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Meeting Abstracts  |   August 2005
Quantifying Mechanical Heterogeneity in Canine Acute Lung Injury: Impact of Mean Airway Pressure
Author Affiliations & Notes
  • David W. Kaczka, M.D. Ph.D.
    *
  • David N. Hager, M.D.
  • Monica L. Hawley, Ph.D.
  • Brett A. Simon, M.D. Ph.D.
    §
  • * Assistant Professor of Anesthesiology and Critical Care Medicine and Biomedical Engineering, † Fellow in Pulmonary and Critical Care Medicine, § Associate Professor of Anesthesiology and Critical Care Medicine, The Johns Hopkins University School of Medicine, Baltimore, Maryland. ‡ Adjunct Assistant Professor of Otolaryngology, University of Maryland School of Medicine, Baltimore, Maryland.
Article Information
Meeting Abstracts   |   August 2005
Quantifying Mechanical Heterogeneity in Canine Acute Lung Injury: Impact of Mean Airway Pressure
Anesthesiology 8 2005, Vol.103, 306-312. doi:
Anesthesiology 8 2005, Vol.103, 306-312. doi:
ACUTE lung injury (ALI) is a complex pathologic process involving a heterogeneous interaction of mechanical and biochemical processes.1 Although there are many etiologies for this syndrome, it is ultimately characterized by respiratory failure in the presence of airway closure and atelectasis, alveolar flooding, increased lung resistance, reduced lung compliance, and impairments in gas exchange.1,2 The current mainstay of treatment is supportive therapy with tracheal intubation and mechanical ventilation.1 Given the heterogeneous nature of this disease, positive-pressure ventilation may expose certain regions of the lung to further injury due to either an unequal distribution of inspired volume, resulting in high alveolar pressures and overdistention, or repetitive end-expiratory derecruitment and reopening.3,4 Because such ventilator-associated lung injury is the direct result of the heterogeneous nature of the injury, the ability to quantify mechanical heterogeneity may be useful in optimizing ventilatory parameters such as positive end-expiratory pressure (PEEP), tidal volume, or frequency.
The forced oscillation method to measure respiratory input impedance (Zrs  ), the complex ratio of pressure to flow at the airway opening as a function of frequency, is gaining increasing acceptance as a valid method for assessing dynamic mechanical properties of the respiratory system.5 When measured over frequencies ranging from approximately 0.1 to 10.0 Hz, Zrs  is a sensitive indicator of serial and parallel airway heterogeneity,6–8 provides insight into the locus of airway obstruction in asthma and chronic obstructive pulmonary disease,9,10 and may be useful in partitioning the mechanical properties of the lung tissues.11 
The goal of this study was to characterize the dynamic mechanical behavior of the respiratory system in a canine model of ALI. We used the frequency-dependent features of Zrs  , specifically respiratory resistance (Rrs  ) and elastance (Ers  ), to assess the degree and severity of lung injury in dogs after administration of oleic acid. Particular emphasis was placed on quantifying mechanical heterogeneity before and after injury at different levels of mean airway pressure (ao  ). The motivation for this work arises from recent morphometric modeling studies that demonstrate a strong association between the heterogeneity of airway constriction and the frequency-dependent features of Rrs  and Ers  .6,8,12–14 Although the structure–function relation of the respiratory system is extremely complex, we reasoned that the heterogeneous changes in regional lung elastances occurring in ALI may affect Zrs  in a specific and predictable manner. Furthermore, we hypothesized that as ao  increased and lung regions became more uniformly expanded, there would be characteristic changes in Zrs  , reflecting the process of recruitment and possibly a reduction in lung heterogeneity.
The specific aims of this study were (1) to measure Zrs  in mongrel dogs at baseline and after ALI induced with oleic acid; (2) to use an inverse distributed modeling approach to characterize Zrs  in terms of different distributions of respiratory tissue heterogeneity; and (3) to investigate the impact of ao  on Zrs  and by extension, on mechanical heterogeneity.
Materials and Methods
Animal Preparation and Measurements
Measurements were made in 11 mongrel dogs weighing between 7.6 and 20.0 kg. The protocol was approved by the Johns Hopkins Animal Care and Use Committee (Baltimore, Maryland) to ensure humane treatment of animals. Each dog was anesthetized with pentobarbital given intravenously (25 mg/kg at induction with 5 mg/kg hourly maintenance), relaxed with pancuronium, orally intubated with an 8.0-mm-ID endotracheal tube, and mechanically ventilated (initial rate 20 min−1and tidal volume 15 ml/kg, titrated to achieve an end-tidal level between 30 and 40 mmHg, and 5 cm H2O PEEP). Oxygen saturation was continuously monitored with a pulse oximeter applied to the tongue. Femoral arterial and venous catheters were inserted via  femoral cut-down, and a pulmonary artery catheter (7.5 French; Edwards Lifesciences LLC, Irvine, CA) was inserted through the femoral vein. Systemic arterial, pulmonary arterial, and central venous pressures were continuously monitored (Tram-Rac 4A; GE Marquette Medical Systems, Milwaukee, WI). Cardiac output was measured in triplicate using the thermodilution technique. Airway flow was measured with a pneumotachograph (Hans Rudolph 4700A; Kansas City, MO) coupled to a pressure transducer (Honeywell DC001NDC4 ± 1 in H2O; Morristown, NJ). Tracheal pressure was measured with an additional pressure transducer (Honeywell DC020NDC4 ± 20 in H2O) attached to a small polyethylene catheter placed through the endotracheal tube and allowed to extend approximately 2 cm into the trachea.15 An intravenous infusion of lactated Ringer's (15 ml · kg−1· h−1) was given for maintenance fluid replacement.
Protocol
To measure Zrs  , each dog was disconnected from the conventional mechanical ventilator and connected to a custom-built servo-controlled pneumatic pressure oscillator.16 This device is based on a proportional solenoid valve (ASCO Posiflow model SD8202G4V; Florham Park, NJ) that adjusts flow in proportion to an applied voltage and is incorporated into a closed-loop arrangement that provides accurate control of ao  during superimposed oscillations. To standardize volume history, a deep inflation to 30 cm H2O was first performed, and then tracheal pressure was reduced to a specified ao  . A discretized broadband pressure excitation signal with energy between 0.078 to 8.9 Hz was generated at a sampling rate of 40 Hz using a digital-to-analog converter (Data Translations DT-2811; Marlboro, MA). This digital signal consisted of nine sinusoids with equivalent amplitudes and random phases, with frequency components chosen to obey a nonsum nondifference criterion to minimize the impact of nonlinearities in computations of Zrs  .17 The digital signal was low-pass filtered at 10 Hz (Frequency Devices 858L8B-2; Haverhill, MA) and presented to the electronic control unit of the proportional solenoid. The net amplitude of the driving signal was adjusted to yield a delivered tidal volume of approximately 50 ml root mean square, as computed using trapezoidal integration of the sampled flow waveform. Corresponding tracheal pressure and flow signals were low-pass filtered at 10 Hz and sampled at 40 Hz by an analog-to-digital converter (Data Translations DT-2811) for subsequent processing. Between oscillatory pressure excitations, the dog was reconnected to the ventilator for a period of 4–5 min. The Zrs  measurements were obtained at ao  levels of 5, 10, 15, and 20 cm H2O applied in random order.
