Towards a mathematical theory of athletic training. Part 3. The theory of adaptation of energy metabolism to physical loads by Nikolai Ivanovich Volkov

Relevance. In the 1960s–1970s the Soviet Union saw the beginning of the active application of mathematical methods in sport. However, the contribution of Nikolai Ivanovich Volkov, Doctor of Biological Sciences, Professor, founder of the research field of “sport bioenergetics” and creator of the corresponding department at the GTSOLIFK, to the construction of a mathematical theory of adaptation to strenuous muscular activity has not yet been systematized. Meanwhile, it was he who was the first to describe the kinetics of bioenergetic processes by a system of differential equations and to introduce the criteria of power, capacity and efficiency of the energy supply mechanisms. Objectives of the work — systematization, interpretation and comparative analysis of the ideas, models and results of N.I. Volkov relating to the theory of adaptation and its mathematical expression; identification of the structure of the model he proposed and of its place among other approaches; establishment of a typology of relations between his constructions and modern models. Methods. The scientific works and biographical materials of N.I. Volkov of 1965–2013 were analysed, including the papers of 1965, 1966 and 1968, his doctoral dissertation (1990) and the monograph “Sport Bioenergetics” (2011). A comparison was made with the approaches of N.N. Engwer, E. Banister and V.N. Seluyanov and with modern quantitative models of the bioenergetics of muscular activity, and the relations between the models were classified. Results. The mathematical model of adaptation of N.I. Volkov has been systematized: 1) a compartmental kinetic model of energy metabolism in which the dynamics of the oxidizable substrate is described by a second-order linear differential equation; 2) biexponential equations of the kinetics of O₂ uptake during work and in recovery and an equation of lactate dynamics in the form of a difference of two exponentials; 3) a system of criteria of power, capacity and efficiency for the alactate, glycolytic and aerobic mechanisms; 4) an apparatus for optimizing the training process — target functions, the zone of optimal loads and the design of an extreme experiment. The approach of N.I. Volkov has been compared with the approaches of N.N. Engwer, E. Banister and V.N. Seluyanov and characterized as mechanistic, proceeding from the kinetics of metabolic processes. The solution of the equations of the model has been compared with current views: the fast and slow exponential components of the kinetics of O₂ uptake correspond to phase II and to the slow component, while the calculated time of the maximum lactate content corresponds to that observed at a load above the maximal lactate steady state. It is established that R. Morton relied on the 1966 paper of N.I. Volkov, and his three-component model is regarded in the literature as a development of the models of N.I. Volkov. A typology of relations is proposed: direct citation, structural coincidence, conceptual correspondence, analogy and commonality of the object. It is shown that the notion of critical power and of the limited work reserve above it corresponds to the criteria of power and capacity of N.I. Volkov, the maximal lactate steady state to the notion of a critical load, and the models of reconstitution of this reserve are analogous to the equations of recovery kinetics. Conclusion. N.I. Volkov created the first integral mathematical theory of the adaptation of energy metabolism to physical loads in Russian sport science, which completes the series of four mathematical models of athletic training considered in the present cycle of papers and supplies in it the mechanistic link — a description at the level of the kinetics of metabolic processes. His equations and criteria retain their significance as a basis for modern models of training process management, while the forecast he formulated — the development of mathematical models of adaptation and automated systems for controlling the physical condition of athletes — defines the research agenda at the present time as well.