Using Differential Equations to Model Heavy Metal Bioaccumulation as part of SCUDEM IX Challenge

Abstract

Research examining heavy metal concentrations in cacao products has raised concerns about potential toxicity for frequent chocolate consumers. In this study, conducted as part of SCUDEM IX Challenge, a nonlinear system of differential equations was developed to predict lead dynamics and bioaccumulation in the body. The model considers time-dependent lead concentrations in the GI tract M, bloodstream A, soft tissue S, and hard tissue H. Lead diffusion between systems follows logistic-based osmosis transport equations. Diffusion parameters were loosely selected from literature and iteratively refined to preserve model stability and ensure outputs align with surveyed values. Two primary consumption patterns, or inhomogeneities, were selected for test cases; daily “stress” and weekend “binge” consumption patterns were defined by a sinusoid and periodic step function, respectively. The model surveyed a year period, additionally considering increased consumption for American holidays and birthdays. Runge-Kutta fourth-order approximations were implemented to plot the solution curves developed via numerical methods. According to model predictions, maintaining daily chocolate consumption below 50g is unlikely to induce lead toxicity, as compared with the CDC Blood Lead Reference Value (BLRV) of 3.5 μg/dL.