MetaboliSim: a Python implementation of the Mader model for dynamic and steady-state simulation of muscular energy metabolism

Abstract

The Mader model is the most widely used mathematical framework for muscular energy metabolism in German-language sport science, underpinning lactate diagnostics, maximal lactate steady state (MLSS) estimation and training prescription. Despite decades of use, neither its dynamic ODE formulation nor its steady-state equations have been available as open code, leaving results based on the model impossible to reproduce independently. We close this gap with MetaboliSim, an open-source Python implementation of both formulations: a dynamic model that integrates the five-variable ODE system (phosphate potential, $\dot{V}\mathrm{O}_2$, muscle and blood lactate, and glycogen) with a fourth-order Runge-Kutta scheme, and a steady-state model that computes MLSS power and the lactate-power relationship in one- and two-compartment variants. We verified implementation correctness against published reference values and assessed physiological plausibility across constant-load, step-test, sprint and running protocols. The implementation reproduces the published reference output within stated tolerances and remains numerically stable throughout (halving the time step changes blood lactate by less than 0.01 mmol/L), with both formulations yielding congruent MLSS estimates. Key physiological behaviour ($\dot{V}\mathrm{O}_2$ on-kinetics, lactate accumulation, PCr dynamics and the sub/supra-MLSS separation) emerges directly from the model equations without protocol-specific tuning, and a sensitivity analysis shows MLSS power varying approximately linearly with $\dot{V}\mathrm{O}_{2\max}$ and nonlinearly with $\dot{V}\mathrm{La}_{\max}$. As the first openly available implementation of the complete Mader model (AGPL-3.0), MetaboliSim lets independent groups reproduce, verify and build on published model-based results. Source code: https://codeberg.org/3phos/metabolisim; Platform: https://metabolisim.org