To model the clinical hazard to a mechanically heterogeneous environment more realistically requires modification of the ‘uniform sphere’ model we previously derived [15] by applying ‘amplification multipliers’ to its expressions of tension (the product of pressure and sphere radius) and surface area change that characterize peripheral stress and strain, respectively. Mathematically, these amplification ‘multipliers’ are the multiplicative co-factors of measures of stress, strain, tension, and energy. This approach assumes that the discontinuous interface occurs where a less flexible region of reduced compliance ‘C2’ meets a region of similar baseline dimension within the surface (‘shell’) of a larger, more flexible sphere having compliance ‘C1’. The volume of the non-uniform sphere is designated V1. Both regions (surface elements) are exposed to the same applied pressure difference but because of their differing compliances experience different tensions (stress) and area expansions (strain) at their interface (Fig. 2). Theoretically, the less flexible, arc-like surface element would naturally form part of the shell of a smaller uniform sphere having volume V2 were that same pressure applied to it in isolation (Fig. 2). The interface may be an arc segment of any length and differing flexibility which is incorporated into the sphere’s ‘shell’ (fused at the intersection). In other words, the segment has a different incremental expansion response to gas pressure than the remainder of the shell but does not form part of a separate (smaller) sphere. It follows that the surface area of that less flexible surface element embedded in the non-uniform sphere would also expand less (and experience greater tension) than a corresponding area of the shell that surrounds it in response to the same applied pressure difference, ∆P. These assumptions ignore deformation of either spherical shape resulting from stretch above relaxed volume. Because tidal compliance is (∆V/∆P), ∆V1/∆V2 = C1/C2. Note that these dissimilar surface elements undergo different amounts of stretch in response to the pressure increment, generate shear stresses at their interface, and store different amounts of elastic energy in the same C1/C2 ratio that applies to volume.

Estimates of how those different elastic energies within the sphere are partitioned into tension (stress) and area (strain) requires estimates of their respective radii, R1 and R2. In a sphere of any dimension, volume equals 4/3 π R3 and area is 4 π R2. Consequently, if considered independently from one another at the same static pressure, both the local tensions and areas of the disparate surface elements can be derived from knowledge of the radii of their respective volumes and compliances (Fig. 2). As detailed in the on-line Additional file 1, the interface stress amplifier then would be estimated as (∆V1/∆V2)1/3 = (C1/C2)1/3 and the strain multiplier as (C1/C2)2/3. Importantly, if we concentrate on the elastic energy input to this non-uniform shell, ∆(P×V) = ∆(T×A), the tension (stress) multiplier (C1/C2)1/3 would be the co-factor of the area (strain) multiplier: (C1/C2)2/3. The product of the tension and area multipliers (stress and strain ratios) is the ratio of elastic energies stored in the flexible (E1) and less flexible (E2) regions of the shell interface: E1/E2 = [(C1/C2)1/3 × (C1/C2)2/3)] = C1/C2. Appropriately, this estimate agrees with ‘P×V determined’ stored energies relating to their compliance-defined volumes. The ratio C1/C2 (equivalent to V1/V2) is a key input to our model’s estimates of amplifiers of stress and strain. Therefore, as a first approximation, its possible range would span the tenfold volume range found histologically by Mead and colleagues [10]. Our conceptual hypothesis can be stated: In a mechanically non-uniform sphere, these compliance-driven expressions are the multipliers that cause stress and strain to focus where different surface elements interface.

To assess its actual VILI hazard, total power—currently defined for clinical purposes as the product of frequency and inflation energy per cycle—needs to be considered in the context of its relation to the micro stresses (tensions) and micro strains (area increments) occurring at the local level. For ARDS, such a VILI hazard would be affected by the relative size of the ventilated ‘baby’ lung, as its reduced aerating capacity concentrates the measured ventilating power. Such concentration may deliver damaging energy beyond the pressure threshold in the form of amplified surface stress and strain [7, 10, 12]. Therefore, viewed selectively from the standpoint of measurable damaging energy that generates intolerable stretch, the concerns are power, critical pressure threshold, and baby lung size. Conceptually, the relative size of the baby lung is reflected by Cobser/Cpred, where Cpred is the patient’s predicted compliance value when healthy [16, 17], and Cobser is the value actually observed during ventilation at ‘optimized’ PEEP [18]. A common convention is to assume that compliance (∆V/∆P) relates more closely to the number of open units than to stiffness of individual units. If so, the ‘relative risk factor’ of power concentration for a given patient’s lung is Cpred/Cobser. Although not commonly measured, the relative proportion of tidal volume compared to actual inspiratory capacity might yield a complementary and measurable estimate of the relative size of the baby lung.

