Regional gas transport in the heterogeneous lung during oscillatory ventilation

Regional ventilation in the injured lung is heterogeneous and frequency dependent, making it difficult to predict how an oscillatory flow waveform at a specified frequency will be distributed throughout the periphery. In this study, we developed a computational model of distributed gas flow and CO₂ elimination during oscillatory ventilation. Our simulations indicate that ventilation distribution in a heterogeneous lung is spatially clustered and frequency-dependent, indicating that oscillatory frequency is an important factor in regional gas exchange.

Abstract

Regional ventilation in the injured lung is heterogeneous and frequency dependent, making it difficult to predict how an oscillatory flow waveform at a specified frequency will be distributed throughout the periphery. To predict the impact of mechanical heterogeneity on regional ventilation distribution and gas transport, we developed a computational model of distributed gas flow and CO₂ elimination during oscillatory ventilation from 0.1 to 30 Hz. The model consists of a three-dimensional airway network of a canine lung, with heterogeneous parenchymal tissues to mimic effects of gravity and injury. Model CO₂ elimination during single frequency oscillation was validated against previously published experimental data (Venegas JG, Hales CA, Strieder DJ, J Appl Physiol 60: 1025–1030, 1986). Simulations of gas transport demonstrated a critical transition in flow distribution at the resonant frequency, where the reactive components of mechanical impedance due to airway inertia and parenchymal elastance were equal. For frequencies above resonance, the distribution of ventilation became spatially clustered and frequency dependent. These results highlight the importance of oscillatory frequency in managing the regional distribution of ventilation and gas exchange in the heterogeneous lung.

NEW & NOTEWORTHY

Regional ventilation in the injured lung is heterogeneous and frequency dependent, making it difficult to predict how an oscillatory flow waveform at a specified frequency will be distributed throughout the periphery. In this study, we developed a computational model of distributed gas flow and CO₂ elimination during oscillatory ventilation. Our simulations indicate that ventilation distribution in a heterogeneous lung is spatially clustered and frequency-dependent, indicating that oscillatory frequency is an important factor in regional gas exchange.

for patients with the acute respiratory distress syndrome (ARDS), flow is distributed throughout the lung periphery in a nonuniform and frequency-dependent manner due to mechanical heterogeneity (3, 10). Patients with ARDS are particularly susceptible to the harmful processes of cyclic overdistension and derecruitment, collectively termed ventilator-induced lung injury (VILI). The ensuing mechanical stresses associated with these processes result in the release of inflammatory mediators that may contribute significantly to multiple organ failure and death (37). Intratidal overdistension and derecruitment often occur simultaneously in different regions of injured lung due to nonuniform distribution of flow (8). Thus optimizing the magnitude and distribution of flow to mitigate VILI is a critical objective of mechanical ventilation in ARDS.

Lung-protective ventilation significantly reduces mortality in patients with ARDS, using high positive end-expiratory pressures to prevent derecruitment and small tidal volumes to avoid overdistension (1). High-frequency oscillatory ventilation (HFOV) is an alternative modality that applies ventilatory rates >2 Hz and is used as a rescue therapy for ARDS patients whose condition is refractory to conventional or lung-protective ventilation (31). Despite the use of smaller tidal volumes compared with lung-protective ventilation, multicenter clinical trials of HFOV in adult ARDS patients failed to demonstrate improvements in mortality (12, 56). This may be due to heterogeneous and frequency-dependent distributions of flow in the periphery, which may contribute to regional lung injury that is not easily observed in patients.

There are several mechanisms by which gas transport occurs during oscillatory flow: direct alveolar ventilation, asymmetric velocity profiles, inspiratory-expiratory differences in velocity profiles, turbulent and oscillatory dispersion, pendelluft phenomenon, cardiogenic mixing, collateral ventilation, and molecular diffusion (9, 31). The relative magnitude of each mechanism’s effect within any particular region of the lung may depend on the combination of velocity profile, oscillatory frequency, and airway geometry (9, 30). Computational modeling of gas transport in the lung allows for useful predictions of the impact of frequency and tidal volume on regional gas transport in critically ill patients. Such predictions may provide insight into derangements of gas exchange in mechanically heterogeneous lungs. Computational models exist for predicting effects of specific gas transport mechanisms on total gas exchange during either conventional mechanical ventilation (32) or HFOV (14). Few models are applicable to gas transport in a branching airway structure over a wide range of frequencies (25, 33), yet these are limited by assumptions of symmetrical airway networks and uniform distributions of flow. Current computational models are limited in their ability to simulate heterogeneous gas transport (e.g., CO₂ elimination) spanning a range of frequencies and tidal volumes encompassing conventional mechanical ventilation and HFOV.

The goal of this study was to develop a computational model of regional flow and gas exchange in a three-dimensional airway network during oscillatory ventilation. Specifically, our objectives were: 1) to generate frequency-dependent distributions of advective flow throughout an anatomically structured airway network, imposing either homogeneous or heterogeneous distributions of tissue elastance to reflect healthy and injured parenchymal mechanics; 2) to simulate gas exchange for any given tidal volume, oscillatory frequency, and flow distribution, based on models of diffusive and advective gas transport phenomena; and finally 3) to assess the frequency-dependence of tidal volumes required for eucapnic ventilation, as well as ventilation heterogeneity for healthy and injured lungs.

