Deconvolution of tracer and dilution data using the Wiener filter

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Abstract
In the study of living systems it is often necessary to inject or infuse a substance into the peripheral circulation and monitor its subsequent concentration in the plasma with time. Examples abound in the pharmacokinetic study of drugs and in the use of the indicator dilution technique for measuring blood flow. Furthermore, it is often necessary to deconvolve one such measured, and hence noisy, data set with another. One of the standard methods for deconvolving noisy signals is the Wiener filter, which is generally derived as a real window in the frequency domain such that the mean squared error between the estimated deconvolved function and the truth, on average, is minimized. Application of the Wiener filter requires some (often crude) model of the noise-to-signal power ratio as a function of frequency. In the pharmacokinetic and indicator dilution situations, however, one invariably has a good model of the actual function to be deconvolved in the form of a sum of decaying exponential functions. Such a model may be employed to calculate the signal-to-noise power ratio for use in the Wiener filter, or alternatively may be directly deconvolved itself. It is shown that better results are achieved with the Wiener filter if the model of the signal is not particularly accurate, whereas with a very accurate model it is better to deconvolve the model itself. The point at which the two deconvolution approaches perform comparably occurs when the error in the model is of a similar magnitude to the noise.