Mathematical complexities in radionuclide metabolic modeling: A review of ordinary differential equation kinetics solvers in biokinetic modeling.

Biokinetic models have been employed in internal dosimetry to model the human body's time-dependent retention and excretion of radionuclides. Consequently, biokinetic models have become instrumental in modeling 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 solving 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 modeling the distribution and dosimetry for internal emitters, highlighting the mathematical frameworks, their capabilities, and their limitations. Further discussion details the mathematical underpinnings of biokinetic solutions in a unique approach paralleling advancements in internal dosimetry with capabilities in available mathematical solvers in computational systems. A survey of ODE forms, methods, and solvers, including state-of-the-art solvers specifically in Python programming language, was conducted to highlight modern capabilities for advancing the utilization of modern toolkits in internal dosimetry. This review is the first of its kind, which provides a comprehensive analysis of biokinetic solving methods and base knowledge for understanding the computational demands, schemes, and implementations for biokinetic modeling, which can be leveraged for an expedited radiation dose assessment.