Glossary

a

Exponent of frequency, in proportional relationship to total CO₂ elimination

ARDS

Acute respiratory distress syndrome

b

Exponent of tidal volume, in proportional relationship to total CO₂ elimination

CO₂

Carbon dioxide

CT

Computed tomography

C_nCO₂

Concentration of CO₂ for nth acinus

d

Diameter of airway segment

D_eff

Effective diffusivity

D_mol

Molecular diffusivity

$D_{\text{dis}}^{\text{lam}}$

Laminar oscillatory dispersion coefficient

$D_{\text{dis}}^{\text{turb}}$

Turbulent oscillatory dispersion coefficient

HFOV

High frequency oscillatory ventilation

f

Oscillation frequency in Hz

f_res

Resonant frequency in Hz

f_min

Frequency in Hz of local minimum normalized acinar flow magnitude

f_max

Frequency in Hz of local maximum normalized acinar flow magnitude

F

Normalized frequency

l

Length of airway segment

Ṁ_nCO₂

Molar flux of CO₂ for nth acinus

${\dot{M}}_{\text{seg}}^{\text{diff}}{\text{CO}}_2$

Diffusive component of molar flux of CO₂ through an airway segment

ṀCO₂

Vector of acinar CO₂ molar fluxes

n

Index through individual acini

N

Total number of acini

P_aoCO₂

Partial pressure of CO₂ at airway opening

P_āCO₂

Partial pressure of CO₂ in mixed arterial blood

P_nCO₂

Partial pressure of CO₂ for nth acinus

ΔPCO₂

Vector of partial pressure differentials between acini and airway opening

Q

Normalized flow cost

r²

Coefficient of determination

R

Universal gas constant

Re

Reynolds number

Re_crit

Critical Reynolds number, transition between laminar and turbulent flow

$R_{\text{T}}^{\text{diff}}$

Diffusive transport resistance

$R_{\text{T}}^{\text{mix}}$

Diffusive transport resistance

$R_{\text{T}}^{\text{tot}}$

Total transport resistance

R_T

Transport resistance matrix

T

Temperature

Ū_RMS

Root-mean-square mean-axial velocity

V_D

Total dead space volume

|V_D,n|

Magnitude of oscillatory dead space volume delivered to nth acinus

|V_F,n|

Magnitude of oscillatory fresh gas volume delivered to nth acinus

V_mix

Oscillatory mixing volume

V_seg

Volume of airway segment

V_T

Tidal volume

$V_{\text{T}}^{\text{euc}}$

Eucapnic tidal ε volume

|V̇_seg|

Magnitude of oscillatory flow through airway segment

V̇_A

Eucapnic total acinar ventilation

${\dot{V}}_{\text{ao}}^{\text{DDM}}{\text{CO}}_2$

CO₂ elimination at the airway opening via diffusion, dispersion, and mixing

V̇_nCO₂

CO₂ elimination at acinus n

${\dot{V}}_n^{\text{DDM}}{\text{CO}}_2$

CO₂ elimination at acinus n via diffusion, dispersion, and mixing

${\dot{V}}_n^{\text{DAV}}{\text{CO}}_2$

CO₂ elimination at acinus n via direct acinar ventilation

V̇_totCO₂

Total CO₂ elimination at the airway opening

${\dot{V}}_{\text{tot}}^{\text{euc}}{\text{CO}}_2$

Eucapnic total CO₂ elimination at the airway opening

VILI

Ventilator-induced lung injury

β₀, β₁, β₂

Power-law regression coefficients

π

Radians in semicircle

ε

Constant of proportionality for mixing transport resistance

v

Kinematic viscosity

METHODS

Model structure.

The model structure is based on a three-dimensional network of a canine airway tree shown in Fig. 1, in which each airway segment was assumed to be an ideal cylinder with dimensions and spatial position based on a thoracic CT scan of a supine dog inflated to total lung capacity (22). Airways <2 mm diameter were algorithmically generated within the constrained boundaries of segmented lung lobes (38). The length (l) and diameter (d) of each segment were then scaled down from total lung capacity to a mean airway opening pressure of 10 cmH₂O, where the size scaling for each airway was determined according to its time-averaged local transmural pressure, defined as the difference between mean lumenal pressure and local pleural pressure (3, 41). Local pleural pressure in the model was assumed to vary with gravity by 0.25 cmH₂O/cm according to the hydrostatic weight of pleural fluid in a supine dog (26), with an average value of –5 cmH₂O. Airway dimensions defined longitudinal and shunt impedances to oscillatory flow, which are briefly summarized here from Colletti et al. (10). Airway segment longitudinal impedance was determined assuming Womersley-type oscillatory flow in a cylinder (42, 54). Airway segment shunt impedance was given by a parallel combination of isothermal gas compression compliance, soft tissue viscoelasticity, and cartilaginous viscoelasticity (10, 24). Each terminal airway in the tree was subtended by a viscoelastic “constant-phase” element representing acinar wall distensibility (17) in parallel with gas compression compliance (10, 24). The arrangements of all longitudinal and shunt impedances in the model were defined according to Kaczka et al. (24). Flow oscillations at the airway opening were distributed throughout the lung periphery using a recursive flow-dividing scheme, apportioning flows at each topological branching point among parallel pathways according to their respective total input impedances (10).

Fig. 1.

Central airways of a canine lung as segmented from an X-ray computed tomographic scan in a supine dog (left) and resulting anatomically based 3-dimensional airway network (right) generated using a space-filling algorithm (38, 39).

Acinar viscoelasticity was determined according to a second-order polynomial function of local transpulmonary pressure (3). Heterogeneous injury was simulated by modulating acinar tissue elastance according to a gravitationally weighted gradient noise distribution using a Perlin method (11), which produces a textured distribution that is smoother and more natural compared with purely random variations in tissue properties (10). Gravitational weighting increased the magnitude of the noise distribution toward the dependent regions of the model, loosely mimicking the effects of fluid accumulation in injured lungs. The resulting distribution of injured acinar tissue elastance demonstrated a mean ± standard deviation of 1.93 ± 1.19 cmH₂O/μl, compared with the healthy case of 0.31 ± 0.02 cmH₂O/μl. Cross-sectional views of acinar tissue elastance are provided in Fig. 2.

Fig. 2.

Transverse cross sections of simulated regional acinar tissue stiffness for healthy (upper) and injured (lower) conditions, spatially interpolated between acinar elements for visualization.

To simplify computation of mechanical impedance and flow distribution in the frequency-domain we did not consider intratidal variations in model structure or mechanical properties (2). Additionally, CO₂ transport throughout the airway network was solved as a steady-state problem, i.e., time-averaged throughout the oscillatory ventilation cycle and with CO₂ elimination equal to CO₂ production.

Diffusive transport.

To approximate CO₂ diffusion in the tree, we assumed that molar flux of CO₂ in the axial direction of each airway segment due only to diffusive phenomena (${\dot{M}}_{\text{seg}}^{\text{diff}}{\text{CO}}_2$) was proportional to its axial partial pressure difference (ΔP_segCO₂):

$${\dot{\text{M}}}_{\text{seg}}^{\text{diff}}{\text{CO}}_2=\frac{1}{R_{\text{T}}^{\text{diff}}}\Delta P_{\text{seg}}{\text{CO}}_2$$ (1)

where $R_{\text{T}}^{\text{diff}}$ is the diffusive transport resistance:

$$R_{\text{T}}^{\text{diff}}=\frac{4\mathcal{R}Tl}{\pi d^2{D}_{\text{eff}}}$$ (2)

The R parameter is the universal gas constant, T is body temperature, and D_eff the effective diffusivity constant for CO₂ contained in the airway segment. Under zero-flow conditions, D_eff is equal to the molecular diffusivity D_mol. However during advective flow, the effective diffusivity D_eff includes dispersive components to account for enhanced diffusion resulting from transient velocity profile changes during laminar oscillatory flow (50) or the mixing of eddies during turbulent oscillatory flow (14, 40):

$$D_{\text{eff}}=\{\begin{array}{cc}{D}_{\text{mol}}+D_{\text{dis}}^{\text{lam}},& \text{Re}<{\text{Re}}_{\text{crit}}\\ D_{\text{mol}}+D_{\text{dis}}^{\text{turb}},& \text{Re}\ge {\text{Re}}_{\text{crit}}\end{array}$$ (3)

where $D_{\text{dis}}^{\text{lam}}$ and $D_{\text{dis}}^{\text{turb}}$ are the laminar and turbulent oscillatory dispersion coefficients, respectively, and Re is the Reynolds number given by the airway segment diameter d, kinematic gas viscosity v, and root-mean-square mean-axial velocity Ū_RMS:

$$\text{Re}=\frac{{\overline{U}}_{\text{RMS}}d}{\nu }$$ (4)

