Estimating the ventilation-perfusion distribution: an ill-posed integral equation problem

The distribution of ventilation-perfusion ratio over the lung is a useful indicator of the efficiency of lung function. Information about this distribution can be obtained by observing the retention in blood of inert gases passed through the lung. These retentions are related to the ventilation-perfusion distribution through an ill-posed integral equation. An unusual feature of this problem of estimating the ventilation-perfusion distribution is the small amount of data available; typically there are just six data points, as only six gases are used in the experiment. A nonparametric smoothing method is compared to a simpler method that models the distribution as a histogram with five classes. Results from the smoothing method are found to be very unstable. In contrast, the simpler method gives stable solutions with parameters that are physiologically meaningful. It is concluded that while such smoothing methods may be useful for solving some ill-posed integral equation problems, the simpler method is preferable when data are scarce.