Liquid ventilation (LV) uses a liquid as a carrier for oxygen and carbon dioxide. Since 1960s, LV has used perfluorochemical (PFC) liquids.^{1} Two LV methods have been studied: total LV (TLV) in which lungs are totally filled with PFC and then ventilated with tidal volume of PFC by using a liquid ventilator, and partial LV (PLV) in which lungs are partially filled with PFC and then gas ventilated.

Liquid ventilation has been tested in a number of preclinical studies including normal mature and immature lungs, and acute lung injury models. Results from those studies unequivocally demonstrate that LV improves gas exchange and lung compliance,^{2,3} has lung protective effects,^{4} allows a lavage of airways and alveolar debris,^{5,6} and reduces inflammation^{7} and hyperoxic lung injury.^{8,9} However, the recent LiquiVent trial using PLV in adult respiratory distress syndrome (ARDS) did not fulfill its promises.^{10} Among all the good reasons that have been advocated to explain this disappointing result, we would like to emphasize the possibility that PLV, an over-simplified method of TLV, was not able to fulfill all the demonstrated preclinical benefits of LV. Indeed, a number of preclinical studies have shown that TLV is superior to every tested ventilation strategy including PLV.^{3,11,12} But, in contrast to PLV, TLV necessitates a dedicated mechanical system (liquid ventilator). Design of a liquid ventilator for experimental and clinical acceptance is an exciting but complex challenge.

### The Chocked Flow Problem in TLV

As opposed to mechanical gas ventilation, the expiration is driven by the liquid ventilator. Because of the liquid density and viscosity, the elastic forces accumulated in the lung during inspiration are not sufficient to expel the inspired liquid out of the lungs. Hence, the expiratory pump generates a driving pressure in the airways to withdraw the inspired volume at the desired flow rate. This flow rate (*vs.* the expiratory time) is determined by a flow profile set by the clinician. If the profile is not set properly, important negative pressure can be measured in the trachea to maintain the desired flow rate and in the worst case, the airway collapses. This is also called the choked flow phenomenon, and it was experimentally observed in TLV.^{13,14} To cope with this limitation, all tidal liquid ventilators have a lower pressure limit, set to avoid strong negative pressure in the trachea. When this pressure limit is reached, the expiratory phase is stopped. On the Inolivent prototype, a supervisor adjusts the inspired volume to the expired volume, so that the FRC is kept constant.^{15}

The expiratory flow rate depends on the breathing frequency (*F*_{r}) and tidal volume (*V*_{t}), necessary to reach an adequate minute ventilation (product of *F*_{r} by *V*_{t}). A minimum value of minute ventilation must be reached to ensure gas exchange in the lungs.^{16} An increase in minute ventilation will inevitably raise the expiratory flow. However, the chocked flow problem will limit the minute ventilation (by decreasing the exchanged volume and/or increasing the breathing frequency).

To address this important limitation in TLV, some authors have suggested that flow, pump type,^{17} initial lung volume, and fluid properties^{18} have an influence on this maximal expiratory flow.^{14,19} Also, Baba *et al*.^{14} have stated that choked flow or flow limitation occurs when pressure drops below −27 cm H_{2}O. Komori *et al*.^{20} have claimed that choked flow occurs around −10 to −15 cm H_{2}O, and Foley *et al*.^{19} reported approximately the same values.

The hypothesis of the presented work is to demonstrate that the collapse can be avoided, and the minute ventilation maximized, by profiling the expiratory volume. For this purpose, the method is based on a mathematical lungs model in TLV, including flow limitation mechanisms. The lungs model, scaled for the newborn lamb model, requires physiologic parameters, and details of the experimental methodology used to identify these necessary parameters are presented. We also present the simulations results obtained and highlight sensitive parameters to deal more efficiently with flow limitations and their consequences. Finally, we demonstrate the existence of an optimal expiratory profile which ensures maximum minute ventilation.