Lung injury was then induced by infusing 0.08 ml/kg oleic acid (Sigma-Aldrich, Inc., St. Louis, MO) into the right atrial port of the pulmonary artery catheter over 20 min. After allowing 90–120 min for the injury to stabilize, clinical signs of severe pulmonary injury were evident, including bilateral crackles and wheezes, and oxygen saturation less than 91% with inspired oxygen fraction equal to 1.0. The Zrs  measurements were then repeated as above. Dogs were killed with an intravenous overdose of pentobarbital (10–20 mg/kg) followed by rapid injection of 50 ml saturated solution of potassium chloride.
Signal Processing
Respiratory impedance Zrs  and its coherence function γ2were determined using an overlap-average periodogram technique.18 Each Zrs  spectrum was computed using a 25.6-s time window with 83% overlap. After neglecting the first 1,000 points in the data record (approximately 25 s) to minimize the influence of transient responses, between 12 and 20 overlapping windows were used to calculate Zrs  for each animal. Total Rrs  was determined as the real part of Zrs  only at those frequencies fk  where input energy was placed: Rrs  (fk  ) = Re{Zrs  (fk  )}. The effective Ers  was calculated from the imaginary part: Ers  (fk  ) =−2πfk  Im{Zrs  (fk  )}. In no instance did we find that γ2was less than 0.95 at any fk  .
Modeling Analysis
To interpret the Zrs  spectra and quantify parallel tissue heterogeneity, we modified the distributed modeling approach of Suki and coworkers.19–22 We modeled the respiratory system as a parallel arrangement of branches with equivalent linear resistive (R  ) and inertial (I  ) elements, with each branch subtended by a unique viscoelastic constant-phase tissue element (fig. 1).23 These branches do not represent distinct anatomical structures, but rather discrete functional compartments that uniquely contribute to the overall mechanical properties of the respiratory system. The R  element is a frequency-independent parameter reflecting both airway resistance as well as the purely Newtonian component of chest wall resistance.24 The I  element accounts primarily for the inertia of gas in the central airways with a small contribution from the mass of the parenchymal tissues and chest wall. The viscoelastic tissue elements are characterized by identical hysteresivity parameters (η), which account for the dynamic pressure–volume hysteresis of the respiratory tissues (i.e.  , tissue resistance), but unique elastance parameters (H1  , H2  ,…, HN  ) that vary from branch to branch according to a continuous probability density function P  (H  ). Therefore, the distribution of tissue elements allow for frequency dependence in Rrs  and Ers  by two distinct mechanisms: viscoelasticity11,23 and parallel heterogeneity.6,8,25 
Fig. 1. Model of the respiratory system with a parallel arrangement of  N  branches, each consisting of equivalent linear resistance (  R  ) and inertance (  I  ) elements and subtended by a viscoelastic tissue element. The tissue elements have equivalent hysteresivity parameters (η) but distinct tissue elastance parameters (  H1  ,  H2  , …,  HN  ) that may vary from branch to branch in a probabilistic manner. The exponent α depends on η: α= (2/π)tan−1(1/η); ω: angular frequency; and  j  : unit imaginary number (  i.e.  ,√−1)  .
Fig. 1. Model of the respiratory system with a parallel arrangement of  N  branches, each consisting of equivalent linear resistance (  R  ) and inertance (  I  ) elements and subtended by a viscoelastic tissue element. The tissue elements have equivalent hysteresivity parameters (η) but distinct tissue elastance parameters (  H1  ,  H2  , …,  HN  ) that may vary from branch to branch in a probabilistic manner. The exponent α depends on η: α= (2/π)tan−1(1/η); ω: angular frequency; and  j  : unit imaginary number (  i.e.  ,√−1) 
	.
Fig. 1. Model of the respiratory system with a parallel arrangement of  N  branches, each consisting of equivalent linear resistance (  R  ) and inertance (  I  ) elements and subtended by a viscoelastic tissue element. The tissue elements have equivalent hysteresivity parameters (η) but distinct tissue elastance parameters (  H1  ,  H2  , …,  HN  ) that may vary from branch to branch in a probabilistic manner. The exponent α depends on η: α= (2/π)tan−1(1/η); ω: angular frequency; and  j  : unit imaginary number (  i.e.  ,√−1)  .
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To obtain closed-form expressions for the model-predicted impedance (1), we assumed P  (H  ) followed either hyperbolic, linear, or uniform distributions defined over minimum (Hmin  ) and maximum (Hmax  ) values (fig. 2). The uniform distribution does not imply a homogeneous lung; rather, it assumes that the estimated distribution of tissue elastances ranging from Hmin  to Hmax  occur with equal probability. Regardless of the form of the distribution function, the model consisted of five independent parameters (R  , I  , η, Hmin  , and Hmax  ), which were estimated using a nonlinear gradient technique that minimized the sum of squared differences between actual Zrs  spectrum and the corresponding model prediction (Matlab version 7.0; The Mathworks, Natick, MA). If the gradient search algorithm converged to a unique solution for two or more model distributions, the most appropriate distribution for a given Zrs  spectrum was established using the corrected Akaike Information Criterion (AICC).26,27 For the model with the lowest AICCscore, we determined its relative likelihood of being the best model among the other candidates using the technique of Akaike weights.28 Based on the mean and SD of this selected distribution function, we computed an effective tissue elastance (μH  ) and heterogeneity index (σH  ) for each Zrs  .
Fig. 2. Continuous probability density functions used for describing tissue heterogeneity with upper and lower tissue elastance bounds  Hmin  and  Hmax  , respectively  .
Fig. 2. Continuous probability density functions used for describing tissue heterogeneity with upper and lower tissue elastance bounds  Hmin  and  Hmax  , respectively 
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Fig. 2. Continuous probability density functions used for describing tissue heterogeneity with upper and lower tissue elastance bounds  Hmin  and  Hmax  , respectively  .
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Statistical Analysis
To assess the impact of ao  and injury on the overall levels of impedance, we computed the average magnitude of Zrs  across all frequencies:
where K  is the total number of frequencies in the pressure excitation signal. A one-way analysis of variance (ANOVA) of the four ao  levels (5, 10, 15, and 20 cm H2O) was used to compare values of Rrs  and Ers  at each fk  , |rs  |, the five model parameters from the most appropriate distribution function, as well as μH  , and σH  before and after injury (SAS version 8.2; SAS Institute Inc., Cary, NC). If significance was obtained with ANOVA, post hoc  analysis was performed using the least significant difference criterion. At each ao  , preinjury and postinjury comparisons of all variables were made using two-tailed paired t  tests. P  < 0.05 was considered statistically significant.