The critical Reynolds number (Re_crit = 30) characterizes the transition from laminar to turbulent oscillatory flow (35). While $D_{\text{dis}}^{\text{lam}}$ is dependent on oscillatory frequency (35, 50), $D_{\text{dis}}^{\text{turb}}$ depends only on the root-mean-square mean-axial velocity that characterizes eddy mixing strength (14). To account for additional advective gas mixing at airway bifurcations, we developed a mixing transport resistance (${\text{R}}_{\text{T}}^{\text{mix}}$) for each segment:

$${\text{R}}_{\text{T}}^{\text{mix}}=\frac{\mathcal{R}T}{f{V}_{\text{mix}}}$$ (5)

where f is the oscillatory frequency of the advective flow waveform in Hz, and V_mix is the amplitude of volume flux through the airway segment. Thus, $R_{\text{T}}^{\text{mix}}$ characterizes net gas transport from mixing the volume of gas passing completely through an airway segment during each period of oscillatory flow. Therefore $R_{\text{T}}^{\text{mix}}$ will be nonzero only when this oscillating volume exceeds the volume of the airway segment, V_seg, given by:

$$V_{\text{mix}}>V_{\text{seg}}=\frac{1}{4}\pi d^{2}l$$ (6)

where V_mix is:

$$V_{\text{mix}}=\frac{|{\dot{V}}_{\text{seg}}\left(f\right)|}{\pi f}-V_{\text{seg}}$$ (7)

and |V̇_seg(f)| is the magnitude of oscillatory flow in the airway segment at frequency f. The mixing transport resistance is derived in more detail in appendix A-1.

The total transport resistance for each airway ($R_{\text{T}}^{\text{tot}}$) is the parallel combination of diffusive and mixing transport resistances:

$$R_{\text{T}}^{\text{tot}}=\frac{1}{\frac{1}{R_{\text{T}}^{\text{diff}}}+\frac{1}{R_{\text{T}}^{\text{mix}}}}=\frac{R_{\text{T}}^{\text{diff}}{R}_{\text{T}}^{\text{mix}}}{R_{\text{T}}^{\text{diff}}+R_{\text{T}}^{\text{mix}}}$$ (8)

An upper bound on $R_T^{\text{tot}}$ is obtained in the zero-flow limiting case when V_mix → 0 and $R_{\text{T}}^{\text{mix}}\hspace{0.17em}\to \hspace{0.17em}\infty$. In this case $R_{\text{T}}^{\text{tot}}=R_{\text{T}}^{\text{diff}}$ and D_eff = D_mol.

Once $R_{\text{T}}^{\text{tot}}$ is computed for each airway in the network, a mesh analysis (appendix A-2) forms a system of equations for solving the molar flux of CO₂ at each acinus, given the partial pressure differential between each acinus and the airway opening:

$$R_{\text{T}}\dot{M}C{O}_2=\Delta P{\text{CO}}_2$$ (9)

where R_T is a full matrix of transport resistances (appendix A-2), ṀCO₂ is a vector of acinar CO₂ molar fluxes (i.e., for the nth acinus: Ṁ_nCO₂), and ΔPCO₂ is a vector of CO₂ partial pressure differentials between acini and airway opening (i.e., for the nth acinus: ΔP_nCO₂ = P_nCO₂ − P_aoCO₂). Each row of the system corresponds to the mesh formed by the airways connecting a single acinus to the airway opening. The partial pressure of CO₂ at the airway opening was assumed to be zero. After solving the system of equations, the CO₂ elimination due to diffusion, dispersion, and mixing (${\dot{V}}_n^{\text{DDM}}{\text{CO}}_2$) for each acinus can be converted from the molar flux using the ideal gas law:

$${\dot{V}}_n^{\text{DDM}}{\text{CO}}_2=\frac{\mathcal{R}\text{T}}{P_{\text{atm}}}{\dot{M}}_n{\text{CO}}_2$$ (10)

The mesh analysis ensures conservation of mass between CO₂ elimination at the airway opening (${\dot{V}}_{\text{ao}}^{\text{DDM}}{\text{CO}}_2)$ and the sum of ${\dot{V}}_n^{\text{DDM}}{\text{CO}}_2$ across all acini.

Advective transport.

The distribution of advective flow throughout the airway tree was computed using a recursive flow divider algorithm (10). Direct acinar ventilation occurs when volume amplitudes at the airway opening are sufficiently large to allow fresh gas mixing directly with acinar gas during inspiration, effectively bypassing $R_{\text{T}}^{\text{diff}}$ and $R_{\text{T}}^{\text{mix}}$ in Eq. 8. To model this mechanism of gas transport, we assume that the volume of fresh gas delivered to each acinus is completely mixed during inspiration and that the same volume of mixed gas is completely removed during expiration, according to Fig. 3. CO₂ elimination due to direct acinar ventilation is computed according to the concept of “personal dead space,” which apportions an amount of inspired fresh gas to each individual acinus (13).

Fig. 3.

Direct acinar ventilation is modeled by assuming complete mixing within each nth terminal acinus of inspired fresh gas (V_F,n), inspired dead space gas (V_D,n), and residual volume (V_A,n). The concept of “personal dead space” is used to distribute the total dead space (V_D) throughout all acini in proportion to the magnitude of the ventilation distribution (13).

In this manner, the dead space of all conducting airways was successively distributed in proportion to the flow distribution through subtending airways until reaching the terminal acini. Thus the distribution of personal dead space is dependent on frequency, in accordance with the distribution of flow. We determined the CO₂ elimination due to direct acinar ventilation (${\dot{V}}_n^{\text{DAV}}{\text{CO}}_2$) for a particular acinar compartment by comparing its personal dead space volume [V_D,n(f)] to the total volume delivered to that acinus over one period of oscillation:

$${\dot{V}}_n^{\text{DAV}}{\text{CO}}_2=\frac{P_n{\text{CO}}_2}{863}f\left|V_{\text{F},n}\right|$$ (11)

where 863 is a conversion factor between partial pressure in mmHg and gas volume fraction,¹ P_nCO₂ is the partial pressure of acinar CO₂, and |V_F,n| is the magnitude of frequency-dependent fresh gas volume delivered to the acinus given by:

$$\left|V_{\text{F},n}\right|=\{\begin{array}{cc}\frac{\left|{\dot{V}}_n\left(f\right)\right|}{\pi f}-\left|V_{\text{D},n}\left(f\right)\right|,& \frac{\left|{\dot{V}}_n\left(f\right)\right|}{\pi f}>\left|V_{\text{D},n}\left(f\right)\right|\\ 0,& \text{otherwise}\end{array}$$ (12)

|V̇_n(f)| is the magnitude of oscillatory flow at frequency f in the terminal acinus. Note that if the total volume of inspired gas delivered to an acinus over one period of oscillation is less than the personal dead space of that acinus, there will be no contribution of direct acinar ventilation to CO₂ elimination from that particular acinus. Total dead space volume (V_D) was calculated as the sum of all cylindrical airway segment volumes.

Total transport and eucapnia.