From the selective perspective of the individual lung unit where stress and strain that arise from intracycle elastic energy may cross the damaging threshold, however, the factors to consider are inflation pressure, tension, stress focusing, and prevalence of interfaces in the ventilated environment. Building an indicator of damaging local stress in stepwise fashion, starting from measured pressure would require estimates for (1) alveolar tension (as described in our prior ‘uniform sphere’ model [15]); (2) interfacial focusing/amplification; and (3) proportion of high-risk ventilated interfaces within the baby lung. For a given baby lung of any size, the proportion of interfaces that experience amplification might range from negligible in an ARDS lung in which an uninterrupted block of ventilated units with normal compliance is completely separated from those that are unventilated, to a fraction that is influenced by the number of refractory units evenly (diffusely) distributed among open ones (Fig. 3). The formulae modeling these stress and power elements of the multi-component hazard risk are summarized in Table 1. Note that an intervention might simultaneously influence one or more of these key hazard components in a direction that opposes the others. For example, raising PEEP might help by recruiting unstable units to increase their number in the ventilated baby lung even as PEEP adversely raises alveolar tension and amplifies interfacial stress (Examples are provided in the Additional file 1).

Two extremes of interface distribution. Left: Complete regional separation of normal from abnormal ventilation (minimal interfacing) versus complete dispersal (maximized interfacing) of ventilating abnormalities among the normally ventilating units of the baby lung (stippled). Right: Illustration of how gas exchange information from venous admixture comprised of true shunt and low V/Q units might help to estimate the proportion of high-risk interfaces within the baby lung. Together, the normal and abnormal units within the ventilated lung (Cobs/Cpred) comprise the indicated non-shunt fraction of total predicted lung volume

While acknowledging the difficulty of such an approximation, we suggest that standard gas exchange formulae that involve measurable variables might help determine the proportion of the aerated baby lung at greatest risk for the interfacial stress amplification discussed earlier (Fig. 3). The venous admixture that gives rise to hypoxemia is generated by both true shunt and open but inadequately ventilated (low V/Q units). Venous admixture is calculated as (CcO2-CaO2)/(CcO2-CvO2), where CcO2 CaO2 and CvO2 are O2 contents of pulmonary capillary, systemic arterial and mixed central venous blood, respectively at a given fraction of inspired oxygen [19]. For our modeling purpose we hypothesize that the proportion of non-aerated units is the ‘true shunt’ fraction and thus exempt from inflation injury, as opposed to the fraction of low V/Q units at greater risk for interfacial stress amplification during tidal expansion. The ‘true shunt’ fraction is traditionally estimated at bedside by re-measuring venous admixture after administering pure inspired oxygen, thereby eliminating any hypoxic contribution from poorly ventilated units [19]. Theoretically, the proportion of the aerated baby lung that has interface exposure would then be: [(CcO2-CaO2)/(CcO2-CvO2) – (true shunt fraction)]. Alternatively, though less appealing but more simply at the bedside, one might assume all ventilated alveoli that comprise the baby lung are normally perfused, and all others (both shunt and low V/Q) are not. The latter are abnormal units and therefore points of mechanical heterogeneity that are scattered evenly and diffusely throughout the ventilated space. The proportion of stress focusing interfaces within the ventilated baby lung would then be simply: (CcO2-CaO2)/(CcO2-CvO2) [see Additional file 1 example].

This conceptual exercise brings to light how variables that are seldom considered by the clinician but are both recognizable and measurable might help gauge the hazard for VILI of applied pressure and power. To our knowledge this simplified, multi-part model is the first attempt to do so for the caregiver who manages the mechanically heterogeneous environment of injured lungs (ARDS). However, we understand and strongly emphasize that our assumptions and modeling are neither precise descriptors of micromechanics nor intended for immediate clinical use. Quite obviously, they have limited correspondence with the complex geometry that characterizes the actual biological environment of the injured lung. Our highly simplified approximations consider only static elastic forces, ignore dynamics and local differences of transpulmonary pressure, and depend on multiple assumptions. For example, thin-walled spheres are assumed in order to apply the LaPlace formula to the interface between surface elements [20]. However, neither the whole lung nor its constituent units are spheres exposed to a single transpulmonary pressure; biological lung unit contours are both irregular and interdependent, with variable topography of corners and interfaces. Moreover, while the transpulmonary pressures and relative compliances of each surface element of an interface are likely to lie within known ranges [10, 21], in actuality, these vary in accordance with their gravitational positions within the lung and immediate local environments. Estimating baby lung size indirectly from respiratory system compliance is clearly another approximation, as is calculating the proportion of the aerated baby lung with high-risk interfaces from gas exchange measurements. Importantly, the assumption of quasi-normal specific compliance of all aerated baby lung subunits [7], while perhaps reasonable in the first edematous stage of ARDS, may not apply in the later stages of organizing ARDS.