Results
Hemodynamic and Gas Exchange Data
Baseline and postinjury hemodynamic and gas exchange data during conventional mechanical ventilation are shown in table 1. After lung injury, we observed significant decreases in heart rate, cardiac output, and the arterial partial pressure of oxygen/fraction of inspired oxygen ratio and significant increases in mean pulmonary arterial pressure, shunt fraction, and hemoglobin concentration.
Table 1. Gas Exchange and Hemodynamic Data for the 11 Dogs at Baseline and after ALI 
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Table 1. Gas Exchange and Hemodynamic Data for the 11 Dogs at Baseline and after ALI 
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Impedance Data
Figure 3shows examples of Rrs  and Ers  spectra in two representative dogs measured at baseline and after lung injury at ao  levels of 5, 10, 15, and 20 cm H2O. Also shown are the corresponding fits to the data from the model of figure 1using the most appropriate distribution function P  (H  ) based on the minimum AICCscore. For dog A, which exhibited a relatively minor response to oleic acid based on the degree of changes in Zrs  , a model comprising a linear distribution of tissue elastances was sufficient to describe the Zrs  spectra at nearly all ao  values. Dog B was considered a more severe responder to oleic acid, with baseline Zrs  at most values of ao  best described by a linear distribution of elastances but postinjury Zrs  better characterized by a uniform distribution. For both dogs, baseline levels of Ers  increased with increasing ao  , regardless of frequency.
Fig. 3. Examples of respiratory resistance (  Rrs  ) and elastance (  Ers  )  versus  frequency in two representative dogs measured at baseline (  open symbols  ) and after lung injury (  closed symbols  ). Data were obtained at mean airway pressure (  ao  ) levels of 5 (  circles  ), 10 (  inverted triangles  ), 15 (  squares  ), and 20 (  diamonds  ) cm H2O. Shown also are model fits from the most appropriate distribution for a particular  Zrs  , for these dogs either linear (  solid line  ) or uniform (  dashed line  ) distributions. Note differences in vertical axes scales between the two dogs  .
Fig. 3. Examples of respiratory resistance (  Rrs  ) and elastance (  Ers  )  versus  frequency in two representative dogs measured at baseline (  open symbols  ) and after lung injury (  closed symbols  ). Data were obtained at mean airway pressure (  P̄ao  ) levels of 5 (  circles  ), 10 (  inverted triangles  ), 15 (  squares  ), and 20 (  diamonds  ) cm H2O. Shown also are model fits from the most appropriate distribution for a particular  Zrs  , for these dogs either linear (  solid line  ) or uniform (  dashed line  ) distributions. Note differences in vertical axes scales between the two dogs 
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Fig. 3. Examples of respiratory resistance (  Rrs  ) and elastance (  Ers  )  versus  frequency in two representative dogs measured at baseline (  open symbols  ) and after lung injury (  closed symbols  ). Data were obtained at mean airway pressure (  ao  ) levels of 5 (  circles  ), 10 (  inverted triangles  ), 15 (  squares  ), and 20 (  diamonds  ) cm H2O. Shown also are model fits from the most appropriate distribution for a particular  Zrs  , for these dogs either linear (  solid line  ) or uniform (  dashed line  ) distributions. Note differences in vertical axes scales between the two dogs  .
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Figure 4shows a summary of the Rrs  and Ers  spectra for all 11 dogs at baseline and after ALI at the four levels of ao  . At baseline, Rrs  demonstrated a frequency-dependent decrease throughout the bandwidth under excitation for all values of ao  . The Ers  demonstrated a slight positive frequency dependence up to approximately 2 Hz, beyond which it demonstrated a frequency-dependent decrease and often became negative at the highest frequencies. This high frequency decrease results from the effect of gas inertia in the central airways on the respiratory system reactance, from which the Ers  spectrum is computed. At baseline, the Ers  at 20 cm H2O was significantly higher over the 0.07- to 4.02-Hz range compared with 5, 10, and 15 cm H2O. After lung injury, both Rrs  and Ers  exhibited increases in their respective mean levels and dependence on frequency compared with baseline for all values of ao  . Significant increases in Rrs  and Ers  after ALI were observed using paired t  tests at each fk  as shown in figure 4. After ALI, Rrs  at 5 cm H2O ao  was significantly higher compared with 10, 15, and 20 cm H2O ao  at all frequencies, although the Ers  at 5 cm H2O was higher only for frequencies between 0.20 and 2.31 Hz.
Fig. 4. Summary of respiratory resistance (  Rrs  ) and elastance (  Ers  ) from 0.078 to 8.9 Hz at baseline (ˆ) and after oleic acid injury (•) at four different mean airway pressures (  ao  ) for all 11 dogs. Values are presented as mean ± SD. Significantly higher from baseline data at same frequency and  ao  using two-tailed paired  t  test. † Significantly higher than corresponding data at 10, 15, and 20 cm H2O at same condition and frequency using analysis of variance and least significant difference criterion. ‡ Significantly higher from corresponding data at 5, 10, and 15 cm H2O at same condition and frequency using analysis of variance and least significant difference criterion  .
Fig. 4. Summary of respiratory resistance (  Rrs  ) and elastance (  Ers  ) from 0.078 to 8.9 Hz at baseline (ˆ) and after oleic acid injury (•) at four different mean airway pressures (  P̄ao  ) for all 11 dogs. Values are presented as mean ± SD. Significantly higher from baseline data at same frequency and  P̄ao  using two-tailed paired  t  test. † Significantly higher than corresponding data at 10, 15, and 20 cm H2O at same condition and frequency using analysis of variance and least significant difference criterion. ‡ Significantly higher from corresponding data at 5, 10, and 15 cm H2O at same condition and frequency using analysis of variance and least significant difference criterion 
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Fig. 4. Summary of respiratory resistance (  Rrs  ) and elastance (  Ers  ) from 0.078 to 8.9 Hz at baseline (ˆ) and after oleic acid injury (•) at four different mean airway pressures (  ao  ) for all 11 dogs. Values are presented as mean ± SD. Significantly higher from baseline data at same frequency and  ao  using two-tailed paired  t  test. † Significantly higher than corresponding data at 10, 15, and 20 cm H2O at same condition and frequency using analysis of variance and least significant difference criterion. ‡ Significantly higher from corresponding data at 5, 10, and 15 cm H2O at same condition and frequency using analysis of variance and least significant difference criterion  .
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The impact of injury and ao  on |rs  | is shown in figure 5. At baseline, |Z̄Zrs  | at 20 cm H2O was significantly higher compared with 5, 10, and 15 cm H2O. After lung injury, significant increases in |rs  | were observed at all values of ao  and |rs  | was significantly increased at ao  = 5 cm H2O compared with all other values of ao  .
