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

2026-06-06

AI Summary

This study addresses the longstanding lack of open-source implementations of the widely used Mader model for skeletal muscle energy metabolism, which has hindered reproducibility in the field. We present MetaboliSim, a Python-based tool that provides the first complete open-source implementation of both the dynamic (a five-variable ODE system) and steady-state (single- and two-compartment) formulations of the model. Employing a fourth-order Runge–Kutta method, MetaboliSim accurately simulates the dynamics of phosphagen stores, oxygen uptake, intramuscular and blood lactate concentrations, and glycogen depletion, while also enabling computation of maximal lactate steady-state (MLSS) power and lactate–power relationships. Validation demonstrates excellent agreement with published data, numerical stability (blood lactate variation <0.01 mmol/L upon halving the time step), concordance between MLSS estimates from dynamic and steady-state models, and physiologically plausible linear and nonlinear relationships of MLSS power with VO₂max and VLamax, respectively—all without protocol-specific parameter tuning.

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

Problem

• Mader model
• muscular energy metabolism
• open-source implementation
• reproducibility
• lactate steady state

Innovation

• Mader model
• muscle energy metabolism
• open-source implementation
• lactate steady state
• numerical simulation