Indirect estimation of physiological distribution functions.

Multicompartment models, such as sums of exponential decays and sums of effects of different ventilation-perfusion ratios, are cast in the form of integrals. Difficulties in obtaining the density function in such an integral from measured values of the integral are attributed to amplification of error in the inversion solution and to the limited number of measurement points. The present approach to control the effect of the error is regularization with the use of a non-negativity constraint on the density function. The answers are sums of the influence or kernel functions of the integral wherever the sum is positive, and zero elsewhere. Such non-negative answers not only ensure that true density functions are obtained but also permit the answer to fall abruptly to zero. For example, a delta function can be much more closely approximated with the non-negativity constraint than without. A rule is developed to choose the value of smoothing parameter so as to minimize an approximate upper bound on the integral of the squared error of the answer. This typically tends to result in some oversmoothing. Functions tested without error and with 2% relative error are as follows: one of the kernel functions (best results); rectangular boxes and delta functions (fair results); and wide boxes (poor results).