INTRODUCTION

TOP ABSTRACT INTRODUCTION QDC THEORY METHODS AND DATA RESULTS DISCUSSION APPENDIX REFERENCES

SINCE THE WORK OF Konno and Mead (6) modeling the chest wall movements with two degrees of freedom and introducing the isovolume maneuver, other methods have been proposed to calculate tidal volume (VT) from measurements of the movements of abdomen and rib cage. They are mostly based on a simultaneous recording of the breathing volume by a spirometer or a pneumotachograph (4, 5, 7, 9, 12). More recently, Sackner et al. (10) described the qualitative diagnostic calibration method (QDC), which is based on spontaneous breathing, and Banzett et al. (2) used standard calibration ratios to determine the appropriate gains for the rib cage and abdominal signals. These calibration methods differ in the information they provide: an absolute calibration against the volume (4, 5, 7, 9, 12) or a relative weighting of the thoracic and abdominal signals (2, 6, 10). Furthermore, the complexity of their implementation varies significantly from the straightforward use of standard gains (2) to the isovolume maneuver requiring active cooperation of the subject (6). From this practical point of view, QDC is very attractive because it is a single-posture method that is carried out during spontaneous breathing without need for a mouthpiece or a facemask. However, this method has been subjected to criticism: Sartene et al. (11) proposed a more rigorous formulation of the QDC coefficient, Thompson (13) concluded from mathematical considerations that QDC is not reliable, and Brown et al. (3) indicated that "a critical evaluation of the mathematics underlying the QDC method, including computer simulations of the potential errors that could arise under circumstances that violate the underlying assumptions, would be extremely valuable."

In this paper, we analyze the principles and the mathematics underlying the QDC method, we test the approximations proposed by its authors to fulfill its implicit assumptions, and we evaluate its limits of application.

	QDC THEORY

TOP ABSTRACT INTRODUCTION QDC THEORY METHODS AND DATA RESULTS DISCUSSION APPENDIX REFERENCES

We will briefly recall the theory underlying QDC (10). The two-compartment model of the respiratory system of Konno and Mead (6) expresses the change of volume measured at the mouth (Va) as the sum of a rib cage (Vrc) and an abdominal (Vab) contribution

(1)

These volume changes are obtained indirectly by measuring the variations of two representative thoracic and abdominal dimensions by inductive plethysmography, magnetometers, or strain gauges. Using the notations of Sackner et al. (10), Eq. 1 can be rewritten as

(2)

where the calibration factor K weights the relative contribution of the uncalibrated (u) electrical signals uVrc and uVab from the thoracic and abdominal compartments and M scales the sum (K uVrc + uVab) to Va. The QDC as well as the isovolume calibration are methods yielding K. The isovolume calibration consists of having the subject shift volume between the thoracic and abdominal compartments while the upper airways are occluded. Hence, Va = 0 and Eq. 2 becomes

(3)

The QDC method (10) works at constant VT instead of Va = 0. It is based on a claimed observation that the breath-to-breath variations of Vrc and Vab are normally distributed (10). With breathing at constant VT, calculating the standard deviation (SD) of each term of Eq. 2 and using SD(VT) = 0 would yield according to Sackner et al. (10)

(4)

Because, during natural spontaneous respiration, breathing at constant VT is not possible, Sackner et al. propose to approximate a constant VT by collecting a large number of breaths and by excluding those whose sums show large deviations from the mean sum µ(uVrc + uVab). The range to be considered is thus

(5)

where x defines the interval width around the mean. According to Sackner et al., this chosen parameter should be between 0.6 and 1.

