Strain, Strain Rate, and Mechanical Power: An Optimization Comparison for Oscillatory Ventilation

Abstract

The purpose of this study was to assess the potential for optimization of mechanical ventilator waveforms using multiple frequencies of oscillatory flow delivered simultaneously, to minimize the risk of ventilator-induced lung injury (VILI) associated with regional strain, strain rate, and mechanical power. Optimization was performed using simulations of distributed oscillatory flow and gas transport in a computational model of anatomically-derived branching airway segments and viscoelastic terminal acini under healthy and injured conditions. Objective functions defined by regional strain or strain rate were minimized by single-frequency ventilation waveforms using the highest or lowest frequencies available, respectively. However, a mechanical power objective function was minimized by a combination of multiple frequencies delivered simultaneously. This simulation study thus demonstrates the potential for multi-frequency oscillatory ventilation to reduce regional mechanical power in comparison to single-frequency ventilation, and thereby reduce the risk of VILI.

Mini-Abstract:

A computational model of distributed oscillatory flow and gas transport was used to assess the potential for optimization of mechanical ventilator waveforms using multiple frequencies of oscillatory flow delivered simultaneously. The risks of ventilator-induced lung injury associated with regional strain or strain rate were minimized by single-frequency ventilation waveforms using the highest or lowest frequencies available, respectively. However injury associated with mechanical power was minimized by a combination of multiple frequencies delivered simultaneously, or multi-frequency oscillatory ventilation.

Graphical Abstract

Introduction

Despite substantial progress in understanding the pathophysiology of the Acute Respiratory Distress Syndrome (ARDS) and ventilator-induced lung injury (VILI), there remains great difficulty in translating such knowledge into effective treatment strategies. The current paradigm of lung-protective mechanical ventilation aims to minimize the dynamic strain applied to lung tissues with low tidal volumes (V_T) or driving pressures (ΔP), while supplying sufficient positive end-expiratory pressure (PEEP) to maintain lung recruitment. Although mechanical ventilation using high strain without PEEP is associated with higher mortality compared to a low-strain high-PEEP strategy [1,2], optimal strategies for managing ventilator settings remain the subject of ongoing debate [2–5].

Nonetheless, mechanical strain imparted to the parenchyma may not be the most physiologically relevant predictor of VILI or mortality in ARDS. Given that lung recruitability is highly variable among patients with different illness severities and etiologies, it should be expected that conservative management of mechanical strain based on body weight or effective compliance may still result in increased risk of VILI or mortality. Several experimental studies indicate that mortality is influenced not only by the total amount of lung strain but also by the rate of strain [6]. Recent studies suggest that mechanical power, or the rate of energy dissipated across the parenchymal tissues, is the most robust predictor of VILI risk [7,8]. Given that the lung is structurally complex and mechanically heterogeneous, and that VILI has numerous associated mechanisms [9], inference on regional pathologic derangements based on a single reference point at the airway opening is fraught with difficulty: optimal ventilator settings for one lung region may be injurious for another [10,11].

Previously, we proposed that multi-frequency oscillatory ventilation (MFOV) may have distinct advantages over conventional lung-protective ventilation modalities, by delivering multiple frequencies of oscillatory flow simultaneously at the airway opening and allowing the mechanically heterogeneous lung regions to selectively filter the broadband input flow [12]. We hypothesized that MFOV may be ‘tuned’ to provide a less injurious ventilation distribution, depending on patient-specific regional mechanical heterogeneity, as well as the balance between regional ventilation and potential for VILI [13].

The limited availability of regional lung mechanical information presents a major challenge for optimal control of mechanical ventilation under clinical or experimental conditions. However, computational simulation allows prediction of distributed ventilation in high-fidelity anatomical lung models. Emergent behavior from structural complexity in models may yield useful insight for guiding the selection and optimization of MFOV waveforms. In this study, we hypothesized that the spectral content of MFOV waveforms could be tuned to minimize VILI in mechanically heterogeneous lungs. To test this hypothesis, we simulated distributed flow and gas transport during MFOV in a computational model of the lungs, and identified MFOV waveforms that minimized cost functions quantifying parenchymal strain, strain rate, and mechanical power.

Materials and Methods

A computational model of porcine lungs was implemented based on an airway tree segmented from a thoracic computed tomographic scan of a pig [14]. The model consisted of 30,959 cylindrical airway segments, with 15,479 viscoelastic acini. Regional oscillatory flow and gas transport throughout the model were simulated in the frequency-domain, according to previously reported techniques [11,13,15]. Briefly, flow was distributed according to the mechanical impedance of individual airway segments and acini [11]. Airway dimensions defined longitudinal and shunt impedances to oscillatory flow [11]. The longitudinal impedance of each airway segment was determined assuming Womersley-type oscillatory flow in a cylinder [16], while the shunt impedance was given by a parallel combination of isothermal gas compression and a combination of soft tissue and cartilaginous viscoelasticity [17]. Each terminal airway in the tree was subtended by a viscoelastic “constant-phase” element representing acinar wall distensibility [18] in parallel with gas compression compliance [10]. Flow oscillations at the airway opening were distributed throughout the model using a recursive flow-divider scheme, based on the serial and parallel combinations of subtending input impedances at each node of the tree [11]. Gas transport was determined by mathematical descriptions of bulk advection, mixing at bifurcations, oscillatory and turbulent dispersion, and molecular diffusion [15]. Lung injury was simulated by applying a heterogeneous distribution of increased acinar tissue elastance, according to a gravitationally weighted gradient noise distribution [15]. Acini with an 8-fold or greater increase in injured acinar tissue elastance relative to the healthy condition were considered derecruited, and were re-assigned a mechanical impedance of infinite magnitude (Figure 1) [19].

Figure 1.

Example porcine computational lung model with pre-existing lung injury. The airway tree is shown in black (a), while the colored acini denote tidal strain (b), mechanical power (c), and CO₂ elimination (d) throughout the model during oscillatory ventilation at 10 Hz and 46.7 mL. The direction of the gravitational field g, assuming supine positioning, is indicated as into the page ($\otimes$) or towards the bottom of the page ($\downarrow$). These oscillatory settings resulted in 5.5 mL min⁻¹ kg⁻¹ of total CO₂ elimination at the airway opening. Derecruited acini are shown in gray (~39% of all acini in the model), and do not participate in gas exchange.

