_{pl}) or total airway conductance is known, the problem of determining peripheral resistance is well defined and has a unique solution. In our data set, neither parameter was measured, but can be extrapolated from a published correlation between specific airway conductance of the entire respiratory system (sG_{aw}) and FEV1. On the basis of a study by Van Noord et al. (33), values of sG_{aw} were determined from FEV1 measured in subjects in a supine position at both baseline and during bronchoconstriction. In the same study, they also found that the same relationship between sG_{aw} and FEV1 existed among different disease groups, such as asthma, chronic bronchitis, and emphysema. This reaffirms the validity for using this correlation to extrapolate values of sG_{aw} in the nonasthmatic group.

Although the required assumptions could reduce the absolute accuracy of our estimates, this model can indicate the general characteristics and relative magnitude of central vs. peripheral airway conductance in asthma and during bronchoconstriction. Further improvements of our model may include an evaluation of the model's stability under conditions of noise in PET and HRCT images, a sensitivity analysis of the model given uncertainties in the calculated model parameters, and errors in underlying assumptions. Although we cannot verify all of our assumptions, the following results suggest that they are quite reasonable.

First, global values of lung specific conductance measured in humans with ages ranging from 4 to 87 yr of age are ∼0.2 [cmH_{2}O/s]^{−1} (1). This compares well with the values of the lung specific conductance calculated from our model as the volume-normalized inverse of the real component of total lung impedance without upper airways (average of 0.10 ± 0.05 SD for NA and 0.08 ± 0.04 for AS at baseline).

A second set of data suggesting that our estimates of *R*_{p} are reasonable was from the estimated values of *R*_{p}/*R*_{T} derived from theoretical calculations of airway resistance per generation by Pedley et al. (24) in a Weibel symmetric tree. The sum of the resistances of the 4th generation and beyond in that model accounted for 0.86 of the total sum of resistances for the full tree assuming laminar flow or ∼0.63 for the case with a flow rate of 10 l/min. Our estimates of *R*_{p}/*R*_{T} at baseline ranged from 0.960 to 0.997 (median of 0.990). Although these estimates are larger than those from Pedley's model, they are not unrealistic because they came from imaging in vivo airways that had a baseline smooth muscle tone. In contrast, airway geometry for Pedley's model was derived from cadaver castings of airways without any smooth muscle tone and likely having lower peripheral resistance.

Techniques using a retrograde catheter, alveolar capsules, and wedged bronchoscope have been used to measure resistance of peripheral airways in human lungs (32, 37), rabbits (25), and dogs (21). Values of peripheral resistance obtained from these techniques are also consistent with ours. For example, using the wedged bronchoscope technique, Wagner and colleagues (37) measured peripheral resistance in asymptomatic asthmatic subjects at baseline and after histamine challenge. They found that *R*_{p} increased significantly after the challenge. Romero and Ludwig (25) used alveolar capsules glued to the pleura in rabbit lungs to measure tissue viscance, which was thought to, in part, include peripheral resistance. They found that the contribution of tissue viscance to total lung resistance was 65 ± 15% at baseline and increased to 84 ± 12% after challenge with MCh aerosols. These data sets give additional support to our estimates of *R*_{p} values. Future studies can greatly benefit from direct measurements of sC and a global dead space (for example using the Fowler technique).

Thus these findings from existing literature support the validity of estimates of regional peripheral resistance derived from our model. In addition, the model is inherently consistent because it is constructed such that regional functional parameters (i.e., ventilation) were derived with a global functional parameter such as FEV1 or specific airway conductance.

In summary, this paper presents an image-based, subject-specific modeling approach including PET and HRCT imaging that links global and local parameters of respiratory structure and function in human lungs at baseline and during bronchoconstriction. Measurements of central airway morphometry, derived from HRCT images, and ventilation distribution, derived from PET scans, were combined to yield estimates of peripheral resistance of anatomically defined sublobar regions. Moreover, these peripheral resistances are derived from and form a bridge between well-studied global parameters, such as FEV1 or specific airway conductance, and regional ventilation heterogeneity in both normal and constricted lungs. This method may further aid the evaluation and study of regional differences in the constriction of peripheral airways in vivo. Data obtained from this approach may also help in identifying mechanisms responsible for the location and strength of peripheral airway obstruction in a subject-specific manner without requiring in vivo measurements of small airways, which are limited by imaging spatial resolution. It may also offer insights into the regional physiology of distal airways in other lung diseases.

## GRANTS

This work was supported by National Institutes of Health Grant R01-HL-068011.

## DISCLOSURES

No conflicts of interest, financial or otherwise, are declared by the authors.

## AUTHOR CONTRIBUTIONS

Author contributions: C.W., R.S.H., T.W., and J.G.V. performed experiments; C.W. and E.G. analyzed data; C.W. and J.G.V. interpreted results of experiments; C.W. prepared figures; C.W. drafted manuscript; C.W., R.S.H., E.G., T.W., and J.G.V. edited and revised manuscript; C.W., R.S.H., E.G., T.W., and J.G.V. approved final version of manuscript; R.S.H., T.W., and J.G.V. conception and design of research.

## ACKNOWLEDGMENTS

We thank Mamary Kone, Tyler Wellman, and Anja Braune, for expert support in the acquisition of data.

## APPENDIX: DERIVATIONS OF SPECIFIC PATHWAY ADMITTANCE

#### Model 1.

Each pathway in *Model 1* consisted of elastic elements with different effective compliance of each sublobar regions (C_{e,i}; all resistive elements were assumed to be zero). Airflow through each pathway (*i*) was assumed to be sinusoidal with maximum flow rate (*V̇*_{max,i}), average flow rate (*f*).

The average pressure drop across C_{e,i} in each pathway (*i* = 1, …, 19) during inspiration (*P*_{pw,i}) to the flow rate through each pathway (*V̇*_{i}).

In the Laplace domain,
*j(2πf)* and

Therefore, the pathway admittance (Y_{pw,i}) of *Model 1* is

#### Model 2.

The network in *Model 2* consisted of conductance of central airways (G_{c,k}) and C_{e} [i.e., peripheral resistance (*R*_{p,i}) was neglected]. The total pressure drop, therefore, P_{pw,i} was the sum of the resistive pressure drop and the pressure drop across the effective compliance.

In the Laplace domain,
*k* is the index of an airway leading to the sublobar region *i*, and *V̇*_{k} is the flow through the airway *k*.

Hence,

#### Model 3.

The network in *Model 3* consisted of G_{c,i}, peripheral resistance (*R*_{p,i}), and C_{e,i} connected in the order listed. A unique set of *R*_{p,i} was determined to satisfy the pressure drop boundary condition both in magnitude and phase.

Similar to the analysis above

Solve for *R*_{p,i} such that P_{pw,i} = –P_{pl}

- Copyright © 2013 the American Physiological Society