## Methods and Materials

### Lung Model Equations

In gas ventilation, numerical models have been built to represent respiratory mechanics during conventional mechanical ventilation and normal breathing. They were developed to underline the important parameters to consider, depending on the pathologic situation encountered. The more complete models use morphological data with collapsible airways and elastic cavities.^{21–23} However, in TLV, although some authors have proposed models^{24,25} based on rigid airways and elastic cavities, flow limitation based on collapsible airways cannot be reproduced with these models In fact, a simplified lumped-parameter theoretical model (alveoli modeled by a compliant sphere and the airways by a collapsible tube) agrees qualitatively well with experimental results.^{18} It thus seemed relevant to enhance this work by developing a complete lung model to simulate flow limitation in the airways during TLV.

#### Assumptions.

To develop the model, different assumptions were considered: 1) The interaction between the parenchyma and airway walls is neglected. Consequently, the pressures in the model are only a function of the flow and the collapsible airways. 2) The pressure gradient between the lungs and the trachea because of the effect of gravity is considered negligible. This case may correspond, for instance, to a patient ventilated in a supine position in TLV.^{26} 3) A symmetric model of the human lungs is sufficient to capture the main behavior of the flow limitation instead of an asymmetric one.^{27} 4) The fluid inertance can be neglected if we consider slow flow variation in time, which is the case during TLV. In other words, the transient inertial contribution is neglected, but the steady inertial contribution is not neglected. Hence, the numerical model is solved for steady flows, but the obtained results can be used in computing slow time varying flow profiles. 5) To study flow limitation, the experimental alveolar pressure *versus* lung volume (called the PV curve) obtained with perfluorooctylbromide (PFOB) can be used for other PFC. The latter considers the worst case because the lungs are more compliant when filled with PFOB than with other PFC liquids.^{18}

#### Lung Geometry.

The lung geometry is modeled according to the symmetric model of Weibel,^{28} schematically represented in Figure 1. Each generation of airways, indexed *z*, is numbered from the trachea (*z* = 0) to the periphery airways (*z* = 16 at the last generation, for infant). For each generation, the airway area *A*_{z} is a function of total lung capacity (TLC), whereas length *L*_{z} is given at 75% of TLC. The number of generation N_{z} doubles for each new generation, N_{z} = 2^{z}.

#### Collapsible Airways.

A collapsible airway is considered as a flexible tube, where its section area depends on the transmural pressure *P*_{tm} (the pressure difference between the airway *P*_{aw} and pleural space *P*_{pl}). The relationship between *P*_{tm} and the airway section is given by the tube law for each generation of the Weibel model.^{22,29}

In the present model, the normalized section α_{z} (bound between 0 and 1) is the ratio of the airway area *A*_{z} over its maximal value A_{mz}. When the transmural pressure is negative (*P*_{tmz} ≤ 0), the normalized section is given by Equation 1a with *P*_{1}=α_{0z}n_{1z}/α̇_{0z}. When the transmural pressure is positive (*P*_{tmz} > 0), the normalized section is given by Equation 1b with *P _{2}=−*n

_{2z}(1−α

_{0z})/α̇

_{0z}.

#### Airway Pressure in the Trachea.

All pressures are referred to pleural pressure. So, the alveolar pressure *P*_{alv} and the airway pressure *P*_{aw} are, respectively, equal to the elastic recoil pressure of the lung *P*_{st} and *P*_{tm}. Assuming steady flow in the airways, the airway pressure *P*_{aw} is equal to the alveolar pressure *P*_{alv} from which kinetic pressure (velocity *U*_{0} in the trachea is zero in the alveolus; ρ the PFC density) and viscous pressure losses (Δ*P*_{v} per unit distance) over the entire length of the airway system *L*=∑_{z=0}^{16}*L*_{2} are subtracted.^{30} In this relationship (Equation 2), the pressure gradient because of gravity is considered negligible. This case may correspond, for instance, to a patient ventilated in a supine position in TLV.^{26}

#### Wave Speed Limit.

The flow velocity in a flexible tube filled with liquid cannot exceed the wave speed limit or the speed at which a small perturbation propagates in the tube. Most likely, this will lead to an airway collapse. In TLV, this classic flow limitation can be considered with the steady flow assumption. Considering an incompressible liquid in a collapsible tube, the wave speed is determined by Equation 3 for each generation *z*.^{31,32}

However, for each generation, this limit is not constant and depends indirectly on transmural pressure. Thus, it is necessary to compute the transmural pressure for each lung generation. For this purpose, Equation 2 is differentiated by *x* for all generation *z*, and after rearrangement using *U*_{z}=V̇* _{e}/A_{z}*, Equation 4 is obtained.