Optimization of mechanical ventilation was formulated as a constrained single-objective problem, with multiple objective functions to be compared. Overall, the problem involves optimizable parameters of the MFOV ventilator waveform, a constraint on total gas exchange, and objective functions relating to tissue strain, strain rate, and mechanical power. During MFOV, the adjustable properties of the ventilatory waveform include the number of discrete frequencies ($K$), the value of each frequency ($f_k$), as well as the amplitude ($|V_k|$ ) and phase ($\angle V_k$ ) associated with the volume (or alternatively, flow or pressure) at each frequency. Thus the input volume waveform at the trachea was constructed as:

$$V_{\text{trachea}}\left(t\right)=\sum _{k=1}^K|V_k|\sin \left(2\text{π}{f}_{k}t+\angle V_k\right).$$ (1)

A recursive flow division algorithm [11] was applied to distribute oscillatory flows in inverse proportion to the total subtending impedance of each pathway at all bifurcations in the model, resulting in a distribution of acinar flow magnitudes and phases (i.e., $|{\dot{V}}_{n,k}|$ and $\angle {\dot{V}}_{n,k}$ for the $n$^th acinus at the $k$^th frequency). Time-varying acinar volume ($V_n$) was computed using trapezoidal integration of acinar flow, such that:

$$V_n\left(t\right)={\overline{V}}_n+\underset{0}{\overset{t}{\int }}\left[\sum _{k=1}^K|{\dot{V}}_{n,k}|\sin \left(2\text{π}{f}_k\tau +\angle {\dot{V}}_{n,k}\right)\right]\text{dτ},$$ (2)

where ${\overline{V}}_n$ is the mean acinar volume at mean airway pressure 10 cmH₂O and $\tau$ is the variable of integration. Calculations for ${\overline{V}}_n$ are provided in the Appendix. Peak tissue strain in each acinus (${\epsilon }_n$) was computed relative to the minimum acinar volume during the ventilatory cycle:

$${\epsilon }_n=\frac{\underset{t}{\max }{V}_n\left(t\right)-\underset{t}{\min }{V}_n\left(t\right)}{\underset{t}{\min }{V}_n\left(t\right)}.$$ (3)

Peak strain rate (${\dot{\epsilon }}_n$) in each acinus was computed by the maximum absolute value of the rate of change in acinar strain, which may be more simply expressed in terms of the advective flow rate that each acinus receives:

$${\dot{\epsilon }}_n=\frac{\underset{t}{\max }|{\dot{V}}_n\left(t\right)|}{\underset{t}{\min }{V}_n\left(t\right)}.$$ (4)

Objective functions for strain (${\text{ϕ}}_{\text{S}}$) and strain rate (${\text{ϕ}}_{\text{SR}}$) throughout the model were computed as the 90^th percentile of ${\text{ε}}_n$ and ${\stackrel{·}{\text{ε}}}_n$, respectively. Thus minimizing ${\text{ϕ}}_{\text{S}}$ or ${\text{ϕ}}_{\text{SR}}$ corresponds to minimizing acinar strain or strain rate across 90% of the lung, without regard for other 10%. Alternatively, we computed the mechanical power in each acinus (${\text{MP}}_n$ ), an index which intrinsically incorporates both strain and strain rate [7]:

$${\text{MP}}_n=\frac{1}{2}\sum _{k=1}^K|{\dot{V}}_{n,k}||P_{n,k}|\cos \left(\angle {\dot{V}}_{n,k}-\angle P_{n,k}\right),$$ (5)

where $|P_{n,k}|$ and $\angle P_{n,k}$ are the magnitude and phase of transpulonary pressure oscillations in the $n$^th acinus at the $k$^th frequency. The derivation of Equation (5) is detailed in the Appendix, as well as a particular solution assuming viscoelastic constant phase tissue impedance [20]. The objective function for mechanical power (${\text{ϕ}}_{\text{MP}}$) throughout the whole lung was computed as the 90^th percentile of ${\mathrm{M}\mathrm{P}}_n$.

The optimization was constrained by the necessity of eucapnic ventilation—that is, the total CO₂ elimination (${\dot{V}}_{{\text{CO}}_2}$), as determined from the sum of all acinar CO₂ eliminations [15], was in equilibrium with the weight-based estimate of metabolic CO₂ production (5.5 mL min⁻¹ kg⁻¹) [21]. The ${\dot{V}}_{{\text{CO}}_2}$ during each of our simulations was then computed based on distributed flow and gas transport mechanisms previously described [13,15]. Eucapnic values of flow amplitudes for any specified oscillatory waveform were determined by iterative optimization. Examples of distributed acinar strains, mechanical power, and CO₂ elimination throughout the model are shown in Figure 1.

To reduce computational complexity, only two MFOV waveforms were considered. For the first waveform, $K$ was fixed at 4, with frequencies of 5, 10, 15, and 20 Hz, selected for consistency with our previous experimental work [12]. For the second waveform, $K$ was fixed at 5, with frequencies of 0.4, 2, 7, 12, and 26 Hz, selected to broadly span the range of conventional and high-frequency ventilation. Hereafter, these MFOV waveforms are referred to as MFOV-4 and MFOV-5, respectively. A randomized sampling of relative phases and amplitudes at each frequency, using a Monte Carlo approach, was used to generate 75,000 candidate MFOV-4 and MFOV-5 waveforms. The dimensionality of the problem was reduced by considering only $K-1$ relative phases, as opposed to $K$ absolute phases—that is, by constraining the phase at the lowest frequency to be zero (i.e., $\angle V_1=0$), and then randomizing values of $\angle V_k-\angle V_1$ over the range from 0 to π radians, for $k\ne 1$. For each waveform, the volume amplitudes at each frequency, $|V_1|\dots |V_K|$, were randomly generated with an initial sum of 100 mL. The volume amplitudes were then collectively scaled up or down proportionately in response to under- or over-ventilation, respectively, until the resulting ${\dot{V}}_{{\text{CO}}_2}$ corresponded to a eucapnic level in mixed arterial blood. An additional 45 single-frequency oscillatory ventilation (SFOV) waveforms were also generated, each consisting of nonzero flow amplitude at a single discrete frequency over the range of 0.1 to 30 Hz. For each objective function, the optimal amplitude distribution for each waveform was determined by rank ordering all candidate waveforms according to their corresponding lowest objective function value. Similarly, objective function values were also computed for all candidate SFOV waveforms.

