Stress & strain in mechanically nonuniform alveoli using clinical input variables: a simple conceptual model

Clinicians currently monitor pressure and volume at the airway opening, assuming that these observations relate closely to stresses and strains at the micro level. Indeed, this assumption forms the basis of current approaches to lung protective ventilation [1]. Nonetheless, although the airway pressure applied under no-flow conditions may be the same everywhere in healthy lungs, the pressures and stresses within a mechanically non-uniform ARDS lung are not [2, 3]. Similarly, the potential for ‘power’ of ventilation to damage the lung has been acknowledged as highly relevant to the risk of ventilator induced lung injury (VILI), and its components of tidal volume, airway pressure, flow and cycling frequency are easily measured [4, 5]. As currently described for clinical purposes, however, ‘power’ is actually the cumulative energy delivered per minute by repeated tidal cycles that generate the mechanical forces needed to ventilate [5]. Therefore, to better assess ventilator-induced lung injury (VILI) hazard, the total ‘power’ monitored in the ventilator’s circuit needs to be considered in relation to the regional micro stresses (tensions at the alveolar boundary) and micro strains (resulting increments of area) that occur at the local level with each inflation cycle (intra-cycle power) [6]. Moreover, in theory, a refined index of ventilating hazard from measured power would be affected not only by the size of the ventilated compartment (‘baby lung’) [7] and its pressure threshold for injury [8, 9], but also by its proportion of interfaces where stress is focused and amplified between tissues having different receptivity to stretching (‘surface element compliances’) [10] (Fig. 1). A more clinically informative model of VILI risk would therefore include not only total power, as currently defined, but also estimates for the concentrated specific power (power applied to the baby lung [6, 11, 12], stress amplification at the alveolar level [3, 10] and the proportion of the ventilated baby lung experiencing such stress-augmented interfaces.

The term ‘mechanical stress’ describes the distribution of forces exerted in a solid or fluid body being deformed (‘strained’) as a result of external loads [13]. For ventilation, stress is analogous to pressure for forces perpendicular to the surface of contact (compressive, radial, or tensile stress) and strain is the resulting expansion (volume change) relative to the baseline condition. Forces within the plane of the surface where the cells and extracellular matrix reside may be considered ‘hoop stresses’ associated with changes of tension and area [14]. It follows that while pressure is force per unit of area, tension is force per unit of length.

In a simplified mathematical model using the clinically relatable variables of pressure and volume, we previously recognized those geometry-defined differences of force distribution to propose a conceptual shift from pressure to tension and from volume to area when considering the stress–strain changes of tissue energy at the alveolar periphery (the “shell”) [15]. While valid for a uniform alveolus modeled as a hollow, thin-walled sphere in which tension (T) is the product of pressure (P) and sphere radius R: (T = PR/2), tidal forces distribute stress and strain unevenly in diseases such as ARDS. Stresses in that setting are focused and amplified at the interface between different surface elements, such as those created by contiguous flexible and less flexible units. The ratio of these interfacial tensions is a ratio of forces, which can be viewed as a numerical indicator of stress amplification, a potential contributor to risk for damage (Fig. 2).

On the basis of volume differences observed in the histology of healthy lung, Jere Mead and colleagues accurately reported a 10:1 relationship between the dimensions (volumes) of an alveolus fully distended by 30 cmH₂O and one that is completely collapsed [10]. According to their estimates, the corresponding amplification of the pressure stress at the boundary of open and atelectatic units at that extreme might be fourfold the measured pressure value. Although it is tempting to translate such modeling calculations directly to the clinical problem of approximating stress and strain imposed on the lung by mechanical ventilation, to our knowledge, such a mathematical representation of heterogeneous stresses using principles and variables familiar to clinicians is not currently available.

The primary purpose of the present thought exercise is to extend our uniform spherical model [15] to characterize the amplifications of stress (tension) and strain (area change) that occur at boundaries between a sphere’s surface elements that have differing compliances. As examples, these may be caused by local external features such as atelectasis, micro thrombosis, consolidation, edema, or fibrosis in other contiguous tissue adjacent to the spherical unit in question. In describing such a non-uniform spherical unit, we make several key assumptions: time invariant (static) conditions, open architecture not subject to tidal re-opening/closure, linear pressure–volume relationships, and unchanging shape morphology during expansion.

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.

In principle, estimating actual tissue stresses and strains that occur in a mechanically non-uniform environment should account for factors beyond the measurements from the ventilator circuit of airway pressures, tidal volume, and mechanical power (Fig. 4). A first step for the clinician requires consideration of lung unit tension, power concentration, and stress focusing. With reasonable approximations, better understanding of the value and limitations of presently used general guidelines for lung protection may eventually be developed for the individual patient from clinical inputs recognized and measured by the bedside caregiver. Although only a conceptual first step, such modeling may help understand what eventually might constitute a true ‘lung protective’ approach to ventilation.