	METHODS AND DATA

TOP ABSTRACT INTRODUCTION QDC THEORY METHODS AND DATA RESULTS DISCUSSION APPENDIX REFERENCES

Our approach is based mainly on simulated data following Eq. 1 and consists of two parts. First, we analyze the mathematics underlying the QDC method. Second, we study in some detail the approximation of Eq. 5, which selects the breaths at constant VT. For this analysis, we start from simulated sets of thoracic and abdominal volumes and construct uncalibrated signals using a predefined value of K. The procedure is illustrated in Fig. 1: the sets of thoracic and abdominal volumes, {Vrc_i} and {Vab_i}, are obtained from a two-dimensional normal distribution, with major axis at angle  with respect to the abscissa and centered around the mean (Vrc, Vab). [Vrc_i and Vab_i correspond to the ith breath (i = 1, ... , n)]. The sets are then divided by two predefined calibration factors Krc and Kab, which yield {uVrc_i} and {uVab_i}. In this operation, Krc/Kab represents the calibration value K that should be found by the QDC procedure (Kab is equal to M in Eq. 2). The simulated data are illustrated in Fig. 2 and Fig. 3, in terms of distributions and Konno-Mead plot (abdominal vs. rib cage), respectively, for three different cases corresponding to breathing at constant VT (the ideal case of QDC theory, Figs. 2A and 3A), breathing at quasi-constant VT (Figs. 2B and 3B), and spontaneous breathing (Figs. 2C and 3C). The constant and quasi-constant breathings are characterized, respectively, by no or minor variations in VT. In these cases,  necessarily equals 45°; i.e., the points (Vrc_i, Vab_i) are aligned along an isovolume line (Fig. 3A) or are symmetrically distributed between two closely spaced isovolume lines (Fig. 3B). There is a negative correlation between the thoracic and abdominal volumes and a highly variable thoracoabdominal partitioning, which reflects the variable breathing strategy used to achieve the constant or quasi-constant VT. The spontaneous breathing is defined by minor changes in the relative abdominal contribution [ = Vab/(Vab + Vrc)] and simulates the breathing pattern observed in normal adult subjects during quiet breathing (2, 11). A mathematical calculation based on  = constant shows that the points (Vrc_i, Vab_i) are positively correlated and define a general orientation (tan ) equal to /(1  ) (Fig. 3C). In this simulation, larger variations are observed for VT even though the thoracic and abdominal volumes show the same amplitude of variations as in the previous cases. The sets of parameters [, SD(Vrc), SD(Vab)] used to construct each simulation are defined in Table 1. In these examples, we assumed a mean VT of 0.75 liters and a mean relative abdominal contribution  of 1/3 (8). Krc and Kab were chosen at 0.7 and 1, respectively. This choice corresponds to M = 1 and K = 0.7. We considered sets of 150 simulated breaths corresponding to the 3- to 10-min calibration period recommended by Sackner et al. (10).

View larger version (16K): [in this window] [in a new window]   Fig. 1.   Construction of the simulated breathing data: the pairs of thoracic and abdominal volumes (Vrci,Vabi) are obtained from a two-dimensional normal distribution, centered around the mean [µ(Vrc),µ(Vab)], with major axis at angle  with respect to the abscissa. The subscript i indicates the breath number in the data set. The sets are divided by two predefined calibration factors Krc and Kab, which yield the uncalibrated thoracic and abdominal volumes (uVrci,uVabi). In this operation, Krc/Kab represents the calibration value K.

View larger version (24K): [in this window] [in a new window]   Fig. 2.   Thoracic and abdominal volumes (Vrc and Vab, respectively) generated by random sampling from a two-dimensional normal distribution (see Fig. 1) in 3 different cases. A: simulation of constant tidal volume (VT). B: simulation of quasi-constant VT. C: simulation of spontaneous tidal breathing. Top: calibrated volumes (V). Bottom: uncalibrated volumes (uV). We assumed for all simulations a mean VT of 0.75 liters and a mean abdominal relative contribution of 1/3. The sets of parameters used to construct the simulations are given in Table 1. Krc and Kab were chosen at 0.7 and 1, respectively. Every set contains 150 data points.

View larger version (20K): [in this window] [in a new window]   Fig. 3.   Calibrated V () and uncalibrated uV (circles) thoracic and abdominal volumes of Fig. 2, plotted in a Konno-Mead representation. A: simulation of constant tidal volume VT. B: simulation of quasi-constant VT. C: simulation of spontaneous tidal breathing. The boundary lines with a slope of (1) delimit the selected breaths () corresponding to the interval [µ(uVrc + uVab) ± 0.6 SD(uVrc + uVab)]. , Breaths outside the selection interval. Values of the different slopes are indicated by . See DISCUSSION for details.