Results

Figure 2 shows strain and strain rate objectives, ${\text{ϕ}}_{\text{S}}$ and ${\text{ϕ}}_{\text{SR}}$, respectively, during SFOV at rates between 0.1 to 30 Hz (or 6 to 1800 min⁻¹). These two objective functions present conflicting optimization goals: the highest possible frequency will minimize strain, yet maximize strain rate. Likewise the lowest possible frequency will minimize strain rate, yet maximize strain. Both objective function values were increased relative to their corresponding values under the simulated injury condition, regardless of frequency. Figure 3 shows the distributions of ${\text{ϕ}}_{\text{S}}$ and ${\text{ϕ}}_{\text{SR}}$ produced by the 75,000 randomly generated MFOV-4 and MFOV-5 waveforms. For all minimizations of ${\text{ϕ}}_{\text{S}}$ and ${\text{ϕ}}_{\text{SR}}$, both MFOV waveforms yielded strains and strain rates that were equal to, or greater than, those for SFOV. Note that a Pareto boundary is formed by the set of SFOV simulations, such that each point along this curve represents an optimal solution to the multi-objective minimization of strain and strain rate. At any point along this boundary, alterations in the frequency content of the oscillatory waveform result in an increase in at least one of the two objectives, ${\text{ϕ}}_{\text{S}}$ or ${\text{ϕ}}_{\text{SR}}$. With only a few minor exceptions, MFOV waveforms were unable to improve beyond this Pareto boundary formed by SFOV waveforms when considering only a weighted combination of strain and strain rate objectives.

Figure 2.

Conflicting objectives of 90^th percentile acinar strain vs. 90^th percentile acinar strain rate during eucapnic single-frequency oscillatory ventilation in the computational model, for frequencies ranging 0.1 to 30 Hz. Symbols represent the objective function values for several arbitrarily selected frequencies along each curve to provide context. Increases in strain rate are concomitant to reductions in strain, and vice versa.

Figure 3.

90^th percentile acinar strain and strain rate objectives during eucapnic single-frequency and multi-frequency oscillatory ventilation in simulated healthy (a, b) or injured (c, d) lungs, using either 5, 10, 15, and 20 Hz (MFOV-4) (a, c), or 0.4, 2, 7, 12, and 26 Hz (MFOV-5) (b, d). 75,000 MFOV waveforms were simulated for each condition, represented by contoured regions shaded according to density within the two-dimensional objective space. Circles represent the objective values during SFOV at each discrete constituent frequency.

Figure 4 shows scatter plots of the objective functions ${\text{ϕ}}_{\text{S}}$, ${\text{ϕ}}_{\text{SR}}$, and ${\text{ϕ}}_{\text{MP}}$ at many randomly sampled points throughout the high-dimensional parameter space. For simplicity, multiple views of each set of sampled points are shown, each view represents a one-dimensional projection defined by the volume amplitude of each frequency. Note that waveforms could be randomly generated with zero or near-zero volume amplitudes at one or more frequencies. In particular, the SFOV waveforms contained only one non-zero volume amplitude. From these plots, it is apparent that the local minimum of ${\text{ϕ}}_{\text{S}}$ occurs when volume amplitude is maximized at the highest frequency, and volume amplitudes are minimized for all other frequencies. Likewise the local minimum of ${\text{ϕ}}_{\text{SR}}$ occurs when volume amplitude was concentrated at the lowest frequency. By contrast, the scatter plots for ${\text{ϕ}}_{\text{MP}}$ demonstrate that concentrating all energy into a single frequency produced a local maximum of ${\text{ϕ}}_{\text{MP}}$. Minimization of ${\text{ϕ}}_{\text{MP}}$ using either the MFOV-4 or MFOV-5 waveforms was obtained by a nontrivial combination of the selected frequencies. For example, a waveform defined by 0.4 Hz with approximately 200 mL volume amplitude was the best single frequency of the five frequencies shown for minimizing ${\text{ϕ}}_{\text{MP}}$, however a MFOV waveform defined by a combination of approximately 150 mL at 0.4 Hz and at 6 mL at 26 Hz yielded an even lower value of ${\text{ϕ}}_{\text{MP}}$.

Figure 4.

Scatter plots of objective function values for 90^th percentile of acinar strain (a), strain rate (b), and mechanical power (c) in the injured model, projected from a high-dimensional parameter space onto several one-dimensional axes, each representing a single parameter. Each translucent data point represents one multi-frequency waveform with randomly sampled parameters. Symbols represent single-frequency waveforms at 0.4 Hz (△), 2 Hz (◌), 7 Hz (□), 12 Hz (▽), and 26 Hz (♢). Potential optimal waveforms were identified by the lowest objective functions values computed.

Figures 5 and 6 show the objective function values and corresponding volume amplitude distributions in each of the 1000 top-ranked MFOV waveforms that minimized each objective function. Consistent with Figure 4, ${\text{ϕ}}_{\text{S}}$ and ${\text{ϕ}}_{\text{SR}}$ were minimized by the highest and lowest frequencies within each MFOV waveform, respectively. Moreover the spectral content for each MFOV waveform converged towards a single-frequency solution, as rank was reduced from 1000 to 1, indicating that the single-frequency solution was optimal for either ${\text{ϕ}}_{\text{S}}$ or ${\text{ϕ}}_{\text{SR}}$. However the spectral content of waveforms minimizing ${\text{ϕ}}_{\text{MP}}$ remained relatively consistent over the top 1000 ranks, and converged towards the multi-frequency solution at Rank 1. This finding indicates that the Rank 1 MFOV waveform represents a robust approximation of the true local minimum, which is not sensitive to small deviations in volume amplitudes or phases.