Fig. 5. Summary of average impedance magnitude  |rs  |at baseline (  white  ) and after oleic acid injury (  black  ) at four different mean airway pressures (  ao  ) for all 11 dogs. Values are presented as mean ± SD. * Significantly higher from baseline data using two-tailed paired  t  test. † Significantly higher than corresponding data at 10, 15, and 20 cm H2O at same condition using analysis of variance and least significant difference criterion. ‡ Significantly higher from corresponding data at 5, 10, and 15 cm H2O at same condition using analysis of variance and least significant difference criterion  .
Fig. 5. Summary of average impedance magnitude 
	|Z̄rs  |at baseline (  white  ) and after oleic acid injury (  black  ) at four different mean airway pressures (  P̄ao  ) for all 11 dogs. Values are presented as mean ± SD. * Significantly higher from baseline data using two-tailed paired  t  test. † Significantly higher than corresponding data at 10, 15, and 20 cm H2O at same condition using analysis of variance and least significant difference criterion. ‡ Significantly higher from corresponding data at 5, 10, and 15 cm H2O at same condition using analysis of variance and least significant difference criterion 
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Fig. 5. Summary of average impedance magnitude  |rs  |at baseline (  white  ) and after oleic acid injury (  black  ) at four different mean airway pressures (  ao  ) for all 11 dogs. Values are presented as mean ± SD. * Significantly higher from baseline data using two-tailed paired  t  test. † Significantly higher than corresponding data at 10, 15, and 20 cm H2O at same condition using analysis of variance and least significant difference criterion. ‡ Significantly higher from corresponding data at 5, 10, and 15 cm H2O at same condition using analysis of variance and least significant difference criterion  .
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Model Analysis
Figure 6shows a summary of the most appropriate distribution functions as selected according to AICCscore for all 11 dogs at each ao  at baseline and after lung injury. At baseline, the Zrs  for most dogs was best described by a linear distribution of tissue elastances. After lung injury, there was an increase in the number of dogs for which Zrs  was best described by a uniform distribution of tissue elastances, although the majority of dogs were still best characterized by a linear distribution. A summary of the Akaike weights for the selected model distributions is shown in table 2, along with the number of dogs for which two or more model distributions converged for a particular Zrs  data set using the nonlinear gradient search algorithm. Under baseline conditions, the likelihoods that the selected distribution generated the data when compared to the alternate candidate distributions ranged from 85.4 to 90.5% as averaged at each ao  . After lung injury, the likelihoods decreased at all values of ao  , with mean values ranging from 60.2 to 77.8%. No clear trend of Akaike weight on ao  was observed. Although at least one distribution function converged for every Zrs  data set examined, the number of dogs for which two or more distribution functions converged decreased after lung injury.
Fig. 6. Summary of tissue elastance distributions obtained for all 11 dogs at baseline and after lung injury at four different values of mean airway pressure (  ao  ). At baseline, the  Zrs  obtained in most dogs was best described by a model incorporating a linear distribution of tissue elastances, regardless of  ao  . After injury, the number of dogs requiring a model with a uniform distribution of tissue elastances increased for 5, 10, and 20 cm H2O  .
Fig. 6. Summary of tissue elastance distributions obtained for all 11 dogs at baseline and after lung injury at four different values of mean airway pressure (  P̄ao  ). At baseline, the  Zrs  obtained in most dogs was best described by a model incorporating a linear distribution of tissue elastances, regardless of  P̄ao  . After injury, the number of dogs requiring a model with a uniform distribution of tissue elastances increased for 5, 10, and 20 cm H2O 
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Fig. 6. Summary of tissue elastance distributions obtained for all 11 dogs at baseline and after lung injury at four different values of mean airway pressure (  ao  ). At baseline, the  Zrs  obtained in most dogs was best described by a model incorporating a linear distribution of tissue elastances, regardless of  ao  . After injury, the number of dogs requiring a model with a uniform distribution of tissue elastances increased for 5, 10, and 20 cm H2O  .
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Table 2. Summary of Akaike Weights for Model Distributions with Lowest AICCScore 
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Table 2. Summary of Akaike Weights for Model Distributions with Lowest AICCScore 
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A summary of the estimated model parameters is shown in figure 7. The R  parameter, reflecting both airway resistance and the Newtonian component of chest wall resistance, demonstrated no significant dependence on ao  during baseline or ALI conditions. Although R  demonstrated considerable intersubject variability after lung injury, there was no significant difference between preinjury and postinjury R  at any ao  . The I  parameter, reflecting the total inertia of the respiratory system, exhibited no significant dependence on ao  at baseline. After lung injury, I  significantly increased compared with baseline at all ao  values. ANOVA demonstrated a significant dependence of postinjury estimates of I  on ao  , with its value at 5 cm H2O significantly higher compared with 15 and 20 cm H2O. Baseline values of the hysteresivity parameter η were within the range reported by previous studies9,11,29 and significantly increased after lung injury at all ao  levels. Neither the Hmin  nor the Hmax  parameters demonstrated any dependence on ao  at baseline. ANOVA demonstrated a significant dependence of Hmax  on ao  after lung injury, with its value at 5 cm H2O significantly higher than that at 10, 15, or 20 cm H2O. After injury, Hmin  increased significantly only at 20 cm H2O, whereas Hmax  increased significantly at 5, 10, and 15 cm H2O.
Fig. 7. Summary of model parameters  R  ,  I  , η,  Hmin  , and  Hmaxversus  mean airway pressure (  ao  ) for all 11 dogs. Data are shown at baseline (ˆ) and after lung injury (•). Values are presented as mean ± SD. * Significantly higher from baseline data at same  ao  using two-tailed paired  t  test. † Significantly higher than corresponding data at 10, 15, and 20 cm H2O at same condition using analysis of variance and least significant difference criterion. § Significantly higher than corresponding data at 15 and 20 cm H2O at same condition using analysis of variance and least significant difference criterion  .
Fig. 7. Summary of model parameters  R  ,  I  , η,  Hmin  , and  Hmaxversus  mean airway pressure (  P̄ao  ) for all 11 dogs. Data are shown at baseline (ˆ) and after lung injury (•). Values are presented as mean ± SD. * Significantly higher from baseline data at same  P̄ao  using two-tailed paired  t  test. † Significantly higher than corresponding data at 10, 15, and 20 cm H2O at same condition using analysis of variance and least significant difference criterion. § Significantly higher than corresponding data at 15 and 20 cm H2O at same condition using analysis of variance and least significant difference criterion 
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Fig. 7. Summary of model parameters  R  ,  I  , η,  Hmin  , and  Hmaxversus  mean airway pressure (  ao  ) for all 11 dogs. Data are shown at baseline (ˆ) and after lung injury (•). Values are presented as mean ± SD. * Significantly higher from baseline data at same  ao  using two-tailed paired  t  test. † Significantly higher than corresponding data at 10, 15, and 20 cm H2O at same condition using analysis of variance and least significant difference criterion. § Significantly higher than corresponding data at 15 and 20 cm H2O at same condition using analysis of variance and least significant difference criterion  .
