Mathematical complexities in radionuclide metabolic modelling: a review of ordinary differential equation kinetics solvers in biokinetic modelling.

Biokinetic models have been employed in internal dosimetry (ID) to model the human body's time-dependent retention and excretion of radionuclides. Consequently, biokinetic models have become instrumental in modelling the body burden from biological processes from internalized radionuclides for prospective and retrospective dose assessment. Solutions to biokinetic equations have been modelled as a system of coupled ordinary differential equations (ODEs) representing the time-dependent distribution of materials deposited within the body. In parallel, several mathematical algorithms were developed for solving general kinetic problems, upon which biokinetic solution tools were constructed. This paper provides a comprehensive review of mathematical solving methods adopted by some known internal dose computer codes for modelling the distribution and dosimetry for internal emitters, highlighting the mathematical frameworks, capabilities, and limitations. Further discussion details the mathematical underpinnings of biokinetic solutions in a unique approach paralleling advancements in ID. The capabilities of available mathematical solvers in computational systems were also emphasized. A survey of ODE forms, methods, and solvers was conducted to highlight capabilities for advancing the utilization of modern toolkits in ID. This review is the first of its kind in framing the development of biokinetic solving methods as the juxtaposition of mathematical solving schemes and computational capabilities, highlighting the evolution in biokinetic solving for radiation dose assessment.