Figure 5.

Objective function values (top panels) for 90^th percentile of acinar strain (a), strain rate (b), and mechanical power (c) in the injured model. Only the 1000 top-ranked MFOV-4 waveforms identified from the randomly sampled parameter space are shown. Each data point for objective function value corresponds to a stacked bar (bottom panels) representing the relative fraction of volume amplitude at each frequency (5, 10, 15, 20 Hz). Logarithmic spacing of waveform rank is used to apportion greater visual weight to waveforms minimizing the objective function.

Figure 6.

Objective function values (top panels) for 90^th percentile of acinar strain (a), strain rate (b), and mechanical power (c) in the injured model. Only the 1000 top-ranked MFOV-5 waveforms identified from the randomly sampled parameter space are shown. Each data point for objective function value corresponds to a stacked bar (bottom panels) representing the relative fraction of volume amplitude at each frequency (0.4, 2, 7, 12, 26 Hz). Logarithmic spacing of waveform rank is used to apportion greater visual weight to waveforms minimizing the objective function.

Figure 7 summarizes the performance of the optimal solution for minimizing ${\text{ϕ}}_{\text{MP}}$ using SFOV in comparison to the Rank 1 MFOV-4 and MFOV-5 waveforms. For the SFOV waveform, the healthy and injured conditions exhibited local minima of ${\text{ϕ}}_{\text{MP}}$ at 0.4 Hz and 0.7 Hz, respectively. Both MFOV waveforms of Rank 1 consistently outperformed every SFOV waveform with discrete frequency equal to one of the MFOV constituent frequencies. Furthermore, the Rank 1 MFOV-5 waveform minimized ${\text{ϕ}}_{\text{MP}}$ to a lower value compared to the best single-frequency waveform, despite individually nonoptimal constituent frequencies. The relative improvement for the Rank 1 MFOV-5 waveform compared the best single frequency was 5.5 % for the healthy condition and 15.4 % for the injured condition. Figure 8 shows the optimal SFOV, MFOV-4, and MFOV-5 waveforms based on minimal ${\text{ϕ}}_{\text{MP}}$, as time-domain representations of volume, flow and pressure measured at the trachea.

Figure 7.

Comparison between 90^th percentile acinar mechanical power objective function values in simulated healthy (a) and injured (b) lungs for SFOV waveforms with discrete frequencies ranging from 0.1 to 30 Hz (solid black line, left panels), and the optimal SFOV, MFOV-4, and MFOV-5 waveforms (bars, right panels) which yielded the lowest 90^th percentile of acinar mechanical power. Bars represent the optimal waveforms identified using either a single frequency (SFOV, black), a combination of 5, 10, 15, 20 Hz (MFOV-4, red), or a combination of 0.4, 2, 7, 12, 26 Hz (MFOV-5, blue). Symbols (squares, circles, crosses) represent corresponding single-frequency constituents of each waveform.

Figure 8.

Oscillatory volumes, flows, and pressures measured at the trachea of simulated healthy (a) or injured (b) lungs during mechanical ventilation using waveforms that minimize the mechanical power objective with a single frequency (black), a combination of 5, 10, 15, 20 Hz (MFOV-4, red), or a combination of 0.4, 2, 7, 12, 26 Hz (MFOV-5, blue).

Discussion

In this study, we demonstrated that the spectral content of MFOV waveforms can be tuned to minimize the mechanical power imparted to the viscoelastic parenchymal tissues of a computational porcine lung model. Moreover, these tuned MFOV waveforms consistently outperform SFOV waveforms over the same frequency bandwidth. By contrast, the minimization of either strain or strain rate could only be achieved with SFOV. Minimization of a linear combination of strain and strain rate objectives was also achieved only with SFOV, regardless of the relative weighting on strain vs. strain rate.

In a previous modeling study, we demonstrated that a dual-frequency ventilation waveform was capable of minimizing strain heterogeneity compared to either single-frequency constituent [13]. However in that study, the objective function was based on the heterogeneity of parenchymal strain [13]. Nonetheless, the reduced ventilation heterogeneity with two frequencies improved gas exchange and minimized extremes of ventilation-to-perfusion mismatching. However, the total magnitude of strain was not minimized by combining two frequencies. Similarly in the present study, we also found that strain and strain rate objective functions can be minimized by simple single-frequency waveforms. The use of multiple simultaneous frequencies in the model led to no improvement in the minimization of strain or strain rate, even in the presence of severe mechanical heterogeneity similar to that observed in ARDS. While the tidal volume required for eucapnia decreases with increasing frequency during SFOV, the heterogeneity of acinar flow and CO₂ elimination also increases [15]. Thus even for the mechanically heterogeneous lung, the 90^th percentile of acinar strain may decrease with increasing frequency only if the increases in flow heterogeneity are small compared to the reductions in total tidal volume.

The selection of a physiologically relevant objective function to determine risk of VILI is not trivial. Associating VILI solely with tissue strain neglects damage to the parenchyma due to rapid rates of strain [22]. Accordingly, our simulations demonstrated that an objective function based solely on strain requires an SFOV waveform of highest possible frequency. Similarly, an objective function based solely on strain rate requires an SFOV waveform of lowest possible frequency, despite simulated acinar strains in excess of 100% (which corresponds to an intratidal doubling of acinar volume). However, both ${\text{ϕ}}_{\text{S}}$ and ${\text{ϕ}}_{\text{SR}}$ provide a very limited perspective of the complexity of tissue damage that occurs during VILI. Cressoni et al. demonstrated that in initially healthy piglets, lung strain above 200% (corresponding to tidal volumes of 38 mL kg⁻¹) were noninjurious at rates 3 to 6 min⁻¹, despite being fatal at 15 min⁻¹ [6]. They proposed that mechanical power imparted to the parenchymal tissues is a more suitable index to stratify the risk of VILI. Since mechanical power incorporates both strain amplitude and strain rate [7], it offers a comparatively more comprehensive alternative to either ${\text{ϕ}}_{\text{S}}$ and ${\text{ϕ}}_{\text{SR}}$.