#### Viscous Pressure Losses.

Along each airway generation there are viscous pressure losses Δ*P*_{vz}. These losses are calculated using the Poisseuille Equation 5, which assumes a laminar fully developed and irrotational flow.

Different empirical equations *Z*e_{z}(x) have been suggested to adapt the Poisseuille formula to the nature of flow observed in the airway branches. This correction was motivated by the observation of secondary flows created by the bifurcations in the airway branches. These secondary flows were measured experimentally^{33,34} and demonstrated numerically.^{35} On the basis of the Rohrer equation, Reynolds, Elad *et al*., and Collins *et al*. have proposed Equation 6, with different coefficients (*a*, *b*, and n) correlated to their experimental data.^{21,36,37}

where

is the Reynolds number. By looking closely, Reynolds coefficients are more conservative with *a* = 1.5, *b* = 0.0035, and n = 1.0. Consequently, these coefficients are used in the numerical model.

#### Bifurcations.

Based on the geometric model proposed, the transition between generations occurs over a very short distance. This hypothesis allows neglecting viscous pressure losses and only considering the change in dynamic pressure. By using the Bernoulli equation, the pressure drop over a bifurcation can be expressed by Equation 7, where the axial flow profile implies the use of a shape factor λ_{z}; but it is set to 1 for any *z*, as a first approximation.^{37}

From a numerical point of view, Equation 7 is used if and only if *A*_{z}(0) < *A*_{z}_{+}_{1}(*L*_{z}_{+}_{1}). In other words, the pressure difference is set to zero if *A*_{z}(0) ≥ *A*_{z}_{+}_{1}(*L*_{z}_{+}_{1}).^{22}

#### PV Curve Relation.

To determine alveolar pressure *P*_{alv}*versus* lung volume, the Venegas relation was used in the computation to represent the PV curve over the entire lung volume.^{38} This relation, given by Equation 8, requires certain parameters identified from an experimental measurement of the PV curve. The parameter *c*_{e} is the inflection point of the PV curve. The parameter *a*_{e} is the lower asymptote volume. The parameter *d*_{e} is proportional to the pressure range within which most of the volume changes occur. Lastly, for a desired *P*_{alv}, the lung volume *V*_{e} can be determined from the experimental data and thus used to calculate the term *b*_{e}.

Finally, the relationship between alveolar pressure and the final generation of airways, *z* = 16, deduced for the Bernoulli equation, is given by Equation 9.

### Methodology

#### Estimation of the Parameters.

The parameters of the developed numerical model (airway sections, lengths, *etc*.) were fitted for a newborn lamb model. To scale down the morphological data available in the literature, both TLC and the PV curve were identified.

The experiments were conducted based on a protocol approved by our Institutional Ethics Committee for Animal Care and Experimentation. The tests were conducted at room temperature. The lambs were sacrificed with a lethal dose of pentobarbital, 90 mg/kg IV. The lungs were not excised, and the thorax was not opened to reproduce as closely as possible the experimental setting in TLV. Before filling the lungs with PFOB, a cuffed ETT (endotracheal tube) was inserted into the trachea and the proximal end connected to a valve. The lamb was placed in a supine position and a micrometallic tube was slid in the ETT, with its extremity extending 1 cm beyond the ETT ending. The other end of the microtube was connected to a pressure sensor, fixed at the same height as the ETT in the trachea. The pressure sensor (Model 1620, Measurement Specialties, Hampton, VA) was used to measure airway pressure *P*_{aw} in the trachea and the pressure signal recorded using a Signal Ranger I/O board (SoftdB, Quebec, Canada).

#### Measurement of the PV Curve.

Before filling the lungs, the thoracic cage was compressed by hand to force the air out of the lungs. Thereafter, the lung was inflated with PFOB, using a piston pump of the total liquid ventilator. During the first filling period, the thoracic cage was massaged and manipulated to facilitate the removal of the remaining air bubbles. The volume increment for the inspiratory portion of the PV curve was 10 mL of PFC, followed by a waiting period of 30 seconds to obtain a stable pressure measurement. Each of the pressure points was recorded and corresponds to the alveolar pressure.^{39} Once the airway pressure reached a plateau or if *P*_{alv} extended beyond 30 cm H_{2}O, this point was defined as the TLC.^{40} Thereafter, the expiratory portion of the PV curve was initiated, after the same methodology. At the end of the experiment, the thorax was opened to verify the integrity of the lungs and the possible presence of a perfluorothorax.