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Figure 8shows the impact of ao  and injury on the derived effective tissue elastance (μH  ) and heterogeneity index (σH  ). At baseline, ANOVA demonstrated no significant dependence of either μH  or σH  on ao  . After ALI, μH  significantly increased at all ao  levels, whereas σH  increased only at 5 and 10 cm H2O. Both μH  and σH  demonstrated dependence on ao  after ALI, with postinjury values at 5 cm H2O significantly increased compared with 10, 15, and 20 cm H2O. The σH  was significantly increased compared with baseline at 5 and 10 cm H2O.
Fig. 8. Effective mean tissue elastance (μ  H  ) and heterogeneity index (σ  H  ) for all 11 dogs. Data are shown at baseline (ˆ) and after lung injury (•). Values are presented as mean ± SD. * Significantly higher from baseline data at same mean airway pressure (  ao  ) using two-tailed paired  t  test. † Significantly higher from corresponding data at 10, 15, and 20 cm H2O at same condition using analysis of variance and least significant difference criterion  .
Fig. 8. Effective mean tissue elastance (μ  H  ) and heterogeneity index (σ  H  ) for all 11 dogs. Data are shown at baseline (ˆ) and after lung injury (•). Values are presented as mean ± SD. * Significantly higher from baseline data at same mean airway pressure (  P̄ao  ) using two-tailed paired  t  test. † Significantly higher from corresponding data at 10, 15, and 20 cm H2O at same condition using analysis of variance and least significant difference criterion 
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Fig. 8. Effective mean tissue elastance (μ  H  ) and heterogeneity index (σ  H  ) for all 11 dogs. Data are shown at baseline (ˆ) and after lung injury (•). Values are presented as mean ± SD. * Significantly higher from baseline data at same mean airway pressure (  ao  ) using two-tailed paired  t  test. † Significantly higher from corresponding data at 10, 15, and 20 cm H2O at same condition using analysis of variance and least significant difference criterion  .
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Discussion
The heterogeneous nature of ALI is characterized by mechanically disparate lung regions, including collapsed nonventilated regions, injured but recruitable regions, normally ventilated regions, and overdistended regions prone to inflation injury. Recently, an “open-lung approach” to ventilation in these patients has been advocated, using PEEP to recruit collapsed lung units and improve oxygenation and low tidal volumes to minimize the risk of overdistention injury.3 However, optimal PEEP and tidal volume for injured lungs are difficult to determine. Although PEEP may improve compliance in some regions of the lung by recruiting alveoli, it may simultaneously decrease compliance in other regions by overstretching units.
In the past, quasi-static pressure–volume curves were thought to provide insight into the “safe” ranges over which lung recruitment and overdistention injuries occur and therefore have been proposed for use in optimizing the mechanical stresses placed on the lungs during positive-pressure ventilation30,31 and more recently during high-frequency ventilation.32,33 However, as static measurements constructed under periods of zero flow, these curves provide no information regarding the dynamic regional mechanics in the lung or insight into the nature and distribution of lung injury during mechanical ventilation. Simple dynamic measure indices of lung mechanics, such as resistance and elastance measured at a single breathing frequency, can provide some insight into gross pathophysiology of the entire respiratory system but can be misleading if used to interpret regional mechanics of the lungs in the complex pathophysiology of ALI or to optimize ventilation strategy under varying conditions of frequency, tidal volume, or PEEP.34–36 More recently, Ranieri et al.  37 characterized the concavity of the airway pressure–time curve during constant flow inflation with a power law expression to infer the balance between intratidal recruitment and overdistention occurring on a whole lung scale. Although this simple “stress index” may have some predictive value for the development of ventilator-associated lung injury, it does not quantify regional mechanical heterogeneity, which may be a more sensitive indicator. Multibreath inert gas washouts have also been used to detect the presence of heterogeneity and follow trends during the development of ALI,38 but these approaches do not directly quantify heterogeneity or changes in the mechanical properties of the respiratory tissues. Finally, other investigators have used high resolution computed tomography to assess heterogeneity in ALI,39,40 although such a technique is not amenable to bedside implementation. Given the paucity of techniques available to assess heterogeneity in ALI, the ability to optimize the delicate balance of recruitment, oxygenation, and overdistention in this syndrome remains a challenge.
Heterogeneity and Respiratory Impedance
In 1956, Otis et al.  25 were the first to demonstrate how parallel time constant heterogeneity in the lung can cause frequency dependence in Rrs  and Ers  . Since then, other groups have shown that the mechanical impedance spectrum of the lungs or total respiratory system can be a sensitive indicator of mechanical heterogeneity.6,8,20,41,42 Using a computer-generated (but anatomically accurate) airway tree, Lutchen et al.  6,8,12,14 demonstrated that two distinct pathophysiologic mechanisms can be uniquely identified from Zrs  : (1) parallel time constant heterogeneity, which results in enhanced frequency dependence of Rrs  and Ers  predominately below 2 Hz; and (2) random airway closure and derecruitment of lung units, which results in less lung tissue in communication with the airway opening and hence an increase in the levels of Rrs  and Ers  throughout the entire bandwidth of interest. Both of these mechanisms can substantially increase Rrs  and Ers  near breathing frequencies, exclusive of large increases in either overall airway resistance or actual tissue elasticity.12 
Our Zrs  data are surprisingly similar to those data reported in many of these earlier modeling studies. At baseline, both Rrs  and Ers  are dependent on frequency, which has been attributed primarily to the viscoelastic properties of the parenchymal tissues and chest wall in the healthy respiratory system,11,29,34,43,44 although some baseline mechanical heterogeneity may also contribute to frequency dependence.7,14,22,45 Under control conditions, we observed significant increases in Ers  and |rs  | with increasing ao  , which may result from overstretching and plastoelastic phenomena.46 After lung injury, we observed significant increases in |rs  | and both the levels and frequency dependence of Rrs  and Ers  at all values of ao  , indicating widespread mechanical heterogeneity and loss of alveolar units in communication with the airway opening as edema fluid fills air spaces. In contrast to baseline conditions, Ers  and |rs  | both decreased with increasing ao  after ALI, consistent with the notion that moderate increases in ao  are beneficial by reducing heterogeneity and recruiting lung units.
Model Analysis
To further quantify mechanical heterogeneity from our Zrs  data, we relied on the distributed modeling approach initially developed by Suki and coworkers19,21 to describe the mechanical properties of the lungs during bronchoconstriction. More recently, this group proposed a similar analysis to describe the heterogeneity of tissue elastances in mouse model of emphysema, which predicts frequency dependence in both Rrs  and Ers  not only by the mechanism of parallel heterogeneity, but also by viscoelasticity of the respiratory tissues.20 They used only a hyperbolic distribution of tissue elastances to characterize Zrs  , whereas our data suggest that other tissue distribution functions may in fact be more appropriate to describe the dynamic behavior of the respiratory system after lung injury in dogs. In normal lungs, the AICCpredicts that the linear distribution of tissue elastances was most appropriate for describing Zrs  in most dogs regardless of ao  . After lung injury, there was an increase in the number of dogs for which Zrs  was best characterized by a uniform distribution of tissue elastances. Nonetheless, most dogs were still best described by a linear distribution of elastances.
