B U L L E T I N O]~ ~¢IATIIENIATICAL B I O L O G Y
VOLVME 39, 1977
THE I/A/Q RESOLUTION OF I N E R T GAS DATA
• R. S. HOWARDand H. B~ADN~R
Scripps Institution of Oceanography, University of California, San Diego, La Jolla, California 92093, U.S.A.
The non-uniqueness of I/A/Q distributions satisfying inert gas retention data without error is studied. The ability of such data to resolve blood flows at particular I/A/Q values is discussed ~hrough the application of linear programming and Backus-Gllbert theory. I t is shown t h a t the resolution deteriorates away from the extremes of low and high
#~/Q.
1. Introduction. Since the inequality of ventilation and blood flow in a lung is responsible for most of the defective gas exchange in pulmonary disease, one of the goals of the respiratory physiologist is to draw the frequency distribution of ventilation-perfusion ratios within the abnormal lung. In recent years, the use of non-respiratory gases as pulmonary indicators has been appreciated, and it has been repeatedly suggested that measurement of multiple inert gas elimination would offer the most information about the comprehensive double distribution of ventilation and blood flow over their ratio. These suggestions appeared to culminate when Wagner et al. (1974a) reported t h a t virtually continuous I~A/Q distributions could be defined using only a few inert gas retention measurements. When the amount of an inert gas in the alveolar gas volume is maintained constant by its continuous intravenous infusion, its relative arterial retention was shown by Farhi (1967) to be simply related to the blood flow distribution over FA/Q by Pae
r ~ ~GQ(VA/Q)
P~G 87
88
R.S. HOWARD
AND H. BRADNER
where Q(~A/(2) is the normalized blood flow per unit FA/(2, and ~ is the Ostwald blood :gas partition coefficient, herein simply referred to as the "solubility", of the inert gas G. The relative excretion of the gas is analogously related to the normalized ventilation, and since the mathematics is identical, the distribution of ventilation will not be discussed. F a r m realized that inert gas elimination would be a useful tool for assessing VA/Q inhomogeneity. He noted that such a technique should detect and group spatia]ly scattered lung units with any particular ventilation-perfusion ratio. This would then complement radio isotope scanning techniques which reveal the topography of averaged relationships b u t cannot detect flows in a particular ratio unless they occupy a definite volume. Should the continuous functional dependence of retention upon solubility be known, e.g. b y curve-fitting the data, then one can formally invert (1) to obtain the blood flow distribution. Setting x = FA/Q:
-
;o Ire Q(x)
]
e -sx e -sx ds dx
(2)
and it follows from the definition of the Laplace transform and the applicability of Fubini's theorem that
R(~) 2
- ff[Se(Q(x))],
(3)
whence
(4) which, in the sense of Lerch's lemma, is a unique solution. The inversion of an integral equation can be stable, as in the case of the inversion of the Fourier transform. However, the inversion of the Laplace transform is unstable, in the sense of Hadamard : the solution does not depend continuously on the data. Small perturbations of the data can lead to large perturbations in the solution. Many techniques and tables are available for computing stable, and correspondingly smooth, inversions of the Laplace transform (Bellman et al., 1966). So we are concerned below with assessing the data inversion, rather than smoothing it. The Laplace transform relationship indicates the intrinsic instability of the inversion, but gives no insight into the resolving power of the finite data set. The distribution of blood flow over ~A/(~ is thought to be continuous because the causes of inhomogeneity are multifaeeted and lung structures so complexly interrelated. Yet Wagner et al. (1974a) chose to consider discrete values for the variable VA/Q, and so compartmentalize the blood flow. With VA/Q values equally spaced logarithmically, they reported blood flows in
T H E 17A/Q R E S O L U T I O N OF I N E R T GAS DATA
89
fifty compartments directly recoverable from six retention data. Using a gradient technique to determine a "minimum length" solution, they suggested that "all the possible solutions lie very close to each other." However, Jaliwala et al. (1975) demonstrated with similar computational methods that when random errors of a magnitude consistent with present analytical techniques were introduced into retention data, recovered distributions differed significantly from original ones. So Evans and Wagner (1976) added to their least-squares fitting procedure an arbitrary amount of the square of the distribution parameters, thereby, in some sense, applying a degree of smoothing, which stabilized the recovery of sample distributions. The resulting analytical method selects a distribution which is then suggested to be "representative" and, presumably, which yields more information than a simple "smooth" curve drawn through the data on a retention-solubility diagram. These analyses have been confusing since a finite number of observations can only establish the values of an equal number of parameters in a linear system, even when constrained by non-negativity (Gass, 1975). So one would anticipate models with far fewer compartments than fifty yielding the same six retention data yet differing significantly from a fifty compartment distribution. Olszowka (1975) constructed several "physiologically reasonable" yet qualitatively different distributions which yield the same retention data, and so again questions the ability of such a technique to distinguish between areas containing shunt and those with low I)A/(2, or between areas containing dead space and those with high VA/Q. This paper discusses such resolving problems in the absence of experimental errors.