The measured PV curve, shown in Figure 2, exhibits the entire hysteresis loop. Therefore, and as illustrated, only a portion of the PV curve (identified as the generated expiratory curves) was used during expiration. This expiratory portion was generated by curve fitting using the experimental data and the Venegas relation (Equation 8).^{38} In this relation, *a*_{e} is equal to 3.9 mL/kg, *c*_{e} is located at 2.1 cm H_{2}O, and the value for *d*_{e} is 2.9 cm H_{2}O. Thus, for a desired *P*_{alv} = positive end-inspiratory pressure (PEIP), the lung volume *V*_{e}(PEIP) can be determined from the experimental data and used to calculate the term *b*_{e}. Finally, the relationship between alveolar pressure and lung volume for a particular PEIP and tidal volume (*V*_{t}) values is obtained. In Figure 2, six generated expiratory PV curves are shown at their respective PEIP level. The PEIP is the starting point of the expiratory PV curve.

With the help of the PV curve, the lengths of each airway generation can be scaled to represent the dimensions of a newborn lamb. Hence, airway lengths are multiplied by a scale factor of the cubic root of lamb (L) to adult (a) human total lung volume (LV_{L}/LV_{a})^{1/3}.^{41} Total lung volume is defined as the sum of TLC and lung tissue volume. Lambert *et al.*^{42} has determined that LV_{a} is equal to 6040 mL. From the PV curve in Figure 2, the TLC (TLC_{L}) is 486 mL for a 4-kg newborn lamb. To estimate the lung tissue volume of the lamb, the infant (i) lung tissue was scaled down from the ratio of TLC_{L}/TLC_{i}, and calculated at 112 mL, for a TLV_{L} of 598 mL. Finally, the scale factor for airway length was 0.463. As lung volume decreases, the length of each airway generation is adjusted to the cube root of lung volume.^{41}

Lambert *et al*.^{42} have proposed some area modifications to more accurately represent infant lungs. From the given value, the infant trachea has a diameter of 8.74 mm. Because the input data are referred to the lamb, the trachea diameter needs to be slightly adjusted. Previous experiments in our laboratories have shown a 5.5 ETT is tight in the trachea of a 4-kg lamb, hence the external diameter of this tube (7.6 mm) was taken as the diameter of the trachea. All airway sections were then multiplied by a scale factor of 0.756. All sections and lengths for each generation are listed in Table 1 and the liquid properties used in TLV are provided in Table 2.

#### Computation.

The initial inputs for the model are alveolar pressure *P*_{alv} and volumetric flow V̇_{e}. To start the computation, Equations 7 and 1 are solved which yield the airway section *A*_{z}(0) and transmural pressure *P*t_{mz}(0) at the start of the 16th generation. The Bernoulli equation (Equation 4) is integrated along this generation using a fourth order Runge Kutta which provides the local transmural pressure *P*t_{mz} (*x*) in the generation. During the integration, the airway section *A*_{z}(*x*) is updated using Equation 1. Once the end of the generation is reached, the bifurcation modeled with Equations 7 and 1 are solved. This method is used for each subsequent generation until the entrance of ETT is reached. Finally, the flow is slightly increased (by increments of 1 mL/s) and the calculation is repeated. Figure 3 illustrates the computation steps.

This computation continues until local flow speed exceeds the wave speed or if the pressure is less than −500 cm H_{2}O (considered the negative infinity). When local flow speed exceeds the wave speed for the first and second time, the flow increment is decreased by a factor of one tenth. After or when the local transmural pressure reaches negative infinity, the calculation is stopped. This computation procedure was performed for the various PFC liquids listed in Table 2 using their respective properties.

## Results

### Isovolume Pressure Flow Curves

Figure 4 presents the isovolume pressure flow curves (IVPF) obtained with the numerical model with PFOB as the PFC liquid. This graph indicates the expiratory flow *versus* the airway pressure for different TLC of PFC.