Total CO₂ elimination (V̇_totCO₂) for the model at a specified frequency was computed from the sum of CO₂ elimination across all N acini:

$${\dot{V}}_{\text{tot}}{\text{CO}}_2=\sum _{n=1}^N{\dot{V}}_n{\text{CO}}_2$$ (13)

where n indexes individual acini, and the CO₂ elimination in each acinus (V̇_nCO₂) is determined by the sum of CO₂ elimination via direct acinar ventilation and by solving the system of transport resistances:

$${\dot{V}}_n{\text{CO}}_2={\dot{V}}_n^{\text{DAV}}{\text{CO}}_2+{\dot{V}}_n^{\text{DDM}}{\text{CO}}_2=\frac{P_n{\text{CO}}_2}{863}f\left|V_{\text{F},n}\right|+\frac{\mathcal{R}T}{P_{\text{atm}}}{\dot{M}}_n{\text{CO}}_2$$ (14)

Equation 14 assumes that each acinar CO₂ partial pressure is equal to its corresponding end-capillary CO₂ partial pressure. Mixed-venous blood was assumed to enter each capillary with 46 mmHg CO₂ partial pressure. However end-capillary CO₂ partial pressure for each acinus is determined by the rates of acinar CO₂ elimination and perfusion. Thus the distribution of P_nCO₂ is dependent on the distribution of acinar CO₂ elimination, and Eq. 14 for total CO₂ elimination must be solved iteratively. An optimization routine minimized the error in conservation of mass for each acinar compartment, such that CO₂ elimination via ventilation (i.e., Eq. 14) was equal to the product of perfusion rate and CO₂ content difference between end-capillary and mixed-venous blood (53). Eucapnic ventilation was assumed to occur when total CO₂ elimination for the model approximately equaled the predicted metabolic CO₂ production for a 25 kg dog (1.93·10⁻³·Ls⁻¹) (18).

Simulations.

The recursive algorithms for computing and storing impedances and determining advective flow division throughout the tree were written in C++. Computation time for generating advective flow distribution at each distinct f was ~3 s on a 64-bit computer with an Intel Core i7-950 processor operating at 3.07 GHz with 12 GB RAM. The gas transport model was written and executed in MATLAB (version 7.13, The Mathworks, Natick, MA). Total computation time for each distinct f and V_T simulation was ~300 s.

The mechanical impedance and ventilation distribution for the model were computed at 115 distinct frequencies, spanning 0.1–100 Hz. The resonant frequency (f_res), at which the reactive component of total lung impedance is zero, was determined by interpolation. Gas transport distributions were computed for a range of volume amplitudes at each frequency between 0.1 and 30 Hz, with emphasis on smaller volume amplitudes at higher frequencies. Volume amplitudes were selected between 5 and 600 ml, chosen to ensure that a simulation representative of eucapnic ventilation was computed at each frequency. The tidal volume required for eucapnic ventilation ($V_{\text{T}}^{\text{euc}}$) as a function of frequency was regressed to a power-law (47):

$$V_{\text{T}}^{\text{euc}}\left(f\right)={\beta }_0+{\beta }_1{f}^{{\beta }_2}$$ (15)

The parameters β₀, β₁, and β₂ were estimated using a nonlinear least-squares technique (MATLAB v7.13).

To quantify the heterogeneity of ventilation distribution, we examined the amplitude and phase histograms of acinar flows as functions of oscillatory frequency for both health and injured conditions, along with the histograms for acinar CO₂ elimination. Acinar flow magnitudes were normalized relative to the theoretical values obtained for a perfectly symmetric, homogeneous lung with rigid airway walls and no gas compression (i.e., perfectly uniform ventilation distribution). Acinar flow phases were normalized relative to the tracheal flow phase, constrained to be within ± 180°. Bin sizing for normalized acinar flow magnitudes, phases, and V̇_nCO₂ were 167 per decade, 1.39 per degree, and 167 per decade, respectively.

RESULTS

Validation.

Figure 4 shows simulation results for V̇_totCO₂ as a function of V_T and f during oscillatory ventilation with a single frequency. Simulations are shown for each frequency with increasing V_T increments of 5 ml, until V̇_totCO₂ exceeded the predicted value for eucapnia. The required CO₂ elimination to maintain eucapnia $({\dot{V}}_{\text{tot}}^{\text{euc}}{\text{CO}}_2$) is also indicated, as well as V̇_totCO₂ produced by molecular diffusion only (i.e., zero-flow conditions) which establishes the lower bound of CO₂ elimination regardless of V_T or f. There are noticeably different behaviors for V̇_totCO₂ with respect to V_T and f. Specifically, there are certain ranges of volume amplitudes for which certain gas transport mechanisms dominate the others: starting from the lower bound molecular diffusion, there are dramatic increases in V̇_totCO₂ with the advent of oscillatory dispersion, followed by turbulent dispersion, advective mixing at bifurcations, and finally direct acinar ventilation. These ranges were identified by iteratively eliminating specific gas transport mechanisms from the model, and then resimulating V̇_totCO₂.

Fig. 4.

Simulation results for V̇_totCO₂ as a function of V_T and f. Solid black horizontal line represent the lower bound of total CO₂ elimination due solely to molecular diffusion (i.e., zero-flow conditions), while the dashed horizontal line represents total CO₂ elimination to achieve eucapnia for a 25 kg dog. Text labels indicate dominant gas transport mechanism during 0.1 Hz oscillation.

Figure 5 shows $V_{\text{T}}^{\text{euc}}$ as a function of frequency over localized frequency domains 0.1 Hz < f < 1 Hz and 2 Hz < f < 30 Hz. Regression curves are extended beyond the fit domains to demonstrate the relative reductions in required volume amplitudes that occur inside each range compared with the extrapolated fit of the other range. Parameter estimates are provided in Tables 1 and 2. Expected values for low-frequency gas exchange (i.e., 0.1 Hz < f < 1 Hz) are derived from standard equations of gas exchange during conventional mechanical ventilation and spontaneous breathing (appendix A-3). Expected values for high frequency gas exchange (i.e., 2 Hz < f < 30 Hz) are obtained from previous experimental studies.

Fig. 5.

Simulation results representing model predictions for eucapnic ventilation at various single frequencies of oscillation (dashed black lines). Dotted vertical black lines represent the resonant frequency (f_res: elastic and inertial components of impedance equal and opposite). Model predictions of tidal volume required for eucapnic ventilation derived from power-law regressions over low frequencies (purple: 0.1 ≤ f ≤ 1 Hz) and high frequencies (blue: 2 ≤ f ≤ 30 Hz).

Table 1.

Power-law regression parameters for Eq. 15 during low frequency oscillatory ventilation

Parameter	Expected Value	Regression Estimate	Deviation from Expected Value, %
β0	VD = 0.169	0.151	10.7
β1	$\frac{863}{P_{\overline{a}}{\text{CO}}_{\text{2}}}{\dot{V}}_{\text{tot}}^{\text{euc}}{\text{CO}}_2=0.042$	0.043	2.2
β2	−1	−0.99	1.1

Table 2.

Ratio of power-law exponents for CO₂ elimination as a function of f ^a and $V_{\text{T}}^b$ during high frequency oscillatory ventilation, as reported in various studies

Study	Species	b/a
Slutsky et al. 1981 (36)*	dog	1.7
Berdine et al. 1984 (7)†	dog	1.9
Jaeger 1991 (19)	dog	1.9
Jaeger et al. 1984 (20)	dog	2.0
Venegas et al. 1985 (45)	dog	2.1
Present study	dog	2.1
Venegas et al. 1986 (47)	various	2.2
Watson, Jackson 1985 (52)	monkey	1.9
Watson, Jackson 1984 (51)	rabbit	2.7

Only the ratio of b to a is presented because the power-law regression in Eq. 15 determines only the relative relationship between frequency and volume amplitude.

^*Values for a and b reported by Venegas et al. (47).