The optimization of ${\text{ϕ}}_{\text{MP}}$ using SFOV led to the selection of very low frequencies: 0.4 Hz (with tidal volume of 9.43 mL kg⁻¹) for the healthy condition and 0.7 Hz (with tidal volume of 9.52 mL kg⁻¹) for the injured condition. Surprisingly, a tidal volume of 9.43 mL kg⁻¹ at a rate of 0.4 Hz (or 24 min⁻¹) corresponds to relatively normal minute ventilation for the assumed 15 kg pig of our computational model. However with our pre-existing lung injury condition, the ideal single frequency of 0.7 Hz (or 42 min⁻¹) is almost twice that of the normal respiratory rate in a pig. However our eucapnia constraint on ${\text{ϕ}}_{\text{MP}}$ mandated a tidal volume of 9.52 mL kg⁻¹ during simulated injury, similar to that of the healthy condition. It is also worth considering that the ‘driving pressure’ for this injury condition, which we may compute as the difference between the maximum and minimum intratidal tracheal pressures, was 13.4 cmH₂O at 0.7 Hz, less than the 15 cmH₂O threshold described by Amato et al. for increased mortality in patients with ARDS [2]. By comparison, ventilating our injured model at the normal respiratory rate of 0.4 Hz instead results in 12.9 mL kg⁻¹ tidal volume and 17.7 cmH₂O driving pressure, both exceeding currently recommended guidelines [23]. Thus our optimal SFOV waveforms based on mechanical power are supported within the current paradigm of lung-protective ventilation [2], at least within the context of normal physiology [21]. High-frequency oscillatory ventilation (HFOV), although theoretically ideal for lung protection under the premise that excessive lung strain is the primary risk factor for VILI [24], resulted in consistently elevated values of ${\text{ϕ}}_{\text{MP}}$ compared to lower, more ‘conventional’ frequencies. This may in part result from increased strain rates and their contribution to mechanical power, as well as the increased heterogeneity of acinar flows at high frequencies. These results support current clinical guidelines advising against routine use of HFOV in ARDS patients [23].

Nonetheless, MFOV results in further minimization of mechanical power, beyond the lowest values of ${\text{ϕ}}_{\text{MP}}$ achieved with SFOV. It is unclear whether a 15% reduction in mechanical power is clinically significant. Direct comparison with the values reported by Cressoni et al. is difficult, due to the extreme strains applied in that experimental study (i.e., between 200% to 350% relative to functional residual capacity in initially healthy piglets) [6], and the associated nonlinear mechanics of strain-stiffening. By contrast, our computational model does not allow intratidal variations in lung mechanics, and we constrained total CO₂ elimination to assess mechanical power during eucapnic ventilation. Our simulations demonstrate the potential for reducing the risk of VILI using MFOV, under the premise that lung injury is associated with mechanical power [6,25]. Reduction of mechanical power during MFOV compared to SFOV was achieved by the addition of higher-frequency components in the ventilator waveform, albeit at the expense of reducing the magnitudes at lower frequency components. Indeed, no SFOV waveform with a frequency contained within the frequency contents of the MFOV-4 and MFOV-5 minimized ${\text{ϕ}}_{\text{MP}}$. Rather, all SFOV waveforms coincided with a local maximum of ${\text{ϕ}}_{\text{MP}}$. The implication of this finding is that heterogeneous distributions of oscillatory flow at any single frequency result in regionally amplified or attenuated acinar mechanical power (Figures 9 and 1). Therefore, large regional flow amplitudes during SFOV result in increased rates of energy dissipation within such regions of amplification. However with MFOV, the application of different simultaneous frequencies with lower amplitudes produce different simultaneous distributions of heterogeneous ventilation. Thus potential amplification regions at one frequency may be complemented by different amplification regions at another frequency, such that delivering both frequencies simultaneously reduces the disproportionate burden of ventilation on either amplification region alone [12,13]. The optimal volume amplitudes depend on not only the relative harm associated with any particular mechanical variable (e.g., strain, strain rate, or mechanical power), but also the regional heterogeneity of that mechanical variable as a function of frequency.

Figure 9.

Illustration of heterogeneous strain and strain rate distributions during oscillatory ventilation for three conceptual lung compartments consisting of different resistive, elastic, and inertial properties. Each lung compartment selective filters the flow delivered at the airway opening in a frequency-dependent manner. Potential for ventilator-induced lung injury within each compartment is assumed to vary by some contribution of strain and strain rate.

The additional high-frequency components delivered simultaneously with a predominantly low-frequency waveform contribute to enhanced gas exchange by a number of transport mechanisms (e.g., turbulent mixing, velocity profile distortions, increased rates of intrapulmonary gas mixing at bifurcations) [15,26,27]. Such high-frequency components may allow for reduction in the low-frequency volume amplitude required to maintain eucapnia. This phenomenon is a key principle in support of high-frequency percussive ventilation, which superimposes high-frequency oscillations over conventional mechanical ventilation [28,29]. High-frequency percussive ventilation and MFOV thus share some similarities with regard to lung protection, although MFOV in principle is not restricted to combinations of one low frequency and one high frequency. More importantly, MFOV is designed to mitigate the effects of mechanical heterogeneity in injured lungs, rather than treating the recruited lung as a single homogeneous entity.

Limitations

The simulations based on our computational model have a number of limitations. Perhaps the most crucial limitation of this model is the lack of intratidal variation in mechanical properties and nonlinear mechanical phenomena. For example, dynamic strain stiffening due to cyclic overdistension and recruitment/derecruitment are common occurrences in mechanically ventilated patients [30,31]. These effects are especially prevalent during mechanical ventilation with large driving pressures at low frequencies, characteristics describing our optimal SFOV and MFOV-5 waveforms. Accounting for such nonlinear phenomena, while still minimizing mechanical power, may produce different optimal waveforms compared to those selected for the linearized model used in this study. Moreover we did not account for any time-varying changes in the mechanical properties of the model for longer durations, as may be seen over the course of hours or days in actual patients with ARDS. Interfacial stresses in partially flooded acini were also not explicitly modeled, yet may contribute to the progression of VILI in a positive feedback loop [32]. The effects of parenchymal interdependence was not accounted for in the model either, apart from the gravitational dependence of acinar stiffness. Such interdependence may result in clustering of so-called “stress concentrators”, which may further contribute to VILI [33–35]. Cardiogenic oscillations were not explicitly modeled in this study, yet may provide an ‘intrinsic’ source of multi-frequency ventilation when combined with normal spontaneous breathing [36]. Depending on subject size, age, and other hemodynamic factors, cardiogenic frequencies may vary between 50 to 180 min⁻¹ (or 0.8 to 3 Hz).