On the abscissa (or at flow equal to zero), the alveolar pressure *P*_{alv} for each IVPF curve can be read. When the expiratory flow is increased, the pressure at the trachea will decrease to create the desired airway flow. This effect is translated on the IVPF figure by a movement of the curves to the left. On this left part of the figure, the curves reach and horizontal value of flow which is considered the maximal expiratory flow for the specified TLC. This means that for a given amount of PFC in the lungs (a given TLC), the expiratory flow must be lower than this value. For example, at a TLC of 66%, the alveolar pressure is around 10 cm H_{2}O (on the abscissa) and a negative tracheal pressure of −10 cm H_{2}O creates a 5.8 mL/s/kg flow. For this particular curve, the expiratory flow must be below 5.8 mL/s/kg. Otherwise, the pressure will decrease rapidly and lead to an airway collapse.

At high TLC, the pressure change in the trachea is more progressive and the asymptotic value of flow is reached at a relatively large negative airway pressure. However, at very low TLC, the tracheal pressure is rapidly decreasing as the expiratory flow is slowly increased. Moreover, the maximum flow is reached at some centimeters of water below zero.

### Maximal Expiratory Flow Curves

Figure 5 presents the maximal expiratory flow (MEF) curves *versus* the alveolar pressure. To consider the effects of viscosity and density of the PFC liquid, MEF are presented for different type of PFC used in experimental researches: perflubron (PFOB) (Alliance Pharmaceutical Corporation, San Diego, CA), Fluorinert (FC-77) (3M, St. Paul, MN), Perfluorodecalin (PFDEC) (F2 Chemical Ltd., Manchester, UK), and Rimar 101 (RM101) (Miteni, Milan, Italy). Table 2 presents the density and viscosity of these PFC liquids.

Whatever the PFC used, the curves show the influence of the alveolar pressure on flow limitation. When the alveolar pressure rises, the available maximal expiratory flow increases significantly. For example (with PFDEC), the expiratory flow must be below 4 mL/s/kg at an alveolar pressure, *P*_{alv}, of 8 cm H_{2}O, compared with 8 mL/s/kg at *P*_{alv} of 20 cm H_{2}O. So, the expiratory flow can be doubled when the alveolar pressure is increased by 12 cm H_{2}O.

Those curves also highlight the effect of viscosity; as it increases, the maximal expiratory flow decreases for the same alveolar pressure. In other words, a higher alveolar pressure is necessary to develop the same expiratory flow for more viscous PFC liquid.

Finally, the curves clearly present slope changes. The main change occurs at about 6 mL/s/kg (8 cm H_{2}O with PFOB, FC-77, and RM101, and about 12 cm H_{2}O for PFDEC). It is the result of a dominant contribution of viscous pressure losses (Equation 2) when the factor *Z*e_{z} (Equation 5) increases.^{30}

### Maximal Flow Mechanism

Figure 6 presents both the IVPF curves and the wave speed limit curves for generations 0–5 and three TLC (34%, 80%). As can be seen on the top chart, the wave speed limit is far from the IVPF curves for all generations. Hence, for the first generations (or the central airways), it seems that the maximal flow is determined by the coupling between viscous pressure losses and airway compliance. From the 4th generation (bottom chart of Figure 6), general observations noted in gas ventilation still hold: at high lung volume, flow is limited by a coupling between airway compliance and dynamic and viscous pressure losses. At low lung volume, flows are limited by viscous pressure losses and airway compliance. The variation in airway length with lung volume does not play an important role in flow limitation. The most sensitive parameter of the model geometry is airway sections; a change by ±5% is traduced by the same change in maximal expiratory flow.^{43}

### Flow Limiting Segment

Figure 7 presents airway generations in which the flow is limited, termed flow limiting segment (FLS), relative to alveolar pressure. The FLS is the trachea for all alveolar pressure values. This analysis highlights the fact that if a choked flow occurs at high and mid lung volume, the airway generation in which the flow is limited (*i.e.*, FLS) will be concentrated in the central airways. An increase or decrease in liquid viscosity does not change the location of the FLS. Bull *et al.*^{44} in a recent study demonstrated the same finding experimentally; the FLS is located in the trachea and the bronchi independently of end-inspiratory lung volume. These considerations reinforce the usefulness of a pressure sensor in the ETT to monitor airway pressure. If a sudden drop in pressure is recorded in the trachea, it could be associated with the choked flow phenomenon. Hence, actions should be taken to correct the situation.