^†Values for a and b reported by Chang (9).

Figure 6 shows a comparison between these interpolated eucapnic simulation results and experimental data from Venegas et al. (47), converted to dimensionless equations of gas exchange:

$$\mathcal{Q}=\frac{f{V}_{\text{T}}^{\text{euc}}\left(f\right)}{{\dot{V}}_{\text{A}}}$$ (16)

$$\mathcal{F}=\frac{f{V}_{\text{D}}}{{\dot{V}}_{\text{A}}}$$ (17)

where Q and F are the normalized flow cost and normalized frequency, respectively, and V̇_A is the total acinar ventilation required for eucapnia:

$${\dot{V}}_{\text{A}}=\left(863\frac{{\dot{V}}_{\text{tot}}^{\text{euc}}{\text{CO}}_2}{P_{\overline{a}}{\text{CO}}_2}\right)$$ (18)

where P_āCO₂ is the eucapnic partial pressure of CO₂ in mixed arterial blood. The low-frequency relationship Q = 1 + F corresponds to theoretical gas exchange primarily via direct acinar ventilation, whereas the high-frequency relationship Q = 0.40F^0.54 is a regression to experimentally acquired data (47). The model accurately predicts the transition from low-frequency to high-frequency gas exchange behavior, and simulation results demonstrate close qualitative and quantitative agreement with the experimental data of Venegas et al. (47), deviating at the highest frequencies (F > 60, or f > 12.8 Hz).

Fig. 6.

Comparison between simulation results and experimental data of Venegas et al. (47). The axes are dimensionless variables of frequency (F) and flow (Q). The black lines correspond to theoretical predictions of low-frequency gas exchange (dashed) and regression to experimental HFOV data (solid). The gray line represents the simulations results interpolated at eucapnic conditions.

Impact of heterogeneity.

Figure 7 shows five transverse model cross sections that are color-coded to depict the distribution of normalized acinar flows, along with acinar CO₂ elimination rates, for frequencies between 0.2 and 30 Hz. Acinar flow magnitudes were normalized relative to their corresponding values for a perfectly symmetric, homogeneous lung with rigid airway walls and no gas compression. The distribution of acinar CO₂ elimination was normalized to the total CO₂ elimination divided by the number of acini. The coefficients of determination between acinar flow magnitude and CO₂ elimination (r²) at each frequency and condition are also shown. Acinar CO₂ elimination was positively correlated with acinar flow magnitude for either condition, although the correlation tended to decrease with increasing frequency. Gravitational gradients were apparent at low frequencies for the healthy condition, yet diminish with increasing frequency. For the injured condition, the distributions of both flow and CO₂ elimination were spatially concordant with the distribution of acinar tissue stiffness (Fig. 2) for the frequencies shown.

Fig. 7.

Transverse cross sections of simulated regional acinar flow magnitude (left) and CO₂ elimination (right) at selected oscillatory frequencies, for healthy (top) and injured (bottom) conditions, spatially interpolated between acinar elements for visualization. Coefficients of determination (r²) between acinar flow magnitude and CO₂ elimination are shown at the far right.

Figure 8 shows two-dimensional histograms of normalized acinar flow magnitude and phase distributions for healthy and injured conditions, along with the acinar CO₂ elimination distribution from 0.1 to 100 Hz. Acinar flow phase was normalized relative to the corresponding tracheal flow phase, constrained to be within ± 180°. The histograms are color-coded based on the percentage of acini in the model. For frequencies less than resonance, ventilation distribution is largely frequency independent. However, when the oscillatory frequency exceeds resonance, ventilation distribution becomes both frequency dependent and heterogeneous, even for the mechanically homogeneous healthy case. The low-frequency ventilation distribution is much more homogeneous for the healthy case compared with the injured case, as expected. However the effect of high frequency on ventilation heterogeneity is comparable between the two conditions, in both cases increasing heterogeneity with increasing f > f_res. Above f_res the mean normalized acinar flow magnitude exceeded unity, indicating that the sum of acinar flow magnitudes was greater than the tracheal flow magnitude. The increase in mean acinar flow magnitudes above f_res is coincident with increased acinar phase variance. Multimodal patterns were also visible in histograms, reflecting large groups of acini with similar frequency-dependent behavior.

Fig. 8.

Simulation representing model predictions for eucapnic ventilation vs. oscillatory frequency. Horizontal axes are aligned, dashed black lines represent the resonant frequency (f_res: elastic and inertial components of impedance equal and opposite). Acinar flow magnitude, flow phase, and CO₂ elimination are normalized as described in the text.

To determine whether such multimodality in ventilation distribution was consistent with anatomic grouping, we color-coded individual acini according to the frequency at which they experienced either a local minimum (f_min) or maximum (f_max) normalized flow magnitude. Figure 9 shows the spatial anatomic distributions of f_min and f_max for the three-dimensional model. Spatially organized clusters of acini with the same f_min or f_max values are immediately apparent, implying that overventilation and underventilation may manifest simultaneously in clusters of the lung, in a frequency-dependent manner. The particular pattern of clustering was similar between the healthy and injured conditions, despite substantial differences in the distribution of tissue elastance (Fig. 2).

Fig. 9.

A: frequencies of maximum and minimum normalized flow magnitude, shown for clarity in the context of a single acinus. B: frequencies of maximum (top) and minimum (bottom) normalized flow magnitude, represented spatially for every acinus, using consistent coloring according to the scale provided in A. Acinar flow magnitude is normalized to the uniform distribution (unity corresponds to tracheal magnitude divided by total number of acini). Acini that do not exhibit a local minimum or maximum within the range of frequencies simulated are shown in black. The gravitational vector points out of the page, toward the reader.

DISCUSSION

In this study, we developed an anatomic computational model of gas exchange during oscillatory ventilation by incorporating mechanical impedance and pertinent gas transport mechanisms over a wide range of frequencies. The dominant mechanism of gas transport exhibits strong scaling dependence on V_T, with transitions from molecular diffusion near zero-flow conditions to laminar oscillatory dispersion, then turbulent dispersion, then advective mixing at bifurcations, and finally direct acinar ventilation at the largest V_T. The transition from laminar dispersion to turbulent dispersion is determined by a critical Reynolds number, which was set at Re_crit = 30 (35). Values of Re are proportional to the root-mean-square axial flow velocity, which implies dependence on both volume amplitude and oscillatory frequency. Therefore, the transition from laminar to turbulent dispersion will occur at different volume amplitudes for each frequency. Laminar oscillatory dispersion alone is not sufficient to provide eucapnic ventilation at a reasonably small V_T for any f in the simulated range between 0.1 and 30 Hz. By contrast, the transition to turbulence yields about an order-of-magnitude increase in V̇_totCO₂, such that for high frequencies (f > 10 Hz), adequate V̇_totCO₂ is achieved primarily via dispersion.

For intermediate frequencies (between 1 and 10 Hz), mixing at bifurcations contributes the most to eucapnic ventilation, nearly doubling V̇_totCO₂ compared with dispersion alone. The transition from dispersion-dominated to bifurcation-mixing-dominated gas transport occurs at a constant V_T between 50 and 60 ml across all frequencies. The trachea of our canine model has a lumenal volume of 56.3 ml, which is by far the largest volume of any airway in the tree: the next largest is 6.5 ml, followed by 2.2 ml. Thus we may safely assume that advective mixing at airway bifurcations in the model has the greatest effect on total gas transport for V_T larger than tracheal volume.