2. Applicability of Linear Programming. After numerical inversion of data for the construction of a model, there is a tendency to believe our knowledge is more precise than it really is. The selection of one model distribution from the cloud of solutions satisfying the data may indeed simplify the classification of various data sets. However, the end of pulmonary physiological studies is not merely taxonomic, but rather to discover something definite about lung structure. Therefore, our approach with inadequate inert gas data abandons the goal of obtaining a distribution; instead, the data are used to provide us with inequalities that must be satisfied by all models, and hence, by the lung itself. Mathematically, linear programming deals with non-negative solutions to underdetermined systems of linear equations, and so would be an appropriate computational algorithm to use when constraining blood flows to be nonnegative. Wagner et al. have repeatedly acknowledged the applicability of such a technique in model construction. Here we extend their suggestion to
90
R . 8. I - I O W A R D A N D I t . Bt~ADNEI%
provide directly simple bounds on all possible solutions. Linear programming is a proper mathematical method in that it guarantees, within a finite number of iterations, the attainment of the sought solution and not merely an asymptotic approximation to it. This important methodical aspect has not been manifest in many of the other techniques discussed above. Linear programming solves the problem of the form : n-1
minimize
~, aml yi t=1
subject to the constraints It--1
(a)
~ a~tyj = a~n,
1
VOLVME 39, 1977
THE I/A/Q RESOLUTION OF I N E R T GAS DATA
• R. S. HOWARDand H. B~ADN~R
Scripps Institution of Oceanography, University of California, San Diego, La Jolla, California 92093, U.S.A.
The non-uniqueness of I/A/Q distributions satisfying inert gas retention data without error is studied. The ability of such data to resolve blood flows at particular I/A/Q values is discussed ~hrough the application of linear programming and Backus-Gllbert theory. I t is shown t h a t the resolution deteriorates away from the extremes of low and high
#~/Q.
1. Introduction. Since the inequality of ventilation and blood flow in a lung is responsible for most of the defective gas exchange in pulmonary disease, one of the goals of the respiratory physiologist is to draw the frequency distribution of ventilation-perfusion ratios within the abnormal lung. In recent years, the use of non-respiratory gases as pulmonary indicators has been appreciated, and it has been repeatedly suggested that measurement of multiple inert gas elimination would offer the most information about the comprehensive double distribution of ventilation and blood flow over their ratio. These suggestions appeared to culminate when Wagner et al. (1974a) reported t h a t virtually continuous I~A/Q distributions could be defined using only a few inert gas retention measurements. When the amount of an inert gas in the alveolar gas volume is maintained constant by its continuous intravenous infusion, its relative arterial retention was shown by Farhi (1967) to be simply related to the blood flow distribution over FA/Q by Pae
r ~ ~GQ(VA/Q)
P~G 87
88
R.S. HOWARD
AND H. BRADNER
where Q(~A/(2) is the normalized blood flow per unit FA/(2, and ~ is the Ostwald blood :gas partition coefficient, herein simply referred to as the "solubility", of the inert gas G. The relative excretion of the gas is analogously related to the normalized ventilation, and since the mathematics is identical, the distribution of ventilation will not be discussed. F a r m realized that inert gas elimination would be a useful tool for assessing VA/Q inhomogeneity. He noted that such a technique should detect and group spatia]ly scattered lung units with any particular ventilation-perfusion ratio. This would then complement radio isotope scanning techniques which reveal the topography of averaged relationships b u t cannot detect flows in a particular ratio unless they occupy a definite volume. Should the continuous functional dependence of retention upon solubility be known, e.g. b y curve-fitting the data, then one can formally invert (1) to obtain the blood flow distribution. Setting x = FA/Q:
-
;o Ire Q(x)
]
e -sx e -sx ds dx
(2)
and it follows from the definition of the Laplace transform and the applicability of Fubini's theorem that
R(~) 2
- ff[Se(Q(x))],
(3)
whence