Figure 7 presents the equal pressure points (EPP) *versus* alveolar pressure. This is the generation where the pressure in the airway equals pleural pressure. As the lung volume decreases, the alveolar pressure decreases from the central airways (*z* < 2) to the periphery (*z* > 4). These results are in agreement with gas ventilation models, although the FLS in TLV are always downstream of the EPP.^{22} Because liquid density and viscosity are important, flow is limited in the airways where flow speed is more important, namely the central airways. An increase in liquid viscosity will slightly move the EPP in the peripheral airways.

### Optimal Expiratory Profile

Figure 8 presents each MEFV curve plotted relative to lung volume for different PEIP. The latter decrease almost linearly as lung volume increases and the gap between each one is caused by the hysteresis on the PV curve. They are also closely similar to those presented by Meinhardt *et al*.^{17} Because, these curves seem to be linear with the same slope for different PEIP values, each one is modeled as a linear function of volume by the relation 10. The term V̇_{e,lim} is used to notify the limit on the expiratory flow allowed by the airways. Consequently, the expiratory flow must be higher than this value (V̇_{e} ≥ V̇_{e,lim}).

The term *A* is the curve negative slope (

, where ΔV̇_{e} is a variation of the maximal expiratory flow and Δ*V*_{e} is a variation of volume in lungs). The offset term *B* is determined from the flow and lung volume at the selected PEIP. It is important to notice that the parameters *A* and *B* are not directly related to the usual terms of resistance and compliance used to characterize the lungs mechanic.

To remove the liquid as quickly as possible from the lungs without causing a choked flow, the marginal case consists in applying the maximal expiratory flow, for a given amount of PFC present in the lungs. So, an analytical expression of the optimal expiratory profile can be computed by integrating the ordinary differential Equation 10 over the time *t*, starting at *t* = 0. This leads to Equation 11, which is an exponential evolution of lung volume in time.^{45}

The parameter *C=V*_{e}(0) + *B/A* is computed from the terms *A*, *B* and the initial lung volume *V*_{e}(0) (given by the PEIP level selected). Figure 9 presents the decrease in lung volume during expiration *versus* time, for different PEIP levels. Because of the exponential nature of the curves, the decrease is more rapid at the start of the expiration and slower at the end.

The term *A* can be associated with a time constant τ_{lim}, which is the negative inverse value of *A* (τ_{lim} = −1/*A*). Despite the fact that the optimal expiratory profile is an exponential profile characterized by a time constant, there is no direct link with the conventional time constant of the lungs (usually defined by the product of resistance with compliance). This can be explained by the fact that the time constant exposed in this section is defined by a marginal case of expiration (at the limit of an airway collapse). In this case, the expiration is forced by the total liquid ventilator pump as opposed to normal gas ventilation where the expiration is passive (the elastic forces present in the lung after inspiration are sufficient to expel the tidal volume).

For the case where the optimal volume profile is given by Equation 11, it is possible to compute the smallest time necessary to exhale a given tidal volume of PFC from the lungs. This minimum expiratory time *T*_{e} deduced from Equation 11 is given by Equation 12.

Because the parameter *C* depends on the initial lung volume, this minimum expiratory time will depend on PEIP (which is a function of the initial lung volume). A larger tidal volume or a low PEIP (low *C*), will increase the expiratory time, whereas a lower tidal volume or higher PEIP (high *C*), will do the opposite.

## Discussion

If the results and observations are transposed in an experimental context, it is not possible to ventilate at the maximal expiratory flow, which is an extreme case. A small difference in lung geometry could lead to unwanted ventilation problems and collapses. A first approach would be to determine an acceptable expiratory flow based on a fraction of the maximal flow. When choosing a flow limit, the MEFV curve in Figure 8 is moved up by the same fraction. The airway pressure at the limited flow can be determined on the IVPF curve and will change with lung volume. However, these approaches imply that the maximal flow is known for each patient, which is not the case. In addition, at low lung volume, the flow saturates rapidly and even a small negative pressure limit plays on the edge of the maximal expiratory flow.

