The respiratory system (rs) static compliance (Crs) and airway resistance (Rrs) are calculated during volume-controlled (VC) mechanical ventilation with a breath-hold maneuver at the end of quiet inspiration (static method) [1]. Under these conditions Crs = Vtidal /(Pplateau - PEEPa), where Vtidal = tidal volume, Pplateau = breath-hold Paw, and PEEPa = applied positive end expiratory pressure. Similarly, Rrs = (Ppeak - Pplateau)/Faw, where Ppeak = peak inspiratory pressure and Faw is airway flow measured just prior to breath-holding [2].
Conversely, estimating Crs and Rrs under airflow conditions is challenging. One approach is the multiple least squares fit (LSF) technique [3, 4], where measures of Paw, Faw, and lung volume change (ΔV) are fitted to the equation of motion of the respiratory system. Another is the expiratory time constant τe method [5] where equations for Crs and Rrs are developed assuming mono-exponential lung volume release [6]. Both methods require the use of complex computational techniques and absent respiratory muscle effort.
Described is a method to calculate Crs and Rrs during insufflation in the presence of airflow (dynamic method) that combines frequency analysis of the airway signals with a novel numerical solution of the equation of motion. The method was validated experimentally with a one-liter test lung and also clinically using previously acquired Faw and Paw signal data from patients on VC ventilation displaying end-inspiratory holds.
Theoretical development. The respiratory system time dependent equation of motion is [7]:
$${{{{P}_{aw}\left(t\right)=P}_{mus}\left(t\right)+ P}_{vent}\left(t\right)=\frac{\varDelta V\left(t\right)}{{C}_{rs}}+ {R}_{rs}{F}_{aw}\left(t\right)+ I\frac{{d}^{2}V\left(t\right)}{d{t}^{2}}+ PEEP}_{a}+ {PEEP}_{i}$$
1
where the measured airway pressure Paw(t) is the sum of the ventilator and respiratory muscles applied pressures, Pvent(t) and Pmus(t), respectively. Opposing them are the elastic, resistive, and inertial components of the respiratory system. V(t) represents time dependent lung volume; ΔV(t) is the insufflation lung volume at time t, equal to \({\int }_{0}^{t}{F}_{aw}\left(t\right)dt\); I is the respiratory system inertia; and PEEPi the intrinsic PEEP [8].
Assuming passive insufflation (Pmus = 0), negligible PEEPI , and ignoring the effect of the inertia term [9], Eq. (1) becomes:
$${P}_{aw}\left(t\right)= {P}_{vent}\left(t\right)= \frac{\varDelta V\left(t\right)}{{C}_{rs}}+ {R}_{rs} {F}_{aw}\left(t\right)+ {PEEP}_{a}$$
2
This is an indeterminate equation with two unknowns, Crs and Rrs. However, it may be solved numerically for Paw(tk), ΔV(tk), Faw(tk), and PEEPa measurements obtained at sequential times tk during insufflation. The numerical solution entails the development of a Paw(t) solution matrix at each tk interval created by substituting the values for ΔV(tk), Faw(tk), and PEEPa into Eqt. 2 and alternately applying physiologically plausible ranges (C1 ... Cn) for Crs and (R1 … Rn) for Rrs.
$${R}_{1} {R}_{2} \cdots { R}_{n}$$
$${P}_{aw}\left(t\right) solution matrix=\left[ \begin{array}{ccc}{P}_{aw }\left({R}_{1,}{C}_{1}\right) {P}_{aw }({R}_{2},{C}_{1})& \cdots & {P}_{aw }\left({R}_{n}{,C}_{1}\right)\\ ⋮& ⋮& ⋮\\ {P}_{aw }\left({R}_{1},{C}_{n}\right) {P}_{aw }({R}_{2},{C}_{n})& \cdots & {P}_{aw }\left({R}_{n}{,C}_{n}\right)\end{array}\right] \begin{array}{c}{C}_{1}\\ ⋮\\ {C}_{n}\end{array}$$
The elements of the solution matrix contain all possible Paw(t) values capable of satisfying Eqt. 2 for a given set of ΔV(tk), Faw(tk), and PEEPa measurements. Figure 1A shows the Paw(t) solution matrix for values of ΔV(tk) = 300 mL, Faw(tk) = 32 L·min− 1, and PEEPa = 5 cmH2O as a three-dimensional solution surface bounded by ranges of Crs from 10 to 50 mL·cmH2O− 1 and Rrs from 0 to 20 cmH2O·s·L− 1.
In terms of Crs and Rrs, the solution of Eqt. 2 must lie on a path traced by the known Paw(tk) on the solution surface. This is shown in Fig. 1 (top) as path (A) for Paw(tk) = 27 cmH2O where point 'a' is the yet unknown solution of Eqt. 2. Projecting path (A) onto the Crs-Rrs plane generates path (B). This function narrows the possible combination of Crs and Rrs values capable of satisfying Eqt. 2. It is developed by noting the values for (Cn, Rn) of solution matrix elements where Paw(Cn, Rn) = Paw(tk).
It remains to identify solution point ‘b’ that uniquely defines Crs and Rrs for the breath under consideration. This is accomplished by generating a series of Crs-Rrs function, one for each set of Paw(t), ΔV(t), Faw(t), and PEEPa measurements made during insufflation. Point ‘b’ is identified by the intersection of these Crs-Rrs function, assuming that Crs and Rrs have constant values.
It is known, however, that Crs and Rrs in ARDS patients vary early in inspiration as unstable alveoli open and conducting airways distend [10]. However, as lung volume increases past a lower inflection point (LIP), defined here as ΔVLIP, both variables achieve steady state values until reaching an upper inflection point (UIP), defined as ΔVUIP, where over-distention might occur. It is reasonable, therefore, to expect all Crs-Rrs functions generated for insufflation lung volumes ΔVLIP < ΔV(t) < ΔVUIP to intersect at a point ‘b’ that uniquely defines Crs and Rrs for the breath.
Figure 1 (bottom) shows a Crs-Rrs function family created with data from 15 different sampling times tk during a breath insufflation. The functions clearly identify Crs and Rrs by their intersection at Crs = 32.8 cmH20·mL− 1 and Rrs = 23.8 cmH2O·s· L− 1. In practice (inset graph Fig. 1B), the Crs-Rrs functions may not intersect exactly on the Crs-Rrs plane, perhaps the result of random variations in measurement or small changes in thoracic volume produced by the heart’s motion. Accordingly, the intersection point is best defined by the smallest standard deviation (σ) of Crs for all Crs-Rrs functions calculated along the Rrs axis.