Quantifying Mechanical Heterogeneity in Canine Acute Lung Injury: Impact of Mean Airway Pressure

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.15An intravenous infusion of lactated Ringer's (15 ml · kg⁻¹· h⁻¹) was given for maintenance fluid replacement.

To measure Z^rs , each dog was disconnected from the conventional mechanical ventilator and connected to a custom-built servo-controlled pneumatic pressure oscillator.16This 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 P̄^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 P̄^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 Z^rs .17The 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 Z^rs  measurements were obtained at P̄^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 Z^rs  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.

Respiratory impedance Z^rs  and its coherence function γ²were determined using an overlap-average periodogram technique.18Each Z^rs  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 Z^rs  for each animal. Total R^rs  was determined as the real part of Z^rs  only at those frequencies f^k  where input energy was placed: R^rs  (f^k ) = Re{Z^rs  (f^k )}. The effective E^rs  was calculated from the imaginary part: E^rs  (f^k ) =−2πf^k  Im{Z^rs  (f^k )}. In no instance did we find that γ²was less than 0.95 at any f^k .

To interpret the Z^rs  spectra and quantify parallel tissue heterogeneity, we modified the distributed modeling approach of Suki and coworkers.19–22We 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).23These 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.24The 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 ,…, H^N ) 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 R^rs  and E^rs  by two distinct mechanisms: viscoelasticity11,23and parallel heterogeneity.6,8,25 

To obtain closed-form expressions for the model-predicted impedance ( appendix), we assumed P  (H ) followed either hyperbolic, linear, or uniform distributions defined over minimum (H^min ) and maximum (H^max ) values (fig. 2). The uniform distribution does not imply a homogeneous lung; rather, it assumes that the estimated distribution of tissue elastances ranging from H^min  to H^max  occur with equal probability. Regardless of the form of the distribution function, the model consisted of five independent parameters (R , I , η, H^min , and H^max ), which were estimated using a nonlinear gradient technique that minimized the sum of squared differences between actual Z^rs  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 Z^rs  spectrum was established using the corrected Akaike Information Criterion (AIC^C).26,27For the model with the lowest AIC^Cscore, we determined its relative likelihood of being the best model among the other candidates using the technique of Akaike weights.28Based on the mean and SD of this selected distribution function, we computed an effective tissue elastance (μ^H ) and heterogeneity index (σ^H ) for each Z^rs .

To assess the impact of P̄^ao  and injury on the overall levels of impedance, we computed the average magnitude of Z^rs  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 P̄^ao  levels (5, 10, 15, and 20 cm H²O) was used to compare values of R^rs  and E^rs  at each f^k , |Z̄^rs |, the five model parameters from the most appropriate distribution function, as well as μ^H , and σ^H  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 P̄^ao , preinjury and postinjury comparisons of all variables were made using two-tailed paired t  tests. P < 0.05 was considered statistically significant.

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.

Figure 3shows examples of R^rs  and E^rs  spectra in two representative dogs measured at baseline and after lung injury at P̄^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 AIC^Cscore. For dog A, which exhibited a relatively minor response to oleic acid based on the degree of changes in Z^rs , a model comprising a linear distribution of tissue elastances was sufficient to describe the Z^rs  spectra at nearly all P̄^ao  values. Dog B was considered a more severe responder to oleic acid, with baseline Z^rs  at most values of P̄^ao  best described by a linear distribution of elastances but postinjury Z^rs  better characterized by a uniform distribution. For both dogs, baseline levels of E^rs  increased with increasing P̄^ao , regardless of frequency.