The final transition in dominant gas transport mechanisms is attributable to the increasing prevalence of direct acinar ventilation. Occurring only at the largest V_T, this mechanism requires a front of fresh gas to move directly between the airway opening and the acinar compartments, producing substantial gas transport via advection. The transition to direct acinar ventilation occurs over a wider range of V_T compared with the other mechanisms, due to differences in the ventilation distribution as well as the distribution of path lengths. The impact of direct acinar ventilation on gas transport is also much larger than that of any other mechanism for CO₂ elimination by almost an order of magnitude.

The last two transitions seem to have constant V_T thresholds, whereas the diffusion and dispersion transitions are f dependent as well as V_T dependent. This can be explained by the physical dependence of direct acinar ventilation and bifurcation mixing on the size and arrangement of dead space in the conducting airways, whereas dispersion depends more strongly on the magnitude, frequency, and cross-sectional profile of the flow waveform.

Pendelluft occurs when lung regions with differing mechanical time constants oscillate out of phase with respect to each other, resulting in increased intrapulmonary gas mixing and homogenization (16, 29). In this model of time-averaged gas transport, pendelluft manifests as increased acinar phase variance and flow magnitude, as shown in Fig. 8. Gas mixing is increased between lung regions oscillating out of phase, due to increased flow magnitude in each region. The simulations performed in this study indicate that pendelluft occurs primarily above the resonant frequency, consistent with experimental findings of phase variance using stroboscopic measurements of the lung surface (27) and alveolar pressure capsules (15).

Validation.

Simulations from this canine lung model, under healthy conditions with near homogeneous mechanical properties, exhibit agreement with experimental results regarding the behavior of V̇_totCO₂ with respect to f, V_T, and V_D (47). Low-f gas exchange is dominated by direct acinar ventilation. The personal dead space and alveolar mixing models demonstrate concordance with expected behavior of CO₂ elimination during conventional mechanical ventilation and spontaneous breathing. Theoretical predictions and empirical findings for power-law behavior of high-f CO₂ elimination are relatively more inconsistent by comparison. In general it is accepted that V̇_totCO₂ depends more strongly on V_T than on f according to:

$${\dot{V}}_{\text{tot}}{\text{CO}}_2\propto f^a{V}_{\text{T}}^b$$ (19)

with a < b (31). The exponent values obtained from these simulations are within the range of values provided in the literature, represented in Table 2 (9, 47).

Considering the normalized representation of flow cost vs. frequency used by Venegas et al. (47), the simulation results are in close qualitative and quantitative agreement with experimental results. However, our simulations underestimated the normalized flow cost Q at the highest frequencies (i.e., F > 60, or f > 12.8 Hz) in animals. This discrepancy may be attributed to an overemphasized effect of turbulent dispersion in the model and may be improved by the fine-tuning of a scaling parameter used to calculate $D_{\text{dis}}^{\text{turb}}$. In previous studies, Fredberg (14) used an order-of-magnitude approximation to set the scaling parameter equal to unity, while Slutsky et al. (35) used an average value obtained from inspiratory and expiratory flows in a mechanical model (34) used an average value obtained from inspiratory and expiratory flows in a mechanical model. It is possible that more accurate values of the scaling parameter during the turbulent flows of HFOV may be obtained either through experiment or through computational fluid dynamics (CFD) simulation.

Impact of heterogeneity and frequency selection during HFOV.

We can make several predictions about the impact of frequency selection on ventilation heterogeneity during HFOV, by jointly observing the frequency dependence of the acinar flow distribution, acinar V̇_nCO₂ distribution, and total eucapnic volume amplitude. The resonant frequency f_res marks the transition from elastically dominated to inertially dominated flow. For frequencies less than f_res, flow distribution heterogeneity is frequency independent and determined primarily by the distribution of resistive and elastic properties. For frequencies greater than f_res, the transition to frequency-dependent flow heterogeneity suggests that local transitions from elastically dominated to inertially dominated flow occur at different frequencies throughout the lung, depending on local mechanical, geometrical, and topological properties. The multimodal behavior of the flow heterogeneity (Fig. 8) and the spatial distributions of f_min and f_max (Fig. 9) suggest that frequency-dependent behavior of the flow distribution is regionally clustered. Note that two acini sharing the same value of f_max does not imply also sharing the same value of f_min, suggesting that subtle differences between mechanical properties of neighboring acini produce noticeable differences in frequency dependence of ventilation distribution. Furthermore, the healthy and injured conditions exhibited similar clustering, suggesting that the frequency-dependent behavior of ventilation distribution is largely determined by airway properties affecting inertance (i.e., airway size, path-length between airway opening and acinus, airway network topology) rather than tissue properties in this particular model of injury.

The transition from low-f to high-f gas exchange, which occurs between 0.5 and 5 Hz in our canine model, is associated with increased acinar V̇_nCO₂ heterogeneity despite no change in flow distribution heterogeneity. This can be explained by the nonuniform distribution of personal dead space across the acini. Even if flow is distributed uniformly, differences in path-length, airway network symmetry, and airway sizes cause some acini to receive less fresh gas per oscillatory period than others. As the rate of ventilation increases, the tidal volume required for eucapnia decreases, and the acini with large personal dead space do not participate in direct acinar ventilation. A bimodal distribution of acinar CO₂ elimination exists over the transition range 0.5 Hz < f < 5 Hz, with one mode corresponding to acini receiving direct acinar ventilation and the other mode correspond to acini eliminating CO₂ only via mixing and dispersion. For example, the fraction of acini receiving direct acinar ventilation in the healthy simulations reduced to 50% at 1.6 Hz and 0% at 5 Hz. In this range of frequencies, increased acinar V̇_nCO₂ heterogeneity gives the appearance of increased ventilation-to-perfusion mismatching. The effects of frequency-dependent transitioning from direct acinar ventilation to mixing and dispersion observed in these simulations are consistent with the results of washout imaging reported by Venegas et al. (49), whose findings were supported by a simplified model with heterogeneous dead space. Achieving increased direct acinar ventilation at higher frequencies, and thereby reducing the tidal volume required for eucapnia, may be possible through the use of specialized ventilation apparatus designed to minimize dead space of the ventilator tubing and upper respiratory tract (44).

The selection of an optimal frequency for delivering HFOV may be described as a multiobjective optimization problem, wherein one objective is to minimize the volume amplitude and another objective is to minimize ventilation heterogeneity. Lower frequencies (f < f_res) require greater volume amplitudes, thus increasing the risk of overdistension in the presence of lung injury. High frequencies (f > f_res) minimize the volume amplitude required for eucapnia yet increase acinar flow heterogeneity and consequently acinar V̇_nCO₂ heterogeneity. Based on these simulations, we predict that optimal HFOV for this canine model is delivered using frequencies slightly less than f_res. This range provides the best compromise between minimizing both volume amplitude and ventilation heterogeneity in the simulated canine lungs.

