ICU Physiology in 1000 Words: Shorthand Equations for Respiratory System Power
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
Figure 1: The original mechanical power equation - see text for details. PEEP is positive end-expiratory pressure; delta V is tidal volume; Pr is pressure of resistance, EL is lung elastance, Pplat is plateau pressure, Ppeak is peak pressure; I:E is inspiratory to expiratory ratio, RR is respiratory rate, Rrs is resistance of respiratory system. This is analogous to a ventilator breath turned on its side - see vodcast here.
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].
Figure 2: Analogous to figure 1. TV is tidal volume, Pr is pressure of resistance, Pdrive is driving pressure, Pplat is plateau pressure, Ppeak is peak pressure. RS is respiratory system
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.
Figure 3: Analogous to figures 1 and 2. See text for details and previous figures for abbreviations. VE is minute ventilation; F is flow in L/min. The three triangles are labeled A, B and C.
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
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