A second approach is to consider an expiratory tracheal pressure reference, where the risk of developing a choked flow is eliminated. During TLV, tracheal pressure can be monitored which favors this approach for experimental applications. Using the numerical model, the flow at the pressure reference can be determined and used to obtain the volume variation over time during expiration. For the rest of the section, the pressure reference is fixed at −2.5 cm H_{2}O. This pressure reference assures a safe distance from the maximal expiratory flow, when the positive end-expiratory pressure level is above 3 cm H_{2}O, which was the case during most of the TLV done by the Inolivent team. For different PFC liquids, PEIP levels, tidal volumes, and a fixed pressure reference (at −2.5 cm H_{2}O), the optimal expiratory time can be calculated with the numerical model. Graphically, the optimal expiratory profile looks like an exponential (as illustrated in Figure 9) because this solution is relatively close to the optimal one.

Figure 9 presents also the effect of a change in the initial lung volume, and consequently of the PEIP level. Lets consider a PEIP of 10 cm H_{2}O (initial lung volume of 70 mL/kg) and a tidal volume of 25 mL/kg. The curve shows that the minimum expiratory time is predicted at 6 seconds. In the same manner, if the PEIP is increased by 2 cm H_{2}O (the initial lung volume is increased of 10 mL/kg), the tidal volume of 25 mL/kg can be retrieved in 5 seconds. This example illustrates that the PEIP level changes the expiratory time for a fixed tidal volume.

To investigate precisely the dependence of the minute ventilation to the parameters, Table 3 presents the computation of the maximal minute ventilation for different PEIP, PFCs, and tidal volumes. The main point is that a small increase in the PEIP level can prevent chocked flow and drastically improve minute ventilation.

The experimental results of Cox *et al*.^{8} suggest that the minute ventilation must be above 80 mL/kg/min to maintain an adequate gas exchange; other studies are based on ∼100 mL/kg/min.^{40,46,47} To reach minute ventilation above 100 mL/min/kg, the PEIP needs to be above 8 cm H_{2}O (except for PFDEC which needs a PEIP above 10 cm H_{2}O). At a PEIP level below 12 cm H_{2}O, minute ventilation rises as tidal volume increases except above 30 mL/kg. Thus, based on the selected PFC, it is presumed that different tidal volumes and PEIP levels should be considered to optimize the ventilation scheme.

## Conclusion

The presented complete model demonstrates that the FLS is the trachea for all alveolar pressure values and is a consequence of a coupling between viscous pressure losses and airway compliance (at same PV curve). These results are consistent with a previous experimental study^{44} and interpretations given by simplified theoretical lumped-parameter model.^{18} The computed IVPF curves demonstrate that a chocked flow is established when the trachea pressure drops below −25 cm H_{2}O for any TLC, ranging from 30% to 80%. This result is consistent with previous experimental works which consider this pressure limit.^{14,19}

The main conclusion of this study is that a more suitable expiratory volume profile would have the shape of an exponential, where the time constant is not directly related to the product of compliance by resistance, but is dependent on flow limitation dynamics in the airways. The exponential profile shortens the expiratory time and decreases the risk of encountering a choked flow; it also allows higher minute ventilation, hence better gas exchange. This result is consistent with the suggestion of Bull *et al*.^{18} “to sculpt” the expiratory flow to be fast initially and slowing down progressively thereafter.

A practical implementation is proposed by performing expiration at a constant value of the expiratory pressure. Hence, the model is used to compute the maximal minute ventilation allowable with an acceptable risk of collapse. Ongoing work with the simulator has demonstrated that the expiratory pressure could be regulated to perform a safe and efficient TLV. This control strategy could replace the volume controlled ventilation, usually used with tidal liquid ventilators. Future studies will address the experimental validation of this pressure controller on newborn lambs.

## Acknowledgment

This work was supported in part by the Quebec Foundation for Research into Children’s Diseases and the Fonds de Recherche sur la Nature et les Technologies. The authors thank Johann Lebon for his technical assistance.

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