Power Distribution Model

[Laila Berri]

Theprimary findings of this model/simulation indicate that: (1) the strain and strain rate experienced within a modeled compartment are nonlinear functions of delivered energy and power, respectively; (2) for a given combination of flow profile and tidal volume, resting compartmental volumes influence their resulting maximal strains in response to breath delivery; (3) flow profile is a key determinant of the maximal strain as well as maximal strain rate experienced within a multi-compartment lung. By implication, different clinician-selected flow profiles not only influence the timing of power delivery, but also spatially distribute the attendant strains of expansion among compartments with diverse mechanical properties. Importantly, the contours and magnitudes of the compartmental ICP, strain, and strain rate curves are not congruent; strain and strain rate do not necessarily follow the compartmental ICP, and the hierarchy of amplitudes among compartments for these variables may not coincide.

Different flow patterns impact how strain and strain rate develop as compartmental volume crests to its final value. Notably, as inflation proceeds, strain rate may rise or fall even as total strain, a monotonic function of volume, steadily (and predictably) rises. Which flow pattern serves best to minimize the maximal strain rate and VILI risk experienced within any sector, therefore, may strongly depend on the nature and heterogeneity of the mechanical properties of the injured lung.

Potentially damaging tidal energy and power distribute asymmetrically within the heterogeneous environment of the acutely injured lung. Clinician-selected flow profiles not only influence the timing of power delivery, but also spatially distribute the attendant strains of expansion.

Schematic diagram of the 5-compartment model. Proximal shared resistances (R) are designated numerically; terminal, compartment-relevant resistances (R) and compliances (C) are designated alphabetically

A mathematical model for the 5-compartment lung system was constructed using pressure and flow balances. (The mathematical details are provided in the Additional file 1). The model incorporates diverse compartmental resistances and compliances along with the resistances of the rigid connectors of those compartments to the airway opening (Table 1). The mathematical model is a system of linear differential equations for the compartmental volumes during inspiration \(\left(0\le t\le t_i\right)\) and expiration \(\left(t_i\le t\le t_\mathrmtot\right)\) that allows pressure-controlled or flow-controlled ventilation at the airway opening. The solutions of the model yield dynamic compartmental volumes and flows during inspiration and expiration, along with end-expiratory (residual) pressures.

Compartmental volumes for a single complete tidal cycle of the same volume and inspiratory times during constant inspiratory flow (CF, A) and constant inspiratory pressure (CP, B). Note the variations among compartments in end-inspiratory volumes, accompanied by variations in deflation curvature

Here we extend the concept of intracycle power (ICP) for a single-compartment model [2, 3] to this 5-compartment model. Our ultimate goal is to connect the intracycle power applied to each compartment to the strain dynamics experienced by that compartment during inflation. To consider tissue strain we focus on total intracycle elastic power \(\left(\mathrmICP_\mathrmelastic\right),\) which we define mathematically as:

where \(V(t)\) is the instantaneous volume in the compartment, \(Q(t)\) is the instantaneous flow rate into the compartment, \(C\) is the compartmental compliance, and \(P_\mathrmex\) is the residual pressure in the compartment at the end of expiration (total PEEP). The volumes, flows, and residual pressures are readily available from the mathematical model and depend on the mode of ventilation applied to the 5-compartment configuration (A through E). These compartments vary in their ranges of compliance and resistance, with A having the highest compliance, D the highest resistance, and the remaining compartments having mixed, intermediate values for resistance and compliance. Each compartment deflates passively at a rate largely governed by its own time constant. To illustrate how \(\mathrmICP_\mathrmelastic(t)\) varies from compartment to compartment and differs with mode of ventilation, we show this intercompartmental variation for three forms of controlled flow ventilation and for constant pressure ventilation, using the same parameters that were chosen for the prior compartment volume simulations (Fig. 3).