Although mechanical power and energy dissipation into lung parenchymal tissues has been correlated with ventilator-induced lung injury, “safe” thresholds for mechanical power, either total or regional, are not fully understood [6,37]. It is conceivable that the same amount of total mechanical power could be distributed homogeneously with all acini below such an injury threshold, or heterogeneously with some acini above the threshold and some below. Some patients may be so severely injured that eucapnic ventilation is not possible without imposing mechanical power above such an injury threshold, and alternative gas exchange techniques must be considered (e.g., extracorporeal membrane oxygenation). The mechanical power objective was nearly ten-fold greater for the injured lung model compared to the healthy, which is consistent with experimental findings of greater resilience to VILI in healthy subjects compared to subjects with additional injury or inflammation [38–40]. In this study, we do not presume a fixed injury threshold for mechanical power, but rather identify oscillatory waveforms which exhibit potential to minimize mechanical power across 90% of the lung.

Finally, this study used a random search of the parameter space to determine approximations of local minima for each objective function. This approach is unlikely to precisely identify true local minima of a multidimensional objective function, especially those located in regions with large gradient magnitudes. The randomly sampled waveforms producing the lowest objective function values are not guaranteed to be local minima of the objective function. Rather, this approach provides an approximation of potential local minima from densely sampled regions of the parameter space. Furthermore it allows for separate characterization of multiple different objective functions, evaluated densely throughout the parameter space using only a single set of simulation data. Despite these limitations, our computational modeling identified plausible candidates for optimal oscillatory waveforms consisting of nonzero volume amplitudes at more than one ventilatory frequency. Introducing fixed regional flow fluctuations to represent cardiogenic motions and/or introducing nonlinear phenomena contraindicating the use of large pressure fluctuations at low frequencies would likely still result in nontrivial optimal MFOV waveforms, albeit with altered spectral content to reflect altered mechanical power associated with each ventilatory frequency.

Conclusion

Frequency content in multi-frequency oscillatory ventilation waveforms may be tuned to minimize the risk of ventilator-induced lung injury associated with tissue mechanical power. Strain and strain rate can be minimized by single-frequency oscillatory ventilation at either the highest or lowest frequency available, respectively. Although the relative risk of VILI associated with strain compared to strain rate is not fully understood, mechanical power provides a convenient physical variable that encapsulates the contributions of both strain and strain rate to energy dissipation. In this computational modeling study, no single frequency of oscillatory ventilation resulted in a lower mechanical power compared to multi-frequency oscillatory ventilation. Rather, optimal oscillatory ventilator waveforms consisted of nonzero contributions from all frequencies considered in the optimization search, resulting in reductions in mechanical power lower than even the best-case of all single frequencies. Consideration of other MFOV waveforms with different frequency distributions may yield even lower associated risks of VILI, especially in the context of the physiologically-relevant costs of strain, strain rate, or mechanical power.

Glossary of Terms

ARDS

Acute respiratory distress syndrome

CO₂

Carbon dioxide

f

Frequency

f₀

Fundamental frequency

HFOV

High-frequency oscillatory ventilation

H

Tissue elastance

I

Tissue inertance

j

Unit imaginary number

K

Number of frequencies in a multi-frequency waveform

k,l

Frequency indexes

MFOV

Multi-frequency oscillatory ventilation

MFOV-4

MFOV waveform consisting of 5, 10, 15, and 20 Hz oscillations

MFOV-5

MFOV waveform consisting of 0.4, 2, 7, 12, and 26 Hz oscillations

MP

Mechanical power

N

Number of acini

n

Acinus index

P

Pressure

${\stackrel{-}{P}}_{\text{aw}}$

Time-averaged mean airway pressure

${\stackrel{-}{P}}_{\mathrm{t}\mathrm{p},n}$

Time-averaged acinar transpulmonary pressure

${\stackrel{-}{P}}_{\mathrm{p}\mathrm{l},n}$

Time-averaged effective acinar pleural pressure

$\left|P\right|$

Pressure amplitude

$\angle P$

Pressure phase

$\mathrm{\Delta }P$

Driving pressure

$\mathbb{P}$

Instantaneous power

PEEP

Positive end-expiratory pressure

R

Resistance

SFOV

Single-frequency oscillatory ventilation

t

Time

VILI

Ventilator-induced lung injury

V

Volume

$\left|V\right|$

Volume amplitude

$\angle V$

Volume phase

$\stackrel{.}{V}$

Flow

$\left|\dot{V}\right|$

Flow amplitude

$\angle \dot{V}$

Flow phase

${\dot{V}}_{{\mathrm{C}\mathrm{O}}_2}$

Carbon dioxide elimination

V_T

Tidal volume

${\stackrel{-}{V}}_n$

Time-averaged acinar volume

V₀

Minimal acinar volume at zero transpulmonary pressure

${\mathbb{W}}_{\mathrm{D}}$

Dissipative work

Z

Mechanical impedance

α

Exponent of frequency in constant phase model

β

Placeholder variable for derivation of mechanical power

γ

Power law exponent of frequency for flow rate required for eucapnic ventilation

$\mathrm{\epsilon }$

Volumetric strain

$\dot{\mathrm{\epsilon }}$

Volumetric strain rate

η

Tissue hysteresivity

π

Number of radians in a semicircle

${\mathrm{\varphi }}_{\mathrm{M}\mathrm{P}}$

90^th percentile of acinar mechanical power

${\mathrm{\varphi }}_{\mathrm{S}}$

90^th percentile of acinar strain

${\mathrm{\varphi }}_{\mathrm{S}\mathrm{R}}$

90^th percentile of acinar strain rate

${\mathrm{\Psi }}_n$

Injured acinar tissue elastance modifier

Appendix

Distribution of Acinar Tissue Elastance, Mean Volume, and Derecruitment

Transpulmonary pressure was defined by the difference between mean airway pressure ${\overline{P}}_{\text{aw}}$ and the effective time-averaged pleural pressure acting on each acinus ${\overline{P}}_{\text{pl},n}$:

$${\overline{P}}_{\text{tp},n}={\overline{P}}_{\text{aw}}-{\overline{P}}_{\text{pl},n}.$$ (A-1)