	RESULTS

TOP ABSTRACT INTRODUCTION QDC THEORY METHODS AND DATA RESULTS DISCUSSION APPENDIX REFERENCES

Analysis of the mathematics of the QDC method. It is shown in the APPENDIX that

(6)

is actually the solution of Eq. 2 when the pairs (uVrc_i, uVab_i) belong to a straight line. Note that there is no minus sign in Eq. 6. In this case, K is interpreted geometrically as the absolute value of the slope of the line defined by the set of uncalibrated volumes (uVrc_i, uVab_i). If the pairs (uVrc_i, uVab_i) do not belong to a straight line but are strongly correlated, K represents an approximation of the absolute value of the slope of the regression line calculated through the set (uVrc_i, uVab_i). This approximation is all the better when the points are strongly correlated.

Approximation of Eq. 5 for simulated breathing at constant VT. Figure 2A shows the distributions of thoracic, abdominal and volumetric data for the simulation, which corresponds to a respiration pattern at perfectly constant volume. The distribution of {VT_i} is thus an infinitesimally narrow peak, generally referred to as a Dirac distribution. Note that the set of uncalibrated sum {uVrc_i + uVab_i} does not follow a Dirac distribution. Figure 3A shows the calibrated and uncalibrated data in a Konno-Mead plot. Because VT is constant, these data define two straight lines with slopes equal to 1 and 0.7, respectively (Eqs. 1 and 2). The selection of breaths according to x = 0.6 is shown in Fig. 3A by the two dotted lines of slope 1. The proportion of breaths that are eliminated corresponds to (1  p) where p is the probability of being inside the interval (µ x SD, µ + x SD). For x = 0.6, p equals 0.45. Hence, more than half of the total number of breaths (55%) are eliminated although they are, by construction, at constant VT. Nevertheless, the points that remain define always the same straight line with the slope of 0.7. This slope thus provides a correct value for K in the particular case x = 0.6. Figure 4 shows K as a function of the selection of breaths defined by x. This curve was obtained by repeating the simulation procedure 100 times and averaging the results. When breathing is at constant VT, the ratio of SD always provides a correct value for K, regardless the value of x, and thus a fortiori in the range 0.6  x  1. 

View larger version (17K): [in this window] [in a new window]   Fig. 4.   The calibration factor K as a function of x that determines the width of the selection interval (Eq. 5) for the 3 examples of Figs. 2 and 3, i.e., at constant VT, at quasi-constant VT, and for spontaneous (irregular) breathing. K = 0.7 is the correct value, by construction of the data. Each curve was obtained by averaging 100 simulations.

Approximation of Eq. 5 for simulated breathing at quasi-constant VT. In the second simulation, some dispersion is added to VT with respect to the ideal case of Fig. 2A. Figure 2B shows the distributions of the simulated data for this case, and Fig. 3B shows the corresponding Konno-Mead plot. It can be seen that {VT_i} is no longer a Dirac distribution. Furthermore, the points are not perfectly aligned along a straight line of slope 1. Again, the selection of breaths according to the criterion of Sackner (Eq. 5 with x = 0.6) yields many breaths (55%) at quasi-constant VT that are eliminated whereas breaths corresponding to a slope of 1 remain. This is highlighted in Fig. 3B by the two dotted lines. According to the value of x and thus to the selected breaths, K will take different numeric values, which tend to 1 when x decreases (Fig. 4). In the present simulation, a selection of breaths with 0.6  x  1 yields a corresponding value K ranging from 0.938 and 0.864 instead of 0.7. Figure 5 shows K as a function of x for the four different cases of Table 2, i.e., with decreasing standard deviation of the rib cage and of the abdomen volume variations. Here the same averaging procedure as in Fig. 4 was performed. It can be seen that if the variability of the contributions of thoracic and abdominal compartments decreases, the accuracy of K decreases for constant x.

View larger version (17K): [in this window] [in a new window]   Fig. 5.   The calibration factor K as a function of x (Eq. 5) for a decreasing variability of the thoracic and abdominal contributions in the model of quasi-constant volume (Figs. 2B and 3B). K = 0.7 is the correct value. We assumed a mean VT of 0.75 liters and a mean abdominal relative contribution of 1/3. The sets of parameters used to construct the different simulations are defined in Table 2. See Table 2 for definition of symbols. Each curve was obtained by averaging 100 simulations.

