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This synopsis will address the basic geometric calculation of respiratory system work for a single breath [in joules] and power [i.e. joules/seconds or watts] which is work multiplied by the respiratory rate. This geometric approach will then be used to explain two shorthand equations – the second of which was recently studied in a cohort of mechanically-ventilated, critically-ill patients.
Basic Geometry of Work
As described in more detail here, the work for a single breath is the area between the pressure-volume curve. Of note, when a patient is passive with the ventilator, in volume control with a square-wave flow delivery, every pressure-time waveform illustrates work; these physics are also described in a vodcast accompanying this post on asthmatic mechanics.
Figure 1 below illustrates that the work for a single breath may be approximated by summing the area of three shapes nestled within the breath:
1.) a triangle (in red) representing the elastic work above positive end expiratory pressure [PEEP]
2.) a parallelogram (in blue) representing dynamic work [e.g. gas flow, moving tissues]
3.) a rectangle (in pink) representing the elastic work of PEEP
When these three geometric figures are added, the rather foreboding equation within the braces at the bottom of figure 1 results. When work [in joules, within the braces] is multiplied by the respiratory rate [outside of the braces], then work over time – or power – is obtained. The 0.098 converts [L x cm H2O] to joules [1, 2].
Shorthand Equation 1
While the equation in figure 1 most accurately measures power applied to the respiratory system, it is clearly cumbersome to calculate at the bedside. Additionally, it requires measuring both the resistance and elastance [i.e. stiffness] of the respiratory system. For this reason, Gattinoni and colleagues proposed a more clinically-friendly equation  [see figure 2].
Here the same geometries are carried from figure 1, however, this approach employs a ‘method of subtraction’ to calculate the breath work. As tidal volume [TV, y-axis] is outside of the brackets in the equation, it is multiplied first with peak pressure [Ppeak] and this gives the area of the entire rectangle bounded by TV and Ppeak [i.e. width x length]. However, when TV is multiplied by the second term within the brackets [highlighted in purple], the area of the purple triangle is given. The base of the purple triangle is geometrically-equivalent to the plateau pressure less the PEEP [i.e. the driving pressure, Pdrive] and the height of the purple triangle is the VT, outside of the brackets. This is divided by 2 to calculate the area of the triangle [i.e. ½ base x height]. Accordingly, this subtraction method then gives the same area as highlighted in figure 1 which is multiplied by respiratory rate and the conversion factor 0.098 to obtain power.
Shorthand Equation 2
While more clinically-intuitive, shorthand equation 1 still requires a ventilator-maneuver; that is, an inspiratory hold to parse out the plateau pressure from the peak pressure. In a recent investigation, Gattinoni’s group proposed an equation that requires no hold maneuver ; thus, real-time power may be displayed continuously.
Figure 3 shows that this approach essentially requires summing 3 separate triangles to obtain the breath’s work. The height of each triangle is tidal volume [TV, y-axis]; consequently, TV is outside of the parentheses and distributed to 3 separate terms within the parentheses – the bases of the 3 triangles. The base of the right triangle [A] is the peak pressure [Ppeak]; the base of the obtuse scalene triangle [B] is PEEP and the base of the obtuse, scalene triangle [C] is ventilatory flow [F] divided by 6.
What is the origin of flow/6? This is the pressure due to resistance [Pr] and requires an assumption in the absence of an inspiratory hold. The assumption is that the resistance is the mean value of resistance in ventilated patients [i.e. 10 cmH2O∙sec/liters] or 1/6 cmH2O∙min/liters ; accordingly, multiplying this assumption by the flow [F, liters/min] will give the estimated Pr [base of C – see figure 3].
Finally, the denominator – 20 – is a constant resulting from ½ [from the area of the triangles] multiplied by the conversion factor 0.098.
Shorthand equation 2 was recently validated in 200 ventilated, ICU patients from 7 previously published trials . These patients had the mechanical power of the respiratory system calculated by the initial method [figure 1] at PEEP values of 5 and 15 cm H2O. At both PEEP levels the R2 comparing shorthand equation 2 to the more complicated equation was quite excellent – 0.98 and 0.97, respectively. The shorthand equation slightly underestimated the power of the respiratory system by 0.5 – 1.35 J/min.
Caveats and Clinical Implications
Importantly, to presume the aforementioned geometries, patients should be passive [i.e. no respiratory muscles contributing to the pressure waveform] and with constant-flow [i.e. square wave] delivery in volume control. Currently, the power threshold for ventilator induced lung injury [VILI] in humans is unknown. Indeed, Marini recently stated that he does not target a certain power . Nevertheless, the VILI threshold for porcine lungs is 12 J/min. Given that human lungs have twice the specific elastance of porcine lungs , a rough, extrapolated threshold is 24 J/min. Critically, these thresholds represent power across the lung only; the equations above calculate the power across the respiratory system [i.e. lungs and chest wall together].
The power across the lung itself is approximated by multiplying the power across the respiratory system by the fraction of lung stiffness to total respiratory system stiffness [i.e. the El/Ers ratio]. Normally, this is 0.5; in pulmonary ARDS, the ratio can rise to 0.8 .
Given that the range of mechanical power of the respiratory system in 95% of the ICU patients reported by Serpa Neto et al. was 11.7–31.2 J/min  – the estimated power across normal lungs [El/Ers of 0.5] in this cohort would be 5.85 – 15.6 J/min. If we assumed this cohort to have significant pulmonary ARDS [El/Ers of 0.8] then the estimated trans-pulmonary power would be 9.36 – 25 J/min.
Try equation 2 for yourself on a ventilated patient and then estimate the power across the lungs!
Dr. Kenny is the cofounder and Chief Medical Officer of Flosonics Medical; he also the creator and author of a free hemodynamic curriculum at heart-lung.org
- Gattinoni L, Tonetti T, Cressoni M et al: Ventilator-related causes of lung injury: the mechanical power. Intensive Care Med 2016, 42(10):1567-1575.
- Cressoni M, Gotti M, Chiurazzi C et al: Mechanical power and development of ventilator-induced lung injury. The Journal of the American Society of Anesthesiologists 2016.
- Giosa L, Busana M, Pasticci I et al: Mechanical power at a glance: a simple surrogate for volume-controlled ventilation. Intensive care medicine experimental 2019, 7(1):61.
- Marini JJ: How I optimize power to avoid VILI. Critical Care 2019, 23(1):326.
- Protti A, Cressoni M, Santini A et al: Lung stress and strain during mechanical ventilation: any safe threshold? American journal of respiratory and critical care medicine 2011, 183(10):1354-1362.
- Gattinoni L, Marini JJ, Collino F et al: The future of mechanical ventilation: lessons from the present and the past. Critical Care 2017, 21(1):183.
- Neto AS, Deliberato RO, Johnson AE et al: Mechanical power of ventilation is associated with mortality in critically ill patients: an analysis of patients in two observational cohorts. Intensive care medicine 2018, 44(11):1914-1922.