Pleural pressure was assumed to vary according to the hydrostatic weight of the lung, such that the mean value of ${\overline{P}}_{\text{pl},n}$ was −5 cmH₂O, and its gradient in the direction of the gravitational field was 0.25 cmH₂O cm⁻¹ [41]. Acinar tissue elastance $H_n$ for the $n^{\text{th}}$ acinus was assumed to be a quadratic function of time-averaged acinar transpulmonary pressure ${\overline{P}}_{\text{tp},n}$:

$$H_n\left({\overline{P}}_{\text{tp},n}\right)=N\cdot \left(0.3258\cdot {\overline{P}}_{\text{tp},n}{}^2-6.6224\cdot {\overline{P}}_{\text{tp},n}+58.3280\cdot {\text{Ψ}}_n\right),$$ (A-2)

where $N$ is the total number of acini, ${\text{Ψ}}_n$ is an injured tissue elastance modifier (i.e., ${\text{Ψ}}_n=1$ for the healthy condition, ${\text{Ψ}}_n\ge 1$ for the injured condition). The coefficients of (A-2) were scaled from values obtained experimentally for canine lungs [42]. Heterogeneous increased injured acinar tissue elastance was simulated by applying a pseudorandom gradient noise distribution to ${\text{Ψ}}_n$ as a function of spatial position [15].

Mean acinar volume ${\overline{V}}_n$ at a given mean airway pressure was determined according to acinar tissue elastance:

$${\overline{V}}_n\left({\overline{P}}_{\text{tp},n}\right)=V_0+\underset{0}{\overset{{\overline{P}}_{\text{tp},n}}{\int }}\frac{P}{H_n\left(P\right)}\text{d}P,$$ (A-3)

where $V_0$ represents the minimal volume of every acinus under zero transpulmonary pressure. Thus $V_0$ was a constant determined by setting the sum of all ${\overline{V}}_n$ to be equal to a specified functional residual capacity at zero ${\overline{P}}_{\text{aw}}$.

Acinar derecruitment was simulated by assigning an infinite magnitude to acinar mechanical impedance, resulting in zero oscillatory flow into the acinus. Derecruitment was determined by a threshold on the injured acinar tissue elastance modifier, i.e., ${\text{Ψ}}_n\ge 8$[19].

Derivation of Acinar Mechanical Power During Multi-Frequency Oscillatory Ventilation

For our lung computational model with $N$ acini, a generalized oscillatory waveform may be constructed with $K$ frequencies, each of which are integer multiples of a fundamental frequency $f_0$. The received flow and transpulmonary pressure within the $n$ th acinus may thus be expressed, respectively, as:

$${\dot{V}}_n\left(t\right)=\sum _{k=1}^K|{\dot{V}}_{n,k}|\sin \left(2\text{π}k{f}_{0}t+\angle {\dot{V}}_{n,k}\right),$$ (A-4)

and

$$P_n\left(t\right)=\sum _{l=1}^K|P_{n,l}|\sin \left(2\text{π}l{f}_{0}t+\angle P_{n,l}\right).$$ (A-5)

where $k$ and $l$ are used to index the discrete frequency harmonics. The instantaneous mechanical power imparted to the $n$th acinus may be defined as:

$${\mathbb{P}}_n\left(t\right)=P_n\left(t\right){\dot{V}}_n\left(t\right).$$ (A-6)

The dissipative energy loss (i.e., work) per the cycle of $1/f_0$ duration may be derived by integrating instantaneous power of Equation (A-6) over time, which results in nonzero values only when ${\dot{V}}_n\left(t\right)$ and $P_n\left(t\right)$ are in phase:

$${\mathbb{W}}_{\text{D},n}=\underset{0}{\overset{1/f_0}{\int }}{\mathbb{P}}_n\left(t\right)\text{d}t,$$ (A-7)

$$=\underset{0}{\overset{1/f_0}{\int }}\left(\sum _{k=1}^K|{\dot{V}}_{n,k}|\sin \left(2\text{π}k{f}_{0}t+\angle {\dot{V}}_{n,k}\right)\right)\left(\sum _{l=1}^K|P_{n,l}|\sin \left(2\text{π}l{f}_{0}t+\angle P_{n,l}\right)\right)\text{d}t,$$ (A-8)

$$=\sum _{k=1}^K\sum _{l=1}^K|{\dot{V}}_{n,k}||P_{n,l}|\left[\underset{0}{\overset{1/f_0}{\int }}\sin \left(2\text{π}k{f}_{0}t+\angle {\dot{V}}_{n,k}\right)\sin \left(2\text{π}l{f}_{0}t+\angle P_{n,l}\right)\text{d}t\right].$$ (A-9)

Since sinusoidal functions with frequencies that are integer multiples are mutually orthogonal, the integral of the product of oscillations at any two frequencies $k{f}_0$ and $l{f}_0$ over a cycle is only nonzero when $k=l$:

$$\underset{0}{\overset{1/f_0}{\int }}\sin \left(2\text{π}k{f}_{0}t+\angle {\dot{V}}_{n,k}\right)\sin \left(2\text{π}l{f}_{0}t+\angle P_{n,l}\right)\text{d}t=\{\begin{array}{ccc}\frac{\cos \left(\angle {\dot{V}}_{n,k}-\angle P_{n,l}\right)}{2f_0}& ,& k=l\\ 0& ,& k\ne l\end{array}.$$ (A-10)

Eliminating the zero terms and consolidating indexes $k$ and $l$ yields:

$${\mathbb{W}}_{\text{D},n}=\sum _{k=1}^K|{\dot{V}}_{n,k}||P_{n,k}|\left[\frac{\cos \left(\angle {\dot{V}}_{n,k}-\angle P_{n,k}\right)}{2f_0}\right].$$ (A-11)