Figure 4shows a summary of the R^rs  and E^rs  spectra for all 11 dogs at baseline and after ALI at the four levels of P̄^ao . At baseline, R^rs  demonstrated a frequency-dependent decrease throughout the bandwidth under excitation for all values of P̄^ao . The E^rs  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 E^rs  spectrum is computed. At baseline, the E^rs  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 R^rs  and E^rs  exhibited increases in their respective mean levels and dependence on frequency compared with baseline for all values of P̄^ao . Significant increases in R^rs  and E^rs  after ALI were observed using paired t  tests at each f^k  as shown in figure 4. After ALI, R^rs  at 5 cm H²O P̄^ao  was significantly higher compared with 10, 15, and 20 cm H²O P̄^ao  at all frequencies, although the E^rs  at 5 cm H²O was higher only for frequencies between 0.20 and 2.31 Hz.

The impact of injury and P̄^ao  on |Z̄^rs | is shown in figure 5. At baseline, |Z̄Z^rs | at 20 cm H²O was significantly higher compared with 5, 10, and 15 cm H²O. After lung injury, significant increases in |Z̄^rs | were observed at all values of P̄^ao  and |Z̄^rs | was significantly increased at P̄^ao = 5 cm H²O compared with all other values of P̄^ao .

Figure 6shows a summary of the most appropriate distribution functions as selected according to AIC^Cscore for all 11 dogs at each P̄^ao  at baseline and after lung injury. At baseline, the Z^rs  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 Z^rs  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 Z^rs  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 P̄^ao . After lung injury, the likelihoods decreased at all values of P̄^ao , with mean values ranging from 60.2 to 77.8%. No clear trend of Akaike weight on P̄^ao  was observed. Although at least one distribution function converged for every Z^rs  data set examined, the number of dogs for which two or more distribution functions converged decreased after lung injury.

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 P̄^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 P̄^ao . The I  parameter, reflecting the total inertia of the respiratory system, exhibited no significant dependence on P̄^ao  at baseline. After lung injury, I  significantly increased compared with baseline at all P̄^ao  values. ANOVA demonstrated a significant dependence of postinjury estimates of I  on P̄^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 studies9,11,29and significantly increased after lung injury at all P̄^ao  levels. Neither the H^min  nor the H^max  parameters demonstrated any dependence on P̄^ao  at baseline. ANOVA demonstrated a significant dependence of H^max  on P̄^ao  after lung injury, with its value at 5 cm H²O significantly higher than that at 10, 15, or 20 cm H²O. After injury, H^min  increased significantly only at 20 cm H²O, whereas H^max  increased significantly at 5, 10, and 15 cm H²O.

Figure 8shows the impact of P̄^ao  and injury on the derived effective tissue elastance (μ^H ) and heterogeneity index (σ^H ). At baseline, ANOVA demonstrated no significant dependence of either μ^H  or σ^H  on P̄^ao . After ALI, μ^H  significantly increased at all P̄^ao  levels, whereas σ^H  increased only at 5 and 10 cm H²O. Both μ^H  and σ^H  demonstrated dependence on P̄^ao  after ALI, with postinjury values at 5 cm H²O significantly increased compared with 10, 15, and 20 cm H²O. The σ^H  was significantly increased compared with baseline at 5 and 10 cm H²O.

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.3However, 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 ventilation30,31and more recently during high-frequency ventilation.32,33However, 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.34–36More recently, Ranieri et al.  37characterized 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,38but 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,39,40although 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.

In 1956, Otis et al.  25were the first to demonstrate how parallel time constant heterogeneity in the lung can cause frequency dependence in R^rs  and E^rs . 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.6,8,20,41,42Using a computer-generated (but anatomically accurate) airway tree, Lutchen et al.  6,8,12,14demonstrated that two distinct pathophysiologic mechanisms can be uniquely identified from Z^rs : (1) parallel time constant heterogeneity, which results in enhanced frequency dependence of R^rs  and E^rs  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 R^rs  and E^rs  throughout the entire bandwidth of interest. Both of these mechanisms can substantially increase R^rs  and E^rs  near breathing frequencies, exclusive of large increases in either overall airway resistance or actual tissue elasticity.12 