Although the model distribution with the lowest AICCscore was most likely to have generated the data, it is important to understand that the AICCdoes not provide information regarding how well a particular distribution outperforms the other candidate distributions or the relative likelihood that this distribution generated the data. We determined this information using the technique of Akaike weights.28 For the majority of dogs, the likelihood that the selected distribution generated the data were unequivocal (i.e.  , > 95%), although for a few dogs, the distinction between distributions was not as clear, particularly after lung injury (table 2).
Surprisingly, our R  parameter demonstrated no significant dependence on ao  .20,44,47 Although the value of R  is composed of both airway resistance and Newtonian chest wall resistance, both may be influenced by thoracic volume to different extents.24 This may mask any detectable relation between R  and ao  . The I  parameter, which in the healthy respiratory system primarily reflects the inertia of gas in the central airways, was significantly increased after ALI. This may result from a net increase in the oscillating mass of lung tissue due to the accumulation of edema fluid in the air spaces.
The progressive decrease in the average dynamic elastance parameter μH  with increasing ao  in injured animals is consistent with lung recruitment and previous reports of changes in the static lung elastance as a consequence of alveolar flooding and airway closure.30,48 Therefore, it is unlikely that changes in μH  reflect alterations in the actual elastic properties of the parenchymal tissues or chest wall. Baseline values of our heterogeneity index σH  demonstrated a decreasing trend with ao  up to 15 cm H2O but increased slightly at 20 cm H2O, although this did not achieve statistical significance. Although this is consistent with lung recruitment and decreases in mechanical heterogeneity, higher levels of ao  may result in overdistention occurring in a somewhat heterogeneous manner. After lung injury, σH  increased significantly compared with baseline at 5 and 10 cm H2O, implying an increase in mechanical heterogeneity. In addition, ANOVA demonstrated a significant dependence of σH  on ao  after ALI, with its highest value at 5 cm H2O. Further increases in ao  seem to reduce σH  , also consistent with lung recruitment.
We observed significant increases in our hysteresivity parameter η after ALI. Hysteresivity has been proposed by Fredberg and Stamenovic29 as an index to describe the hysteretic pressure–volume relation of the lung tissues and can be thought of as a ratio of energy dissipation to energy storage during cyclic changes in lung volume. It has long been held that η is relatively constant across lung volume, breathing frequency, and tidal volume.29 However, these assumptions have been called into question in recent studies,47,49 especially in animal models of ALI.50,51 A key assumption of our modeling analysis is that η is constant throughout the respiratory tissues regardless of disease condition.20,29 Our observed increases in η after ALI may reflect actual changes in the coupling of energy dissipation and storage in the respiratory tissues.29 Hysteresivity is known to be influenced by several factors, including surface forces of the air–tissue interface, Coulomb friction between collagen and elastin fibers, and cross-bridge cycling in airway smooth muscle or other contractile elements in the lung parenchyma, any of which may be influenced by ALI. Moreover, there may be additional airway or tissue heterogeneity that is not appropriately described by our distribution functions, resulting in modeling error with an artifactually high estimate of η.9,11,41 
Limitations
Despite the apparent utility of our approach to quantify mechanical heterogeneity in the respiratory system, these techniques rely on a few key assumptions that must be considered when using them to quantify regional mechanics. First, for mathematical convenience and simplicity, we evaluated only three simple tissue distribution functions to describe respiratory system heterogeneity as assessed at the airway opening. Although it seems that the Zrs  of most dogs is best described by a linear distribution of tissue elastances, we cannot be certain how accurately these functions describe actual tissue variability. Such information is probably more accurately obtained using functional computed tomography to quantify specific elastances.39 Future studies may incorporate similar modeling approaches with imaging data to obtain more accurate tissue distribution functions for describing Zrs  . Therefore, although the current accuracy of our heterogeneity estimates may be somewhat biased because of modeling error, σH  may still be a useful marker to assess the degree of heterogeneity in the lungs.
In addition, our modeling approach assumes that tissue heterogeneity is randomly distributed throughout the lungs and, as such, does not provide specific anatomic information on regional mechanics. Tissue heterogeneity in ALI may be distributed in a more deterministic manner, depending on the etiology of the injury, orientation in the gravitational field, local pulmonary blood flow, or proximity to the pleura or other organs. Whether such additional anatomical information would be useful in optimizing ventilatory parameters is unknown.
Finally, we recognize that our modeling analysis accounts only for tissue heterogeneity. Airway heterogeneity may also contribute to the enhanced frequency dependence in Rrs  and Ers  ,6,12,19,21,41 although the degree to which it does so in ALI is not clear. A previous study by Barnas et al.  44 using alveolar capsules in dogs demonstrated that increases in the frequency dependence of lung resistance and elastance after oleic acid–induced pulmonary edema were due primarily to changes in parenchymal tissue mechanics, implying that changes in airway resistance are not significant in the oleic acid model of lung injury. Similarly, our model analysis demonstrated no significant differences between preinjury and postinjury R  . Therefore, we believe it is unlikely that airway heterogeneity contributed significantly to the frequency dependence we observed in Rrs  and Ers  after ALI.
Summary
In summary, these data demonstrate that Zrs  can provide specific information about the mechanical heterogeneity of the lungs and the impact of ao  . After lung injury, both the level and frequency dependence of Rrs  and Ers  increase compared with baseline, indicating the presence of widespread mechanical heterogeneity. These differences seem to be reduced by increases in ao  , consistent with the recruitment of lung units and minimization of the impact of heterogeneity on dynamic respiratory mechanics. Model analysis of Zrs  demonstrate that both effective tissue elastance and mechanical heterogeneity increase after ALI and decrease with increasing ao  . These noninvasive approaches may be useful in identifying optimal PEEP levels and tidal volume during conventional mechanical ventilation and possibly allow for the development of ventilation protocols to optimize regional lung mechanics in patients with ALI. Future research should be directed toward the development of more accurate tissue distribution functions to quantify the heterogeneity of injured lungs, which will be essential in the validation of this technique as a clinical diagnostic tool.
The authors thank H. Pierre Burman (Senior Animal Technician) and Kathleen K. Kibler (Laboratory Manager, Anesthesiology and Critical Care Medicine, Johns Hopkins University, Baltimore, Maryland) for technical assistance in this study.
