An analytical solution set for a four-compartment mixed mammillary/catenary model

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Abstract
The purpose of this study was to extract the intercompartmental rate constants analytically from the coefficients and exponents of the exponential sum fitted to tracer data points for a unique four-compartment mixed mammillary/catenary (extended mammillary, radial) model, with losses solely from the central (blood) compartment. This model is useful in estimating myocardial and skeletal blood flow. Using Laplace transforms, the transfer functions of the exponential sum and the differential equations describing the model were obtained and the corresponding coefficients equated sequentially, which yielded the intercompartmental rate constants. Three solution sets of rate constants were found for the model, each of which satisfies both transfer functions. A program written in BASIC is provided, which requires only the coefficients and exponents of the exponential terms as input; the output is the three sets of rate constants. An example is given as an aid in debugging. Only one solution set of the three was physiologically realizable in the example, but the bounds defining these limits are not known. The advantages and limitations of the method are discussed. The method is suitable for initializing a non-linear least-squares fitting program for the rate constants. The appropriate physiological solution set for this model yields fractional and total myocardial or skeletal blood flow in a subject in the steady state.

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