Thorsten Dahmen

Applications of Mathematical Models of Road Cycling

Abstract This contribution discusses several use cases of mathematical models for road cycling. A mechanical model for the pedaling forces is the basis for an accurate indoor ergometer simulation of road cycling on real-world tracks. Together with a simple physiological model for the exertion of the athlete as a function of his/her accumulated power output, an optimal riding strategy for time trials on mountain ascents is computed. A combination of the two models leads to a mathematical optimization problem that can be solved numerically by discretization. The physical model depends sensitively on an accurate estimation of the road slope on the course. For this purpose, we also present a new method that combines model-based slope estimations with noisy measurements from multiple GPS signals of differing quality. Altogether, we provide a means to simulate road cycling on real-world tracks, to analyze cyclist performance, to identify and quantify potential performance improvement, as well as to instruct the athlete exactly where and how to change his/her pacing strategy to achieve these gains.

Modeling Oxygen Dynamics under Variable Work Rate

Measurements of oxygen uptake and blood lactate content are central to methods for assessment of physical fitness and endurance capabilities in athletes. Two important parameters extracted from such data of incremental exercise tests are the maximal oxygen uptake and the critical power. A commonly accepted model of the dynamics of oxygen uptake during exercise at constant work rate comprises a constant baseline oxygen uptake, an exponential fast component, and another exponential slow component for heavy and severe work rates. We generalized this model to variable load protocols by differential equations that naturally correspond to the standard model for constant work rate. This provides the means for prediction of oxygen uptake response to variable load profiles including phases of recovery. The model parameters were fitted for individual subjects from a cycle ergometer test. The model predictions were validated by data collected in separate tests. Our findings indicate that oxygen kinetics for variable exercise load can be predicted using the generalized mathematical standard model, however, with an overestimation of the slow component. Such models allow for applications in the field where the constant work rate assumption generally is not valid.

Kinetic analysis of oxygen dynamics under a variable work rate

Measurements of oxygen uptake are central to methods for the assessment of physical fitness and endurance capabilities in athletes. Two important parameters extracted from such data of incremental exercise tests are the maximal oxygen uptake and the critical power. A commonly accepted model of the dynamics of oxygen uptake during exercise at a constant work rate comprises a constant baseline oxygen uptake, an exponential fast component, and another exponential slow component for heavy and severe work rates. We have generalized this model to variable load protocols with differential equations that naturally correspond to the standard model for a constant work rate. This provides the means for predicting the oxygen uptake response to variable load profiles including phases of recovery. The model parameters have been fitted for individual subjects from a cycle ergometer test, including the maximal oxygen uptake and critical power. The model predictions have been validated by data collected in separate tests. Our findings indicate that the oxygen kinetics for a variable exercise load can be predicted using the generalized mathematical standard model. Such models can be applied in the field where the constant work rate assumption generally is not valid.