Our Z^rs  data are surprisingly similar to those data reported in many of these earlier modeling studies. At baseline, both R^rs  and E^rs  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,11,29,34,43,44although some baseline mechanical heterogeneity may also contribute to frequency dependence.7,14,22,45Under control conditions, we observed significant increases in E^rs  and |Z̄^rs | with increasing P̄^ao , which may result from overstretching and plastoelastic phenomena.46After lung injury, we observed significant increases in |Z̄^rs | and both the levels and frequency dependence of R^rs  and E^rs  at all values of P̄^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, E^rs  and |Z̄^rs | both decreased with increasing P̄^ao  after ALI, consistent with the notion that moderate increases in P̄^ao  are beneficial by reducing heterogeneity and recruiting lung units.

To further quantify mechanical heterogeneity from our Z^rs  data, we relied on the distributed modeling approach initially developed by Suki and coworkers19,21to 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 R^rs  and E^rs  not only by the mechanism of parallel heterogeneity, but also by viscoelasticity of the respiratory tissues.20They used only a hyperbolic distribution of tissue elastances to characterize Z^rs , 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 AIC^Cpredicts that the linear distribution of tissue elastances was most appropriate for describing Z^rs  in most dogs regardless of P̄^ao . After lung injury, there was an increase in the number of dogs for which Z^rs  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 AIC^Cscore was most likely to have generated the data, it is important to understand that the AIC^Cdoes 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.28For 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 P̄^ao .20,44,47Although 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.24This may mask any detectable relation between R  and P̄^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 μ^H  with increasing P̄^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.30,48Therefore, it is unlikely that changes in μ^H  reflect alterations in the actual elastic properties of the parenchymal tissues or chest wall. Baseline values of our heterogeneity index σ^H  demonstrated a decreasing trend with P̄^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 P̄^ao  may result in overdistention occurring in a somewhat heterogeneous manner. After lung injury, σ^H  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 σ^H  on P̄^ao  after ALI, with its highest value at 5 cm H²O. Further increases in P̄^ao  seem to reduce σ^H , also consistent with lung recruitment.

We observed significant increases in our hysteresivity parameter η after ALI. Hysteresivity has been proposed by Fredberg and Stamenovic29as 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.29However, these assumptions have been called into question in recent studies,47,49especially in animal models of ALI.50,51A key assumption of our modeling analysis is that η is constant throughout the respiratory tissues regardless of disease condition.20,29Our observed increases in η after ALI may reflect actual changes in the coupling of energy dissipation and storage in the respiratory tissues.29Hysteresivity 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 

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 Z^rs  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.39Future studies may incorporate similar modeling approaches with imaging data to obtain more accurate tissue distribution functions for describing Z^rs . Therefore, although the current accuracy of our heterogeneity estimates may be somewhat biased because of modeling error, σ^H  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 R^rs  and E^rs ,6,12,19,21,41although the degree to which it does so in ALI is not clear. A previous study by Barnas et al.  44using 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 R^rs  and E^rs  after ALI.

In summary, these data demonstrate that Z^rs  can provide specific information about the mechanical heterogeneity of the lungs and the impact of P̄^ao . After lung injury, both the level and frequency dependence of R^rs  and E^rs  increase compared with baseline, indicating the presence of widespread mechanical heterogeneity. These differences seem to be reduced by increases in P̄^ao , consistent with the recruitment of lung units and minimization of the impact of heterogeneity on dynamic respiratory mechanics. Model analysis of Z^rs  demonstrate that both effective tissue elastance and mechanical heterogeneity increase after ALI and decrease with increasing P̄^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.

The model-predicted impedance (ẑ^rs ) 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 ẑ^rs  as

where P^ΔH  represents a histogram function of bin width ΔH , H^m  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 ẑ^rs  that depends on a continuous probability density function P  (H ) integrated over the upper and lower bounds H^min  and H^max :

Regardless of the form of P  (H ), we can define an effective mean tissue elastance, μ^H , 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 σ^H  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 H^min  and H^max  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