Direct deconvolution algorithms based on Laplace transforms in nuclear medicine applications

A set of algorithms is presented for direct deconvolution of the residue signal of an organ with the input signal to the organ. The deconvolution process yields the residual impulse response from which the distribution of transit times and the important mean transit time can be readily determined. The deconvolution method is based on the Laplace transform and it requires that the input signal can be fitted with an expression consisting of one, two or three exponentials with or without a bolus term at zero time or a constant term. These types of exponential expressions for the input signal cover a wide range of the input signals encountered in nuclear medicine applications. Simulation studies of the residue signal by convolution of various input signals with a number of residual impulse response models yielded an excellent accuracy of the deconvoluted residual impulse response for a suitably small sampling time. The simulations provide an opportunity to understand further the shapes of the residue curves depending on the shape of the input signal and the distribution of transit times. Simulations with Gaussian-distributed noise and noise spikes superimposed on the residue signal were also made to investigate the robustness of the direct deconvolution algorithm using apparently real-life data.