A least squares algorithm to determine the mechanical time constant distribution of the lung during forced expiration

A method to determine the mechanical time-constant distribution of the lung during a forced expiration manoeuvre is proposed. The method is based on a least squares algorithm constrained to give reasonably smooth non-negative solutions. The smoothing constraint was imposed by minimizing the second derivative of the distribution function in accordance with the physiological meaning of the time-constant distribution. Nevertheless, the obtained solution depends greatly on the relative weights of the two terms in the objective function to be minimized i.e., the error on the fit of the volume signal and the smoothness of the distribution function. To select the optimum smoothing weight, a criterion based on the stability of the reconstructed distribution shape was defined. The performance of the algorithm and that of the defined criterion were evaluated by using simulated signals of forced expired volume. The error of reconstructed distributions was quantified by means of the area enclosed between this distribution and the original one used to generate the simulated volume signal. The results obtained showed that for all the analyzed signals: (1) There is a value of the weight of the smoothing constraint which gives rise to a solution that is optimum in a least squares sense. (2) The proposed stabilization criterion enables us to approach this optimum solution from experimental signals.