A dual-threshold muscular breathing inequality model for continuous analysis of ventilation during treadmill cardiopulmonary exercise testing

OBJECTIVE: To develop and validate a continuous, mathematically well-defined model for representing ventilatory dynamics during incremental treadmill exercise that overcomes the limitations of conventional linear, exponential, hyperbolic sine, and breakpoint-based approaches, while retaining interpretability and robustness at the signal level for modeling and system identification tasks. APPROACH: We propose the Dual-threshold Muscular Breathing Inequality (MBI) model, a parametric formulation that represents minute ventilation as a function of oxygen uptake using a linear baseline combined with two logistic transition components. This structure enables explicit parameterization of transition timing, magnitude, and sharpness within a single smooth and differentiable function. Model parameters were estimated using a hybrid global-local optimization strategy designed to ensure numerical stability and practical identifiability. The model was evaluated using a large dataset of 981 maximal treadmill cardiopulmonary exercise tests, comprising over 420 000 breath-by-breath data points. Performance was benchmarked against established linear, exponential, hyperbolic sine, piecewise-linear, and continuous nonlinear models using goodness-of-fit metrics, information criteria, and non-parametric statistical comparisons at both pooled and individual levels. MAIN RESULTS: The proposed MBI model provided a stable and accurate representation of ventilatory dynamics across the full range of exercise intensity. It achieved strong descriptive performance at both population and individual levels, maintained robustness across intensity domains, and demonstrated statistically significant advantages over competing formulations. The model yielded interpretable parameters that separated transition location from transition magnitude and sharpness, revealing conserved features of ventilatory organization alongside highly variable response kinetics. Systematic differences between MBI-derived transition midpoints and conventional breakpoint estimates were observed, reflecting differences in mathematical definitions rather than inconsistencies in the underlying data. SIGNIFICANCE: By modeling ventilatory transitions as continuous processes rather than discrete events, the dual-threshold MBI framework provides a compact, model-based approach for quantitative analysis and parameter estimation of ventilatory time-series data in treadmill cardiopulmonary exercise testing. The proposed formulation supports reproducible system identification and comparative analysis across individuals and conditions, offering a practical engineering-oriented tool for measurement-based analysis of treadmill exercise ventilation.