Thus the dissipative work per cycle within each acinus is then the sum of dissipative work at each frequency:

$${\mathbb{W}}_{\text{D},n}=\frac{1}{2f_0}\sum _{k=1}^K|{\dot{V}}_{n,k}||P_{n,k}|\cos \left(\angle {\dot{V}}_{n,k}-\angle P_{n,k}\right).$$ (A-12)

The average rate of energy dissipation over time (i.e., dissipative mechanical power) is then:

$${\text{MP}}_n=f_0{\mathbb{W}}_{\text{D},n},$$ (A-13)

$${\text{MP}}_n=\frac{1}{2}\sum _{k=1}^K|{\dot{V}}_{n,k}||P_{n,k}|\cos \left(\angle {\dot{V}}_{n,k}-\angle P_{n,k}\right).$$ (A-14)

For the case of a single frequency waveform, Equation (A-14) simplifies to:

$${\text{MP}}_n=\frac{1}{2}|{\dot{V}}_n||P_n|\cos \left(\angle {\dot{V}}_n-\angle P_n\right).$$ (A-15)

If $|{\dot{V}}_n|$ is expressed in units of L s⁻¹, and $|P_n|$ is expressed in units of cmH₂O, then the multiplicative conversion factor (10⁻³ m³ L⁻¹)(98.0665 N m⁻² cmH₂O⁻¹)(60 s min⁻¹)(10⁶ μJ J⁻¹) may be used to express ${\mathrm{M}\mathrm{P}}_n$ in units of μJ min^-1.

Mechanical Power in Viscoelastic Tissues with Constant Phase Impedance

For the case of wall distension of a single acinus modeled by a constant phase tissue impedance in series with a tissue inertance term ($I$), the frequency-dependent relationship between pressure and flow is given by:

$$Z\left(f\right)=\frac{P\left(f\right)}{\dot{V}\left(f\right)}=j2\text{π}fI+\frac{\left(\text{η}-j\right)H}{{\left(2\text{π}f\right)}^{\text{α}}}$$ (A-16)

where:

$$\text{α}=\frac{2}{\text{π}}{\tan }^{-1}\left(\frac{1}{\text{η}}\right)$$ (A-17)

The H parameter describes tissue elastance, and η describes tissue hysteresivity. The dissipative component is therefore the real part of impedance:

$$R\left(f\right)=\frac{\text{η}H}{{\left(2\text{π}f\right)}^{\text{α}}}$$ (A-18)

with the magnitude of tissue impedance given by:

$$|Z\left(f\right)|=\frac{|P\left(f\right)|}{|\dot{V}\left(f\right)|}=\sqrt{{\left(\frac{\text{η}H}{{\left(2\text{π}f\right)}^{\text{α}}}\right)}^2+{\left(2\text{π}fI-\frac{H}{{\left(2\text{π}f\right)}^{\text{α}}}\right)}^2}$$ (A-19)

and phase given by:

$$\angle Z\left(f\right)=\angle P\left(f\right)-\angle \dot{V}\left(f\right)={\tan }^{-1}\left(\frac{2\text{π}fI-\frac{H}{{\left(2\text{π}f\right)}^{\text{α}}}}{\text{η}\frac{H}{{\left(2\text{π}f\right)}^{\text{α}}}}\right)$$ (A-20)

Using Equation (A-15), the dissipative mechanical power of acinar tissue distension per MFOV cycle can be expressed as:

$$\text{MP}=\frac{1}{2}\sum _{k=1}^K{|{\dot{V}}_k|}^2\sqrt{{\left(\frac{\text{η}H}{{\left(2\text{π}k{f}_0\right)}^{\text{α}}}\right)}^2+{\left(2\text{π}k{f}_{0}I-\frac{H}{{\left(2\text{π}k{f}_0\right)}^{\text{α}}}\right)}^2}{\text{β}}_k$$ (A-21)

where:

$${\text{β}}_k=\cos \left({\tan }^{-1}\left(\frac{2\text{π}k{f}_{0}I-\frac{H}{{\left(2\text{π}k{f}_0\right)}^{\text{α}}}}{\text{η}\frac{H}{{\left(2\text{π}k{f}_0\right)}^{\text{α}}}}\right)\right)$$ (A-22)

Applying trigonometric identities, (A-22) may be rewritten as:

$${\text{β}}_k={\left(\sqrt{1+{\left(\frac{2\text{π}k{f}_{0}I-\frac{H}{{\left(2\text{π}k{f}_0\right)}^{\text{α}}}}{\text{η}\frac{H}{{\left(2\text{π}k{f}_0\right)}^{\text{α}}}}\right)}^2}\right)}^{-1}$$ (A-23)

Substituting this form of ${\text{β}}_k$ into (A-21) and simplifying yields the result:

$$\text{MP}=\frac{1}{2}\sum _{k=1}^K{|{\dot{V}}_k|}^2\frac{\text{η}H}{{\left(2\text{π}k{f}_0\right)}^{\text{α}}}$$ (A-24)

wherein only the dissipative terms of (A-18) remain, such that (A-24) can be rewritten instead as:

$$\text{MP}=\frac{1}{2}\sum _{k=1}^K{|{\dot{V}}_k|}^{2}R\left(k{f}_0\right)$$ (A-25)

Note that $R\left(f\right)$ decays with increasing frequency as $f^{-\text{α}}$ with $\text{α}<1$. For example, a value of $\mathrm{\eta }=0.20$ corresponds to α$=0.87$. However, during high-frequency oscillatory ventilation, flow magnitudes increase nearly linearly with frequency due to the requirement for eucapnic ventilation [43], i.e., $|\dot{V}|\propto f^{\text{γ}}$ with $\text{γ}\approx 1$. As a result, $\text{MP}\propto f^{{\text{γ}}^2-\text{α}}$ with $\left({\text{γ}}^2-\text{α}\right)>0$. Therefore despite reductions in dissipative tissue damping at high frequencies, mechanical power is expected to increase with increasing frequency during HFOV.