Approximation of Eq. 5 for simulated spontaneous breathing. In the third simulation, we consider spontaneous breathing (Figs. 2C and 3C). In that case, the slopes of the calibrated and uncalibrated data in the Konno-Mead plot are positive. The values of K obtained from QDC according to x are shown in Fig. 4. Without any selection, K equals [0.7 µ(Vab)/µ(Vrc)]. For 0.6  x  1, the values of K range from 0.492 to 0.42. K tends to 1 when x decreases.

	DISCUSSION

TOP ABSTRACT INTRODUCTION QDC THEORY METHODS AND DATA RESULTS DISCUSSION APPENDIX REFERENCES

Underlying theory and basic assumptions. In this paper, we use the two-compartment model of Konno and Mead (Eq. 1), and we assume the validity of the observation of Sackner et al. (10) that Vrc and Vab are normally distributed. This allowed us to focus on the mathematics underlying QDC. The idea of working at constant VT by analogy with the isovolume calibration is ingenious. We show in the APPENDIX that the formula for K in the original paper (10) is correct although the mathematical demonstration can be criticized (3, 11, 13). In fact, we were intrigued by the negative sign of Eq. 4 and by the simplicity of the mathematical derivation of K by taking the SD of each term of Eq. 2 and stating that the SD of a sum of statistical variables is equal to the sum of the SD of each variable. From a physiological point of view, it is clear that K should be positive. In fact, when the thoracic and abdominal compartments are modeled by two cylinders and uVab and uVrc are considered as variations of their cross-sectional areas, K represents the ratio of the heights of each compartment, which are positive values. In the APPENDIX, it is shown that the QDC-calculated K value can geometrically be interpreted as the approximation of the absolute slope of the regression line through the set of uncalibrated thoracic and abdominal volumes plotted in the Konno-Mead representation.

Breath selection for constant VT. The critical point of the QDC procedure is the hypothesis of constant VT in situations in which this parameter is unknown. Sackner et al. (10) proposed to collect the breaths that fall in an interval around the mean µ(uVrc + uVab). We demonstrated that this selection is not needed for breathing at constant or quasi-constant VT (Figs. 3, A and B, and 4). In these cases, the selection procedure even biases the results by favoring in the Konno-Mead plot breaths that correspond to a slope of 1 instead of breaths that are at constant or quasi-constant VT. When the respiratory pattern is spontaneous, K takes a value that depends on the selection criterion and that tends to 1 with decreasing x. The latter observation is due to the selection procedure, which favors breaths falling on a line with a negative unitary slope. Hence, the breaths selected define a specific constant volume VT = uVrc + uVab related to the situation of K = 1, which is not the actual value of K. Furthermore, it was shown that if the variability of the contributions of thoracic and abdominal compartments decreases, the accuracy of K decreases for constant x. In fact, the selected points become more focused and more easily define a straight line with a slope corresponding to K = 1 instead of its real value. Finally, the limitation of the method used to keep breaths at constant VT is highlighted by the fact that multiplying the gain of one channel (rib cage or abdomen) does not induce the same variation of the QDC-calculated K. This observation is explained by the fact that the modification of gain induces a rotation of the points (uVrc_i,uVab_i) in the Konno-Mead plot while the selection interval of slope (1) remains identical. Hence, this operation modifies the breaths selected and thus the calculation of K.

Applications of QDC. In spite of these serious reservations, QDC was demonstrated to yield, in adults in supine posture, results equivalent to the other calibration procedures, namely isovolume and multilinear regression methods (10, 11). The best results were obtained when estimating VT during uninstructed tidal breathing. Other situations, such as natural preferential thoracic or abdominal breathing and voluntary changes of VT, yielded less accurate VT estimations. These results can be explained by the observation that the variability of thoracoabdominal partitioning in normal adult subjects is very small during quiet breathing (2, 11). Hence the rib cage signal is proportional to the abdominal one. In that case, uVrc =  uVab and Eq. 2 becomes

(7)

It can be seen in Eq. 7, where  is a proportionality constant, that a correct VT will be obtained with an appropriate choice of M, regardless of the value of K. This can explain why all the calibration techniques, including the one using standard ratios (2), provide a good evaluation of the VT for quiet breathing, whereas greater discrepancies in the evaluation of VT are observed when changes in breathing pattern occur. In the latter case, two degrees of freedom are used and, hence, the accuracy of K becomes more critical.