Exact parameter determination for Parkinson's disease diagnosis with PET using an algebraic approach

The mechanism of Parkinson's disease can be investigated at the molecular level by using radio-tracers. The concentration of dopamine in the brain can be observed by using a radio-tracer, 6-[¹⁸F]fluorodopa (FDOPA), with positron emission tomography (PET), and the dopamine kinetics can be described as compartmental models for tissues of the brain. The models for FDOPA kinetics are solved explicitly, but the solution shows a complicated form including several convolutions over time domain. Owing to the complicated form of the solution, graphical analyses such as Logan or Patlak analysis have been utilized as conventional methods over past decades. Because some kinetic constants for Parkinson's disease are estimated in the graphical analyses with the slope or intercept of the line obtained under various assumptions, only a limited set of parameters have approximately been estimated. We have analysed the compartmental models by using the Laplace transformation of differential equations and by algebraic computation with the aid of Gröbner base constructions. We have obtained a rigorous solution with respect to the kinetic constants over the Laplace domain. Here, we first derive a rigorous solution for the parameters, together with a discussion about the merits of the derivation. Next, we describe a procedure to determine the kinetic constants with the observed time-radioactivity curves. Last, we discuss the feasibility of our method, especially as a criterion for diagnosing Parkinson's disease.