(4) which, in the sense of Lerch's lemma, is a unique solution. The inversion of an integral equation can be stable, as in the case of the inversion of the Fourier transform. However, the inversion of the Laplace transform is unstable, in the sense of Hadamard : the solution does not depend continuously on the data. Small perturbations of the data can lead to large perturbations in the solution. Many techniques and tables are available for computing stable, and correspondingly smooth, inversions of the Laplace transform (Bellman et al., 1966). So we are concerned below with assessing the data inversion, rather than smoothing it. The Laplace transform relationship indicates the intrinsic instability of the inversion, but gives no insight into the resolving power of the finite data set. The distribution of blood flow over ~A/(~ is thought to be continuous because the causes of inhomogeneity are multifaeeted and lung structures so complexly interrelated. Yet Wagner et al. (1974a) chose to consider discrete values for the variable VA/Q, and so compartmentalize the blood flow. With VA/Q values equally spaced logarithmically, they reported blood flows in
T H E 17A/Q R E S O L U T I O N OF I N E R T GAS DATA
89
fifty compartments directly recoverable from six retention data. Using a gradient technique to determine a "minimum length" solution, they suggested that "all the possible solutions lie very close to each other." However, Jaliwala et al. (1975) demonstrated with similar computational methods that when random errors of a magnitude consistent with present analytical techniques were introduced into retention data, recovered distributions differed significantly from original ones. So Evans and Wagner (1976) added to their least-squares fitting procedure an arbitrary amount of the square of the distribution parameters, thereby, in some sense, applying a degree of smoothing, which stabilized the recovery of sample distributions. The resulting analytical method selects a distribution which is then suggested to be "representative" and, presumably, which yields more information than a simple "smooth" curve drawn through the data on a retention-solubility diagram. These analyses have been confusing since a finite number of observations can only establish the values of an equal number of parameters in a linear system, even when constrained by non-negativity (Gass, 1975). So one would anticipate models with far fewer compartments than fifty yielding the same six retention data yet differing significantly from a fifty compartment distribution. Olszowka (1975) constructed several "physiologically reasonable" yet qualitatively different distributions which yield the same retention data, and so again questions the ability of such a technique to distinguish between areas containing shunt and those with low I)A/(2, or between areas containing dead space and those with high VA/Q. This paper discusses such resolving problems in the absence of experimental errors.
2. Applicability of Linear Programming. After numerical inversion of data for the construction of a model, there is a tendency to believe our knowledge is more precise than it really is. The selection of one model distribution from the cloud of solutions satisfying the data may indeed simplify the classification of various data sets. However, the end of pulmonary physiological studies is not merely taxonomic, but rather to discover something definite about lung structure. Therefore, our approach with inadequate inert gas data abandons the goal of obtaining a distribution; instead, the data are used to provide us with inequalities that must be satisfied by all models, and hence, by the lung itself. Mathematically, linear programming deals with non-negative solutions to underdetermined systems of linear equations, and so would be an appropriate computational algorithm to use when constraining blood flows to be nonnegative. Wagner et al. have repeatedly acknowledged the applicability of such a technique in model construction. Here we extend their suggestion to
90
R . 8. I - I O W A R D A N D I t . Bt~ADNEI%
provide directly simple bounds on all possible solutions. Linear programming is a proper mathematical method in that it guarantees, within a finite number of iterations, the attainment of the sought solution and not merely an asymptotic approximation to it. This important methodical aspect has not been manifest in many of the other techniques discussed above. Linear programming solves the problem of the form : n-1
minimize
~, aml yi t=1
subject to the constraints It--1
(a)
~ a~tyj = a~n,
1
