Generalized conservation equation for multicompartmental systems

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Abstract
A projection operator technique is used to derive an equation for local tissue tracer content Q(t), assuming linear multicompartmental kinetics for tracer utilization. The resulting equation has the form (d/dt) Q(t)=FCa(t) −ΛQ(t)dτ [ξ(τ)Ca(t−τ)−ψ(τ)Q(t−τ)], where F and Ca(t) denote local blood flow and concentration of tracer, respectively. Tissue complexity is contained within the new parameters Λ, ξ(t), and ψ(t), where the time dependent coefficients are expressed as sums of exponentials. Two simple applications are considered: tissue heterogeneity and internal trapping of tracer. The relationship to effective single compartmental analysis, as is used for local cerebral blow flow determination, is evaluated.