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Stress & strain in mechanically nonuniform alveoli using clinical input variables: a simple conceptual model
Critical Care volume 28, Article number: 141 (2024)
Abstract
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. Nonetheless, although the airway pressure applied under static conditions may be the same everywhere in healthy lungs, the stresses within a mechanically nonuniform ARDS lung are not. Estimating actual tissue stresses and strains that occur in a mechanically nonuniform environment must account for factors beyond the measurements from the ventilator circuit of airway pressures, tidal volume, and total mechanical power. A first conceptual step for the clinician to better define the VILI hazard requires consideration of lung unit tension, stress focusing, and intracycle power concentration. With reasonable approximations, better understanding of the value and limitations of presently used general guidelines for lung protection may eventually be developed from clinical inputs measured by the caregiver. The primary purpose of the present thought exercise is to extend our published model of a uniform, spherical lung unit to characterize the amplifications of stress (tension) and strain (area change) that occur under static conditions at interface boundaries between a sphere’s surface segments having differing compliances. Together with measurable ventilating power, these are incorporated into our perspective of VILI risk. 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.
Background and main text
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 noflow conditions may be the same everywhere in healthy lungs, the pressures and stresses within a mechanically nonuniform 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 ventilatorinduced 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 (intracycle 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 stressaugmented 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 geometrydefined 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, thinwalled 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 cmH_{2}O 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 nonuniform spherical unit, we make several key assumptions: time invariant (static) conditions, open architecture not subject to tidal reopening/closure, linear pressure–volume relationships, and unchanging shape morphology during expansion.
Finally, as another conceptual step toward better defining the VILI hazard under dynamic conditions, we also describe the relevant place of specific elastic power (as opposed to total power measured in the external circuit) in generating such damaging forces within the baby lung.
Model using a mechanically ventilated nonuniform sphere to quantify focused stress & strain
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 cofactors of measures of stress, strain, tension, and energy. This approach assumes that the discontinuous interface occurs where a less flexible region of reduced compliance ‘C_{2}’ meets a region of similar baseline dimension within the surface (‘shell’) of a larger, more flexible sphere having compliance ‘C_{1}’. The volume of the nonuniform sphere is designated V_{1}. 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, arclike surface element would naturally form part of the shell of a smaller uniform sphere having volume V_{2} 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 nonuniform 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), ∆V_{1}/∆V_{2} = C_{1}/C_{2}. 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 C_{1}/C_{2} 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, R_{1} and R_{2}. In a sphere of any dimension, volume equals 4/3 π R^{3} and area is 4 π R^{2}. 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 online Additional file 1, the interface stress amplifier then would be estimated as (∆V_{1}/∆V_{2})^{1/3} = (C_{1}/C_{2})^{1/3} and the strain multiplier as (C_{1}/C_{2})^{2/3}. Importantly, if we concentrate on the elastic energy input to this nonuniform shell, ∆(P×V) = ∆(T×A), the tension (stress) multiplier (C_{1}/C_{2})^{1/3} would be the cofactor of the area (strain) multiplier: (C_{1}/C_{2})^{2/3}. The product of the tension and area multipliers (stress and strain ratios) is the ratio of elastic energies stored in the flexible (E_{1}) and less flexible (E_{2}) regions of the shell interface: E_{1}/E_{2} = [(C_{1}/C_{2})^{1/3} × (C_{1}/C_{2})^{2/3})] = C_{1}/C_{2}. Appropriately, this estimate agrees with ‘P×V determined’ stored energies relating to their compliancedefined volumes. The ratio C_{1}/C_{2} (equivalent to V_{1}/V_{2}) 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 nonuniform sphere, these compliancedriven expressions are the multipliers that cause stress and strain to focus where different surface elements interface.
A multicomponent hazard index of VILI & ‘damaging’ power
Energy and power component
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 C_{obser}/C_{pred}, where C_{pred} is the patient’s predicted compliance value when healthy [16, 17], and C_{obser} 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 C_{pred}/C_{obser}. 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.
Stress and strain component
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 highrisk 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 multicomponent 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).
Proportion of high stress interfaces component
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 (Cc_{O2}Ca_{O2})/(Cc_{O2}Cv_{O2}), where Cc_{O2} Ca_{O2} and Cv_{O2} are O_{2} 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 nonaerated 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 remeasuring 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: [(Cc_{O2}Ca_{O2})/(Cc_{O2}Cv_{O2}) – (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: (Cc_{O2}Ca_{O2})/(Cc_{O2}Cv_{O2}) [see Additional file 1 example].
Limitations
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, multipart 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, thinwalled 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 highrisk interfaces from gas exchange measurements. Importantly, the assumption of quasinormal 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.
Conclusion
In principle, estimating actual tissue stresses and strains that occur in a mechanically nonuniform 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.
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Abbreviations
 A:

Area
 C:

Tidal lung unit compliance
 C_{obs} :

Observed compliance
 C_{pred} :

Predicted (normal) compliance
 C_{1} :

Compliance of the more flexible interfacial surface
 C_{2} :

Compliance of the opposing less flexible interfacial surface
 Ca_{O2} :

Oxygen content of systemic arterial blood
 Cv_{O2} :

Oxygen content of mixed venous (systemic) blood
 Cc_{O2} :

Oxygen content of capillary blood in normally ventilated and perfused lung units
 P:

Transpulmonary pressure
 R:

Sphere radius
 T:

Tension of an alveolar surface
 V:

Lung unit volume
 V/Q:

Ventilation to perfusion ratio
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PRMR is supported by the Brazilian Council for Scientific and Technological Development (CNPq) 408124/20210 and the Rio de Janeiro, State Research Foundation (FAPERJ) E26/010.001488/2019. No further institutional or external funding.
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JJM conceived the project, collaborated in manuscript development, and supervised the varied phases of this work. PRMR provided valued intellectual input and aided in manuscript development. LTT was closely involved with the development of the project and manuscript at all stages. PC worked closely with JJM in developing the core concepts, provided vital intellectual input and assured the accuracy of its predictive equations. All authors reviewed and agreed with the final version of this manuscript.
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Supplementary Information
Additional file 1:
Derivation of Amplifiers.
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Marini, J.J., Rocco, P.R.M., Thornton, L.T. et al. Stress & strain in mechanically nonuniform alveoli using clinical input variables: a simple conceptual model. Crit Care 28, 141 (2024). https://0doiorg.brum.beds.ac.uk/10.1186/s1305402404918y
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DOI: https://0doiorg.brum.beds.ac.uk/10.1186/s1305402404918y