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Appendix
The model-predicted impedance (ẑrs  ) of the network of figure 1can be defined as the reciprocal of the sum of admittances of each branch:
with
and where j  is the unit imaginary number, ω is the angular frequency, and N  is the number of branches in the network. Alternatively, if we assume that the variation in tissue elastance between branches varies in a probabilistic manner, we can approximate ẑrs  as
where PΔH  represents a histogram function of bin width ΔH  , Hm  denotes the center value of elastances stored in bin m  , and M  is the total number of bins in the distribution. In the limit as the number of branches tends toward infinity and ΔH  approaches zero, we obtain an expression for ẑrs  that depends on a continuous probability density function P  (H  ) integrated over the upper and lower bounds Hmin  and Hmax  :
Regardless of the form of P  (H  ), we can define an effective mean tissue elastance, μH  , which provides a rough estimate of the overall elastance of the respiratory tissues:
To quantify the heterogeneity of the respiratory tissues, we relied on the SD σH  of P  (H  ):
To obtain simple, closed-form expressions for equation 4, we defined a probability density function P  (H  ) a priori  to be of the form
where L  is a constant which depends on Hmin  and Hmax  as well as the function P  (H  ). We assume λ=−1, 0, 1, which define models corresponding to linear, uniform, and hyperbolic distributions of tissue elastances, respectively (fig. 3). An expression for L  can be obtained by assuming that the area under the probability density function P  (H  ) is equal to 1:
Thus, using equation 7with λ=−1 to solve for equations 4, 5, and 6, we obtain for the linear distribution,
and with λ= 0 for the uniform distribution
and finally with λ= 1 for the hyperbolic distribution,
Note that for equations 9, 12, and 15, we define the natural logarithm of any arbitrary complex number z  =x  +jy  as
Fig. 1. Model of the respiratory system with a parallel arrangement of  N  branches, each consisting of equivalent linear resistance (  R  ) and inertance (  I  ) elements and subtended by a viscoelastic tissue element. The tissue elements have equivalent hysteresivity parameters (η) but distinct tissue elastance parameters (  H1  ,  H2  , …,  HN  ) that may vary from branch to branch in a probabilistic manner. The exponent α depends on η: α= (2/π)tan−1(1/η); ω: angular frequency; and  j  : unit imaginary number (  i.e.  ,√−1)  .
Fig. 1. Model of the respiratory system with a parallel arrangement of  N  branches, each consisting of equivalent linear resistance (  R  ) and inertance (  I  ) elements and subtended by a viscoelastic tissue element. The tissue elements have equivalent hysteresivity parameters (η) but distinct tissue elastance parameters (  H1  ,  H2  , …,  HN  ) that may vary from branch to branch in a probabilistic manner. The exponent α depends on η: α= (2/π)tan−1(1/η); ω: angular frequency; and  j  : unit imaginary number (  i.e.  ,√−1) 
	.
Fig. 1. Model of the respiratory system with a parallel arrangement of  N  branches, each consisting of equivalent linear resistance (  R  ) and inertance (  I  ) elements and subtended by a viscoelastic tissue element. The tissue elements have equivalent hysteresivity parameters (η) but distinct tissue elastance parameters (  H1  ,  H2  , …,  HN  ) that may vary from branch to branch in a probabilistic manner. The exponent α depends on η: α= (2/π)tan−1(1/η); ω: angular frequency; and  j  : unit imaginary number (  i.e.  ,√−1)  .
×
Fig. 2. Continuous probability density functions used for describing tissue heterogeneity with upper and lower tissue elastance bounds  Hmin  and  Hmax  , respectively  .
Fig. 2. Continuous probability density functions used for describing tissue heterogeneity with upper and lower tissue elastance bounds  Hmin  and  Hmax  , respectively 
	.
Fig. 2. Continuous probability density functions used for describing tissue heterogeneity with upper and lower tissue elastance bounds  Hmin  and  Hmax  , respectively  .
×
Fig. 3. Examples of respiratory resistance (  Rrs  ) and elastance (  Ers  )  versus  frequency in two representative dogs measured at baseline (  open symbols  ) and after lung injury (  closed symbols  ). Data were obtained at mean airway pressure (  ao  ) levels of 5 (  circles  ), 10 (  inverted triangles  ), 15 (  squares  ), and 20 (  diamonds  ) cm H2O. Shown also are model fits from the most appropriate distribution for a particular  Zrs  , for these dogs either linear (  solid line  ) or uniform (  dashed line  ) distributions. Note differences in vertical axes scales between the two dogs  .
Fig. 3. Examples of respiratory resistance (  Rrs  ) and elastance (  Ers  )  versus  frequency in two representative dogs measured at baseline (  open symbols  ) and after lung injury (  closed symbols  ). Data were obtained at mean airway pressure (  P̄ao  ) levels of 5 (  circles  ), 10 (  inverted triangles  ), 15 (  squares  ), and 20 (  diamonds  ) cm H2O. Shown also are model fits from the most appropriate distribution for a particular  Zrs  , for these dogs either linear (  solid line  ) or uniform (  dashed line  ) distributions. Note differences in vertical axes scales between the two dogs 
	.
Fig. 3. Examples of respiratory resistance (  Rrs  ) and elastance (  Ers  )  versus  frequency in two representative dogs measured at baseline (  open symbols  ) and after lung injury (  closed symbols  ). Data were obtained at mean airway pressure (  ao  ) levels of 5 (  circles  ), 10 (  inverted triangles  ), 15 (  squares  ), and 20 (  diamonds  ) cm H2O. Shown also are model fits from the most appropriate distribution for a particular  Zrs  , for these dogs either linear (  solid line  ) or uniform (  dashed line  ) distributions. Note differences in vertical axes scales between the two dogs  .
×
Fig. 4. Summary of respiratory resistance (  Rrs  ) and elastance (  Ers  ) from 0.078 to 8.9 Hz at baseline (ˆ) and after oleic acid injury (•) at four different mean airway pressures (  ao  ) for all 11 dogs. Values are presented as mean ± SD. Significantly higher from baseline data at same frequency and  ao  using two-tailed paired  t  test. † Significantly higher than corresponding data at 10, 15, and 20 cm H2O at same condition and frequency using analysis of variance and least significant difference criterion. ‡ Significantly higher from corresponding data at 5, 10, and 15 cm H2O at same condition and frequency using analysis of variance and least significant difference criterion  .
Fig. 4. Summary of respiratory resistance (  Rrs  ) and elastance (  Ers  ) from 0.078 to 8.9 Hz at baseline (ˆ) and after oleic acid injury (•) at four different mean airway pressures (  P̄ao  ) for all 11 dogs. Values are presented as mean ± SD. Significantly higher from baseline data at same frequency and  P̄ao  using two-tailed paired  t  test. † Significantly higher than corresponding data at 10, 15, and 20 cm H2O at same condition and frequency using analysis of variance and least significant difference criterion. ‡ Significantly higher from corresponding data at 5, 10, and 15 cm H2O at same condition and frequency using analysis of variance and least significant difference criterion 
	.