Comparison of inspiratory compartmental intracycle elastic power (ICPelastic) for all labeled modes of ventilation: constant inspiratory flow (CF), decelerating inspiratory flow (DF), constant inspiratory pressure (CP), sinusoidal inspiratory flow (SF)

To relate ICP to VILI, we propose a simple model that relates compartmental strain at time t (and strain rate, \(\mathrmstrain^^\prime\left(t\right)\)) to compartmental ICP. The model assumes that the compartments are inflatable uniform spheres with strain related to the ratio of volume at time t to rest (unstressed) volume at inflation onset (Vrest). Note that in this macro-level model each of the five compartments is assigned the same initial space when unstressed, independently of the compliance parameter that characterizes it. We show in the Additional file 1 that the dynamic strain and the rate of strain at time t on the walls of the compartments are influenced by delivered elastic energy at time t, A(t), and by resting compartment volume subjected to zero pressure:

We now investigate the effects of PEEP on the dynamic strain in each compartment. In particular, we calculate the level and rate of strain for each compartment for a spectrum of applied PEEPs: \(0\le \mathrmPEEP\le 12.75 \mathrmcm \mathrmH_2\mathrmO\). We repeat these calculations for each of the four modes of ventilation using the same compartmental parameters and ventilator settings and plot the maximum strain for each compartment against PEEP settings (Fig. 5). As illustrated, while maximal strain predictably rises with PEEP in each compartment, its influence on maximal strain varies between them in all modes and distributes differently for pressure as opposed to flow-regulated ventilation.

In ventilating a mechanically heterogeneous system the clinician seeks settings, e.g., mode of ventilation, frequency, tidal volume, inspiratory time that achieve the ventilation goal while minimizing risk for damage to the lung. In the Additional file 1, we carry out in silico experiments over wide ranges of maximal flow and tidal volumes to determine which modes of ventilation minimize the maximum compartmental strain for different tidal volumes and maximum flow settings on the ventilator. The latter is of particular interest, as crossing a strain threshold is theoretically needed before energy and power become damaging [3, 4, 9]. As expected, strain rises monotonically with VT in each compartment. However, for DF and SF ventilation, we detect some nonlinearity in the behavior of the maximum strain as functions of set tidal volume and set maximum flow, rising more rapidly at higher tidal volumes for a specified \(Q_\mathrmmax\). This behavior is primarily due to the required increase in inspiratory time, which in turn increases the auto-PEEP, a component of the dynamic strain. Finally, there are variations in the maximum compartmental strains encountered with a given tidal volume according to which mode of ventilation is used.

As expected, all maximal compartmental strains increase as the rest volumes decrease. However, how much the maximal compartmental strains change as function of Vrest depends on the R & C parameters of the 5-compartment model and on the clinician-set ventilator parameters, as well. The same type of graph was created with decelerating flow ventilation and sinusoidal flow ventilation (not illustrated). There were very few qualitative or quantitative differences between them and Fig. 6A and B depicting CF and CP.

The above simulation can be adapted for calculating the distribution of maximal compartmental strains when compartments are assigned different rest volumes. For example, if we set different resting compartmental volumes for compartments with high (e.g., A) and low (e.g., D) compliances (Table 1), then the distribution of compartmental maximum strains varies, as well (Fig. 6C). In the illustrated case of CF ventilation, Compartment A encounters the highest maximal strain and Compartment D the lowest for these compartmental rest volumes, using our standard R and C parameter settings that are designated in Table 1.

The primary findings of this model/simulation indicate that: (1) the strain and strain rate experienced within a modeled compartment are nonlinear functions of delivered energy and power, respectively; (2) for a given combination of flow profile and tidal volume, resting compartmental volumes influence their resulting maximal strains in response to breath delivery; (3) flow profile is a key determinant of the maximal strain as well as maximal strain rate experienced within a multi-compartment lung. By implication, different clinician-selected flow profiles not only influence the timing of power delivery, but also spatially distribute the attendant strains of expansion among compartments with diverse mechanical properties.

In recent modeling we considered the lung as a single compartment with uniform resistance and compliance characteristics; nonetheless, the maximum ICP value and the energy delivered per breath above an arbitrarily designated pressure threshold (set for the entire lung) varied considerably with the selected flow profile [3, 4]. Our current work extends that line of investigation by showing how ICP and strain might distribute spatially over time in the mechanically heterogeneous environment of the acutely injured lung. To our knowledge this study represents the first attempt to develop and use a multi-compartment mathematical model focused on regional power and strain primed with clinically familiar input variables.

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