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.

Mathematical solutions in internal dose assessment: A comparison of Python-based differential equation solvers in biokinetic modeling.

In biokinetic modeling systems employed for radiation protection, biological retention and excretion have been modeled as a series of discretized compartments representing the organs and tissues of the human body. Fractional retention and excretion in these organ and tissue systems have been mathematically governed by a series of coupled first-order ordinary differential equations (ODEs). The coupled ODE systems comprising the biokinetic models are usually stiff due to the severe difference between rapid and slow transfers between compartments. In this study, the capabilities of solving a complex coupled system of ODEs for biokinetic modeling were evaluated by comparing different Python programming language solvers and solving methods with the motivation of establishing a framework that enables multi-level analysis. The stability of the solvers was analyzed to select the best performers for solving the biokinetic problems. A Python-based linear algebraic method was also explored to examine how the numerical methods deviated from an analytical or semi-analytical method. Results demonstrated that customized implicit methods resulted in an enhanced stable solution for the inhaled60Co (Type M) and131I (Type F) exposure scenarios for the inhalation pathway of the International Commission on Radiological Protection (ICRP) Publication 130 Human Respiratory Tract Model (HRTM). The customized implementation of the Python-based implicit solvers resulted in approximately consistent solutions with the Python-based matrix exponential method (expm). The differences generally observed between the implicit solvers andexpmare attributable to numerical precision and the order of numerical approximation of the numerical solvers. This study provides the first analysis of a list of Python ODE solvers and methods by comparing their usage for solving biokinetic models using the ICRP Publication 130 HRTM and provides a framework for the selection of the most appropriate ODE solvers and methods in Python language to implement for modeling the distribution of internal radioactivity.