Fig. 4. Summary of respiratory resistance (  Rrs  ) and elastance (  Ers  ) from 0.078 to 8.9 Hz at baseline (ˆ) and after oleic acid injury (•) at four different mean airway pressures (  ao  ) for all 11 dogs. Values are presented as mean ± SD. Significantly higher from baseline data at same frequency and  ao  using two-tailed paired  t  test. † Significantly higher than corresponding data at 10, 15, and 20 cm H2O at same condition and frequency using analysis of variance and least significant difference criterion. ‡ Significantly higher from corresponding data at 5, 10, and 15 cm H2O at same condition and frequency using analysis of variance and least significant difference criterion  .
×
Fig. 5. Summary of average impedance magnitude  |rs  |at baseline (  white  ) and after oleic acid injury (  black  ) at four different mean airway pressures (  ao  ) for all 11 dogs. Values are presented as mean ± SD. * Significantly higher from baseline data using two-tailed paired  t  test. † Significantly higher than corresponding data at 10, 15, and 20 cm H2O at same condition using analysis of variance and least significant difference criterion. ‡ Significantly higher from corresponding data at 5, 10, and 15 cm H2O at same condition using analysis of variance and least significant difference criterion  .
Fig. 5. Summary of average impedance magnitude 
	|Z̄rs  |at baseline (  white  ) and after oleic acid injury (  black  ) at four different mean airway pressures (  P̄ao  ) for all 11 dogs. Values are presented as mean ± SD. * Significantly higher from baseline data using two-tailed paired  t  test. † Significantly higher than corresponding data at 10, 15, and 20 cm H2O at same condition using analysis of variance and least significant difference criterion. ‡ Significantly higher from corresponding data at 5, 10, and 15 cm H2O at same condition using analysis of variance and least significant difference criterion 
	.
Fig. 5. Summary of average impedance magnitude  |rs  |at baseline (  white  ) and after oleic acid injury (  black  ) at four different mean airway pressures (  ao  ) for all 11 dogs. Values are presented as mean ± SD. * Significantly higher from baseline data using two-tailed paired  t  test. † Significantly higher than corresponding data at 10, 15, and 20 cm H2O at same condition using analysis of variance and least significant difference criterion. ‡ Significantly higher from corresponding data at 5, 10, and 15 cm H2O at same condition using analysis of variance and least significant difference criterion  .
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Fig. 6. Summary of tissue elastance distributions obtained for all 11 dogs at baseline and after lung injury at four different values of mean airway pressure (  ao  ). At baseline, the  Zrs  obtained in most dogs was best described by a model incorporating a linear distribution of tissue elastances, regardless of  ao  . After injury, the number of dogs requiring a model with a uniform distribution of tissue elastances increased for 5, 10, and 20 cm H2O  .
Fig. 6. Summary of tissue elastance distributions obtained for all 11 dogs at baseline and after lung injury at four different values of mean airway pressure (  P̄ao  ). At baseline, the  Zrs  obtained in most dogs was best described by a model incorporating a linear distribution of tissue elastances, regardless of  P̄ao  . After injury, the number of dogs requiring a model with a uniform distribution of tissue elastances increased for 5, 10, and 20 cm H2O 
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Fig. 6. Summary of tissue elastance distributions obtained for all 11 dogs at baseline and after lung injury at four different values of mean airway pressure (  ao  ). At baseline, the  Zrs  obtained in most dogs was best described by a model incorporating a linear distribution of tissue elastances, regardless of  ao  . After injury, the number of dogs requiring a model with a uniform distribution of tissue elastances increased for 5, 10, and 20 cm H2O  .
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Fig. 7. Summary of model parameters  R  ,  I  , η,  Hmin  , and  Hmaxversus  mean airway pressure (  ao  ) for all 11 dogs. Data are shown at baseline (ˆ) and after lung injury (•). Values are presented as mean ± SD. * Significantly higher from baseline data at same  ao  using two-tailed paired  t  test. † Significantly higher than corresponding data at 10, 15, and 20 cm H2O at same condition using analysis of variance and least significant difference criterion. § Significantly higher than corresponding data at 15 and 20 cm H2O at same condition using analysis of variance and least significant difference criterion  .
Fig. 7. Summary of model parameters  R  ,  I  , η,  Hmin  , and  Hmaxversus  mean airway pressure (  P̄ao  ) for all 11 dogs. Data are shown at baseline (ˆ) and after lung injury (•). Values are presented as mean ± SD. * Significantly higher from baseline data at same  P̄ao  using two-tailed paired  t  test. † Significantly higher than corresponding data at 10, 15, and 20 cm H2O at same condition using analysis of variance and least significant difference criterion. § Significantly higher than corresponding data at 15 and 20 cm H2O at same condition using analysis of variance and least significant difference criterion 
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Fig. 7. Summary of model parameters  R  ,  I  , η,  Hmin  , and  Hmaxversus  mean airway pressure (  ao  ) for all 11 dogs. Data are shown at baseline (ˆ) and after lung injury (•). Values are presented as mean ± SD. * Significantly higher from baseline data at same  ao  using two-tailed paired  t  test. † Significantly higher than corresponding data at 10, 15, and 20 cm H2O at same condition using analysis of variance and least significant difference criterion. § Significantly higher than corresponding data at 15 and 20 cm H2O at same condition using analysis of variance and least significant difference criterion  .
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Fig. 8. Effective mean tissue elastance (μ  H  ) and heterogeneity index (σ  H  ) for all 11 dogs. Data are shown at baseline (ˆ) and after lung injury (•). Values are presented as mean ± SD. * Significantly higher from baseline data at same mean airway pressure (  ao  ) using two-tailed paired  t  test. † Significantly higher from corresponding data at 10, 15, and 20 cm H2O at same condition using analysis of variance and least significant difference criterion  .
Fig. 8. Effective mean tissue elastance (μ  H  ) and heterogeneity index (σ  H  ) for all 11 dogs. Data are shown at baseline (ˆ) and after lung injury (•). Values are presented as mean ± SD. * Significantly higher from baseline data at same mean airway pressure (  P̄ao  ) using two-tailed paired  t  test. † Significantly higher from corresponding data at 10, 15, and 20 cm H2O at same condition using analysis of variance and least significant difference criterion 
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Fig. 8. Effective mean tissue elastance (μ  H  ) and heterogeneity index (σ  H  ) for all 11 dogs. Data are shown at baseline (ˆ) and after lung injury (•). Values are presented as mean ± SD. * Significantly higher from baseline data at same mean airway pressure (  ao  ) using two-tailed paired  t  test. † Significantly higher from corresponding data at 10, 15, and 20 cm H2O at same condition using analysis of variance and least significant difference criterion  .
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Table 1. Gas Exchange and Hemodynamic Data for the 11 Dogs at Baseline and after ALI 
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Table 1. Gas Exchange and Hemodynamic Data for the 11 Dogs at Baseline and after ALI 
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Table 2. Summary of Akaike Weights for Model Distributions with Lowest AICCScore 
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Table 2. Summary of Akaike Weights for Model Distributions with Lowest AICCScore 
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