Fitting Sum of Exponentials to Experimental Data: Linearization by Numerical Integration Approximation

A solution to important and difficult problem of resolving the components of multi-exponential decay is presented. Often the techniques are chosen that are noise sensitive and their error poorly understood. A general, numerically robust and fast (real time) solution method has been developed. Noise attenuation and flexibility of the method are analyzed in detail. Its relevance for signal analysis from different relaxation processes, in particular NMR medical imaging is discussed. Multi-exponential decays are most often results of parallel, independent relaxation processes, decay of a mixture of radionuclides, parallel chemical reactions of the first order, etc. Inverse problem of finding the components with close decay constants in multi-exponential signal is inherently ill-posed because of non-orthogonality of exponential functions. The result is very sensitive to noise and chosen optimization methodology. The general solution for the problem of separating exponentials based on linear approximation through repeated numerical integration has been developed. The algorithm is based on the least squares method with the possibility of using non-negative constrains. Using linear approximation avoids problem with supplying a good initial guess, nonproductive iterations, and local minima. Algorithm also includes implementation of global method by Knutson.