N
2 clearance curves obtained by a multiple breathing method were analysed on the basis of 50 multiple compartments by means of a mathematical technic using a computer, using the following theoretical function reported by Lewis.
C
j/Co=Σ
mj=1Vi⋅(1/1+Si)
j i=1, 2, ...., 50
(C
j: end tidal N
2 concentration at jth breath, Co: initial N
2 concentration, Voi: volume of each compartment of lung at the FRC level, V
A: alveoral ventilation (V
A=V
T-V
D), V
D: dead space, Vi: fraction of ventiation of each compartment, Si: ventilation/volume. (Si=Vi⋅V
A/Voi))
The data recovered from subjects were analyzed on the basis of the following expression, with specified compartments equally spaced on a log scale of Si from 0.005 to 10.
minimize Σ
nj=1[Σ
mj=1Vi⋅(1/1+Si)
j-Cj/Co]
2subject to Σ
mj=1Vi=1
Vi≥0
i=1, 2, ...., m, j=1, 2, ...., n
In this paper, the above expression was considered as a nonlinear programing problem, and a new method for solving this problem was developed, starting from an initial feasible point, and locating new feasible points in an iterative manner by moving a direction conjugate in the previous direction of search. As such, it may be considered a new extension of the work of Rosen on problems involving linear constraints and the work of Hestenes and Siefel on unconstrained problems.
Using this technique, the recovery based on error-free data calculated from known distribution was studied. Distribution patterns in 10 normal subjects and 31 patients with chronic pulmonary diseases were discussed.
1) The intrapulmonary distribution of inspired gas could be shown visually and quantitatively, and the distribution patterns of both groups were distinctly different.
2) The recovery based on error-free data calculated from unimodal known Gaussian distribution almost completely fitted given data, and the recovery from bimodal or trimodal distribution bore a close resemblance to given data.
3) The stable distribution patterns were based on 1000 iterative calculations, and the sum of the squares was near 10
-5. The time requred for 1000 iterative calculation was about 8 minutes on an IBM 4341.
4) From 50 values of fraction of ventilation (Vi) to ventilation/volume (Si), the mean (Si-Mean) and standard deviation (Si-SD) of ventilation/volume were calculated. Si-Mean is a parameter of overall ventilation/volume ratio, and Si-SD is a parameter of unevenness of intrapulmonary distribution of inspired gas.
5) Si-SD correlate with FEV
1.0% (r=-0.672, p<0.01), %MVV (r=-0.698, p<0.01), RV/TLC% (r=0.568, p<0.01) and I.D.I. (r=0.485, p<0.05). This results suggested that the more severe the degree of obstruction and the greater the RV/TLC% ratio, the more uneven was the distribution. There was no correlation with PaO
2 and PaCO
2.
From these results, it was concluded that the analytical method reported in this paper is useful to evaluate the unevenness of intrapulmonary distribution of inspired gas.
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