The spatial distributions shown in Fig. 7 demonstrate spatial concordance between regional acinar flow and CO₂ elimination, consistent with experimental studies combining positron emission and X-ray computed tomographies (21, 43, 55). The positive correlation between flow and CO₂ elimination decreased at higher frequencies, likely due to the diminishing influence of direct acinar ventilation and bulk transport on CO₂ elimination. Simulations from the healthy condition demonstrate a transition from elastically dominated flow and CO₂ elimination at low frequencies (as indicated by the influence of gravity on these distributions) to variable regional ventilation at higher frequencies. That is, the same lung region can experience a larger share of total ventilation at one particular frequency, yet smaller shares at others. This phenomenon yields the clustering of regionally preferred frequencies f_max as shown in Fig. 9. This finding suggests that a combination of multiple frequencies delivered simultaneously may produce more uniform overall ventilation (23).

Limitations

Despite the complexity of our model in terms of structural geometry, topology, fluid flow, and gas transport, there are numerous assumptions and simplifications that limit the physiological interpretation of these results. The pathophysiological features of ARDS are not realistically described by a simple altered distribution of tissue elastances. The model also does not account for derecruitment, perfusion shunting, or any changes in airway mechanics. Furthermore, the model ignores intratidal variations in mechanics, which are especially relevant to overdistension and cyclic derecruitment (2, 5).

Another assumption, that mixed-venous blood has a constant partial pressure of CO₂ in all simulations, restricts the applicability of these simulations to eucapnic conditions and short time scales; that is, all simulations are executed to calculate an instantaneous V̇_totCO₂ under the premise that mixed-venous CO₂ partial pressure is uniformly 46 mmHg throughout the lung. If the value of instantaneous V̇_totCO₂ calculated in this manner for any waveform is equal to the rate of CO₂ production by metabolism, then that waveform will maintain equilibrium of arterial CO₂ partial pressures. Also the assumed boundary condition used at the airway opening (i.e., P_aoCO₂ = 0 mmHg) neglects the additional dead space of the upper airways or an endotracheal tube. Thus we expect our results to slightly underestimate the volume amplitudes required for eucapnic ventilation. The close agreement between simulation results and the experimental data of Venegas et al. (47) may be due in part to the latter’s use of specialized oscillatory apparatus that delivered fresh gas directly to the trachea, bypassing the dead space upper airways (44). Boundary conditions at the acinar level do not include the effects of acinar interdependence (28), collateral ventilation (4), or pleural surface interactions at the chest wall, diaphragm, or between lobes (48).

The model of airway segment impedance used in this study (10) was intended to characterize flow phenomena up to the lower end of the acoustic range (i.e., up to 100 Hz), assuming Womersley-type oscillatory flow (42, 54) and isothermal gas compression. Admittedly, the lumped longitudinal and shunt airway segment impedances are simplifications of actual fluid mechanical descriptions of oscillatory flow. The isothermal gas compression compliance used in our model is also a limiting case of the full thermodynamic expression (6). The model is also limited in that it does not include other potentially influential gas transport mechanisms, such as asymmetric velocity profiles at bifurcations, differences between inspiratory and expiratory velocity profiles, or cardiogenic mixing (9). Furthermore, the included transport mechanisms are assumed to superimpose linearly, except for direct acinar ventilation (which is inherently nonlinear due to complete CO₂ removal at the airway opening). Despite limiting the scope of these modeled mechanisms for the sake of parsimony, this work does provide comparable results with previous experimental work using HFOV in dogs (15, 27, 45).

Aside from these limitations, there are several important considerations regarding the applicability of these canine simulations to mechanical ventilation in humans. Differences in anatomic structure of the canine airway tree compared with that of humans, such as increased branching asymmetry and dead space relative to body weight, will also yield differences in airway resistance, resonant frequency, and the transition frequency from conventional ventilation to HFOV (47). Moreover, the influence of frequency on the pressure cost of ventilation, as well as on the distribution of gas transport, may be altered in humans compared with dogs particularly at high frequencies (46). Clinically relevant predictions for optimal frequency selection during HFOV may be obtained using this gas transport model in airway structures based on human anatomy.

Conclusion

Oscillatory frequency has a substantial impact on ventilation distribution. The total tidal volume required to maintain eucapnia decreases with frequency. Thus increasing oscillatory frequency may minimize the risks associated with excessive volume distension. However increasing frequency may not be a universally appropriate ventilator management strategy, since frequency-dependent transitions from elastically dominated to inertially dominated ventilation distribution results in overventilation and underventilation simultaneously in different lung regions. Transitions in gas transport mechanisms also induce frequency-dependent heterogeneity in ventilation distribution, which may result in ventilation-to-perfusion mismatching and impairments in gas exchange. Thus it is essential to consider these multiple interdependent factors when selecting frequency during oscillation of the injured lung.

GRANTS

This work was supported in part by the Medical Technologies Centre of Research Excellence at the University of Auckland (M. H. Tawhai), National Institutes of Health Grant UM1 HL-108724 (D. W. Kaczka), US Department of Defense Grant PR151761 (D. W. Kaczka), and the University of Iowa Department of Anesthesia (D. W. Kaczka, J. Herrmann).

DISCLOSURES

D. W. Kaczka and J. Herrmann are coinventors on a pending patent involving multifrequency oscillatory ventilation (MFOV). In addition, they are cofounders and shareholders of Oscillavent, Inc.

AUTHOR CONTRIBUTIONS

J.H., M.H.T., and D.W.K. conception and design of research; J.H. and M.H.T. performed experiments; J.H., M.H.T., and D.W.K. analyzed data; J.H., M.H.T., and D.W.K. interpreted results of experiments; J.H. and D.W.K. prepared figures; J.H. and D.W.K. drafted manuscript; J.H., M.H.T., and D.W.K. edited and revised manuscript; J.H., M.H.T., and D.W.K. approved final version of manuscript.

APPENDIX A-1

Derivation of mixing transport resistance

Consider an example bifurcation given by three cylindrical airway segments meeting at a central node. The CO₂ concentration at the central node is given by C₀CO₂, with concentrations at the distal end of each of the three adjoining airway segments (i = 1, 2, 3) given by C₁CO₂, C₂CO₂, and C₃CO₂. In the absence of diffusion and dispersion, the time-averaged value of C₀CO₂ is given by a weighted average of the neighboring concentrations C_iCO₂, for i = 1,2,3. Also assume that the weighting of each neighboring CO₂ concentration is proportional to the rate at which the mixing volume (V_mix,i) passes completely through its respective airway segment. The V_mix,i is given by the (strictly) positive difference between the magnitude of total oscillating volume in the segment |V_osc,i| and the segment lumenal volume itself (V_seg,i):

$$V_{\text{mix},i}=\max \left(\left|V_{\text{osc},i}\right|-V_{\text{seg},i}\hspace{0.17em},\hspace{0.17em}0\right)$$ (A1)

where max () indicates the maximum value of the enclosed arguments. The weighted averaged concentration at the central node is then:

$$C_0{\text{CO}}_2=\frac{\sum _{i=1}^{3}f{V}_{\text{mix},i}{C}_i{\text{CO}}_2}{\sum _{i=1}^{3}f{V}_{\text{mix},i}}$$ (A2)

where f is the oscillation frequency. This weighting scheme ensures that 1): airway segments receiving greater proportions of incoming oscillatory gas flow provide greater contributions to mixing at the bifurcation, and 2) airway segments receiving oscillatory gas volumes less than their respective lumenal volumes do not contribute to mixing at the bifurcation. Converting CO₂ concentrations to partial pressures based on the ideal gas law, we obtain:

$$P_0{\text{CO}}_2=\frac{\sum _{i=1}^3\frac{f{V}_{\text{mix},i}}{\mathcal{R}T}{P}_i{\text{CO}}_2}{\sum _{i=1}^3\frac{f{V}_{\text{mix},i}}{\mathcal{R}T}}$$ (A3)

where R is the universal gas constant and T is the gas temperature. We assume that this mixing transport mechanism can be characterized by a transport resistance, such that the CO₂ molar flux in any airway segment (${\dot{M}}_i^{\text{mix}}{\text{CO}}_2$) is proportional to the mixing transport resistance for that airway ($R_{\text{T},i}^{\text{mix}}$). The axial CO₂ partial pressure difference from the distal end of each airway segment to the central node is given by P_iCO₂ − P₀CO₂. Since each CO₂ partial pressure differential is defined relative to the central node, conservation of mass of the CO₂ molar flux at the central node yields:

$$\sum _{i=1}^3{\dot{M}}_i^{mix}{\text{CO}}_2=\sum _{i=1}^3\frac{P_i{\text{CO}}_2-P_0{\text{CO}}_2}{R_{\text{T},i}^{\text{mix}}}=0$$ (A4)

Rearranging Eq. A4 to solve for the CO₂ partial pressure at the central node yields:

$$P_0{\text{CO}}_2=\frac{\sum _{i=1}^3\left[\left(\frac{1}{R_{\text{T},i}^{\text{mix}}}\right)P_i{\text{CO}}_2\right]}{\sum _{i=1}^3\left(\frac{1}{R_{\text{T},i}^{\text{mix}}}\right)}$$ (A5)

Combining Eqs. A3 and A5, it can be demonstrated that $R_{\text{T},i}^{\text{mix}}$ is inversely proportional to the weighting factor:

$$R_{\text{T},i}^{\text{mix}}=\epsilon \frac{\mathcal{R}T}{f{V}_{\text{mix},i}}$$ (A6)

where ε is a constant of proportionality characterizing the completeness with which V_mix,i is homogenized between neighboring nodes. Very large ε yields $R_{\text{T},i}^{\text{mix}}$ approaching infinity, resulting in no mixing and no transport. In the current study we have assumed complete homogenization of V_mix,i such that ε is unity, and Eq. A6 becomes:

$$R_{\text{T},i}^{\text{mix}}=\frac{\mathcal{R}T}{f{V}_{\text{mix},i}}$$ (A7)

APPENDIX A-2

Mesh analysis of transport resistance network

The transport resistance $R_{\text{T}}^{\text{tot}}$ of an airway segment relates the molar flux Ṁ and CO₂ partial pressure differential ΔPCO₂ in the axial direction. For a network of branching airways, a system of linear ordinary differential equations can be easily solved. A mesh analysis of the system ensures conservation of mass, such that the sum of molar fluxes through the airway opening and every terminal airway segment is zero. The molar fluxes at the proximal node of every acinus are chosen as state variables, and the system of equations is expressed in matrix notation as:

$$R_T\dot{M}C{O}_2=\Delta PC{O}_2$$ (A8)

where each row of the system corresponds to the mesh formed by the airways connecting a single acinus to the airway opening. Accordingly, each term in ΔPCO₂ corresponds to total CO₂ partial pressure loss between an nth acinus and the airway opening P_nCO₂ − P_aoCO₂, and the corresponding nth row of the matrix product R_TṀCO₂ corresponds to the sum of partial pressure losses due to transport resistance in each airway between that nth acinus and the airway opening.

The matrix R_T is completely dense, because the transport resistance of the trachea $R_{\text{T},\text{trachea}}^{\text{tot}}$ is included in every mesh. In other words, every path from an acinus to the airway opening includes the same partial pressure loss over the trachea, which is equal to $R_{\text{T},\text{trachea}}^{\text{tot}}{\displaystyle \sum }_{n=1}^{N_{}}{\dot{M}}_n{\text{CO}}_2$ where n indexes all individual acini. A simple example of an airway network and corresponding system of equations is provided in Fig. 10. Because matrix R_T is completely dense, memory cost may be substantial for large numbers of airways. However R_T is symmetric and can be stored and solved efficiently using dedicated algorithms for symmetric matrices.

Fig. 10.

Example airway network and corresponding system of equations for solving molar flux of CO₂ out of each acinus, according to Eq. A8. This network comprises 5 airways, terminating in 3 acini.

An alternative to mesh analysis is nodal analysis, which involves choosing the state variables as the partial pressures at the proximal node of every airway segment. Nodal analysis forms a system of equations using conservation of mass applied to each individual node, which produces a sparse coefficient matrix. However, small numerical inaccuracies at each node can result in substantial discrepancies between the flux at the airway opening and the total flux across the acini. Thus, mesh analysis is preferred.

APPENDIX A-3

Expected low-frequency gas exchange behavior

Expected values for low-frequency gas exchange (i.e., 0.1 Hz < f < 1 Hz) are derived from standard equations of gas exchange physiology during conventional mechanical ventilation and spontaneous breathing. Given the assumption that CO₂ elimination is dominated by direct alveolar ventilation and is therefore proportional to alveolar ventilation, we have:

$${\dot{V}}_{\text{tot}}^{\text{euc}}{\text{CO}}_2\propto f\left(V_{\text{T}}^{\text{euc}}-V_{\text{D}}\right)$$ (A9)

Alveolar partial pressure CO₂ is related to the eucapnic alveolar ventilation (V̇_A) and CO₂ elimination rate (${\dot{V}}_{\text{tot}}^{\text{euc}}{\text{CO}}_2$) and is assumed to be equal to mixed arterial CO₂ partial pressure (P_¯aCO₂):

$$\frac{{\dot{V}}_{\text{tot}}^{\text{euc}}{\text{CO}}_2}{{\dot{V}}_{\text{A}}}=\frac{{\dot{V}}_{\text{tot}}^{\text{euc}}{\text{CO}}_2}{f\left(V_{\text{T}}^{\text{euc}}-V_{\text{D}}\right)}=\frac{P_{\overline{a}}{\text{CO}}_2}{863}$$ (A10)

where 863 is a conversion factor between partial pressure in mmHg and gas volume fraction, accounting for the difference between ${\dot{V}}_{\text{tot}}^{\text{euc}}{\text{CO}}_2$ in l/s at standard temperature and pressure dry air, and V̇_A in l/s at body temperature and pressure air saturated with water vapor. Equation A10 can be rearranged in the form of the power-law regression in Eq. 15, such that:

$$V_{\text{T}}^{\text{euc}}=V_{\text{D}}+\left(\frac{863}{P_{\overline{a}}{\text{CO}}_2}{\dot{V}}_{\text{tot}}^{\text{euc}}{\text{CO}}_2\right)f^{-1}$$ (A11)

Thus, the expected values for the power-law regression parameters for Eq. 15 are β₀ = V_D, ${\beta }_1=\left(\frac{863}{P_{\overline{a}}{\text{CO}}_2}{\dot{V}}_{\text{tot}}^{\text{euc}}{\text{CO}}_2\right)$, and β₂ = −1.