Abstract
his study compared two physiological modeling approaches, the Peronnet-Thibault (P- T) model and the Minimal Power (MP) model, to identify key parameters representing individual physiological characteristics and to explore their applications in running training. Model parameters were estimated using nonlinear least squares fitting, and predictive performance was evaluated by the mean absolute error (MAE). Results from the World Running Records (WRR) indicated that the MP model generally outperformed the P-T model in linking running performance with physiological variables, demonstrating greater capability in extracting physiological parameters. Further validation using the British Runner Records (BRR) showed that the MP model achieved MAE values of 3.02% for males and 3.47% for females, reflecting strong generalization to real running performance. Furthermore, descriptive analyses of the relationships between MP model parameters and running performance further support its potential value in personalized training and performance prediction. Keywords:running performance; Peronnet-Thibault model; Minimum Power Consump- tion model; endurance; personalized training 1. Introduction With the accelerated pace of life and increased health awareness, running has be- come a common way to improve fitness and health due to its simplicity
the relationships between MP model parameters and running performance further support its potential value in personalized training and performance prediction. Keywords:running performance; Peronnet-Thibault model; Minimum Power Consump- tion model; endurance; personalized training 1. Introduction With the accelerated pace of life and increased health awareness, running has be- come a common way to improve fitness and health due to its simplicity and efficiency. Technological advances and the popularity of wearable devices have gradually made running more specialized and data-driven, enabling the collection of physiological and performance-related data such as heart rate,VO2max , and cadence [1,2]. In recent years, running performance analysis based on physiological parameters has gradually become a hot research topic. By integrating the athlete’s physiological state, training data, re- searchers try to reveal the relationship between physiological parameters and running performance, to provide theoretical support for personalized training and optimization of runners’ performance [3,4]. Over the past century, researchers have proposed a variety of models to predict running performance. Kennelly demonstrated a relationship between time (T) and distance (d), withT∝d 9/8 , based on competition data from animals and humans [5]. However, for distances between 100 m and 50 miles, the relative error can reach up to 9%. Hill [6] Appl. Sci.2025,15, 10720 https://doi.org/10.3390/app151910720
Appl. Sci.2025,15, 10720 2 of 18 proposed the hyperbolic power model, which reveals the maximum power output that athletes can maintain under different durations and lays a theoretical foundation for the correlation between performance and metabolism, and Keller [7,8] introduced the optimal control theory to simulate the speed distribution during a race through the variational method. Ward-Smith [9], based on the first law of thermodynamics, developed an energy consumption model from the perspective of mechanical work and energy output. This model integrates aerobic and anaerobic energy supply mechanisms, incorporating processes such as energy conversion, storage, heat dissipation, and resistance overcoming, to explain energy utilization and speed changes in high-intensity competitions. However, these models did not adequately capture the relationship between physio- logical parameters and athletic performance, prompting the development of new models. Daniels [10] proposed the VDOT model to quantify an athlete’s aerobic capacity and running economy, thereby enabling the prediction of race performance across different distances. However, it depends on precise values of maximal oxygen uptake (VO2max ), which limits the model’s applicability. Peronnet and Thibault proposed a model that incor- porates physiological parameters derived from the aerobic and anaerobic energy supply mechanisms, in order to link them with actual running performance and establish a balance between energy input and output [11]. Alvarez-Ramirez [12] introduced additional time scales into the P-T model to improve its predictive performance; however, this enhance- ment involves more parameters, which increases the model’s complexity and reduces its practicality. Emig et al. proposed a Minimum Power Consumption model to divide metabolic phases on a time scale, analyze the relationship between maximum average power, additional power demand and instantaneous power, and introduce an endurance index to improve performance prediction [13]. Subsequently, Emig and co-workers [14] validated the predictive accuracy of the model and further explored the evolution of world records and physiological parameters to refine the models of physiological mechanisms for running performance prediction. Physiological parameters (e.g., maximal oxygen up- take, aerobic endurance, etc.) employed in model construction can be used to develop personalized training programs aimed at optimizing athlete performance under varying physiological conditions. In
the predictive accuracy of the model and further explored the evolution of world records and physiological parameters to refine the models of physiological mechanisms for running performance prediction. Physiological parameters (e.g., maximal oxygen up- take, aerobic endurance, etc.) employed in model construction can be used to develop personalized training programs aimed at optimizing athlete performance under varying physiological conditions. In this study, we first systematically compare two physiologically based models, the Peronnet-Thibault (P-T) model and the Minimum Power Consumption (MP) model, to iden- tify key parameters that are more representative of an individual’s physical performance characteristics. To evaluate the applicability of the MP model to real-world running, we fitted the model using the personal best records of British runners, and confirmed through prediction results its effectiveness in capturing individual physiological parameters under real conditions. In addition, the study also attempted to explore the association between the model parameters and actual running performance, and further analyzed the distribution characteristics of these parameters in the population and their representativeness. By revealing the variability of physiological abilities among individuals, it aims to provide a theoretical basis and data support for achieving personalized and scientific training. 2. Method 2.1. The Peronnet-Thibault Model During running, the energy required by the body is supplied by two main systems: the aerobic system and the anaerobic system. The aerobic system supplies energy steadily through oxidative processes, while the anaerobic system releases energy rapidly through lactate metabolism and glycolysis during high-intensity exercise. Based on this physiologi- cal mechanism, the Peronnet-Thibault (P-T) Model calculates the power generated by the aerobic and anaerobic systems separately and integrates the two to obtain the total power
Appl. Sci.2025,15, 10720 3 of 18 output of the human body for different durations, which is used to simulate the energy expenditure and exercise performance during running. The average powerPTin the model is obtained based on a combination of theoretical derivation and empirical observation, and is expressed as the sum of the average aerobic power (Paer) and the average anaerobic power (Panr), as follows: PT=Paer+Panr= ≤ S T ⊆ 1−e − T k 2 ⊇≥ + 1 T Z T 0 h BMR+B ≍ 1−e −t/k 1 ≡⟩ dt (1) whereSdenotes the total energy currently available for anaerobic metabolism,BMRis the basal power,Bis the difference between peak and basal power (B=P peak−BMR ). Two constants are introduced to characterize the dynamics of the energy systems.k1(∼30s) characterizes the time constant of aerobic metabolism, reflecting the time scale over which aerobic power gradually increases due to the lag between cardiopulmonary circulation and mitochondrial oxidative reactions [15,16], whilek2(∼20s) represents the time constant of anaerobic metabolism, describing the timescale over which anaerobic energy supply declines exponentially due to limited energy reserves and the accumulation of metabolic by-products [11]. The time (T MAP) for which elite endurance athletes maintain maximum aerobic power (MAP) is finite, usually about 420 s [17–19]. For running races with durations equal to or less than that time, peak power (P peak) is approximately equal toMAP, whereas for races with durations greater than that time,P peakdecreases linearly withlnTas time increases [20] and the magnitude of this decrease is expressed asE. This decline occurs because, as the contribution of anaerobic metabolism diminishes over time, the limited capacity of the aerobic system cannot fully compensate for the loss, resulting in a progressive reduction in the power output. Thus, the parameterBcan be expressed as B= ( MAP−BMR,T<T MAP MAP−BMR+[E(lnT−lnT MAP)],T>T MAP (2) Peronnet and Thibault proposed that it can be assumed that the total anaerobic metabolic energy (A) can be used in its entirety for any race longer than 120 s to 150 s [11], but shorter thanT MAPin duration. BeyondT MAP, the anaerobic metabolic energy (S) gradually decays as the duration of the run
( MAP−BMR,T<T MAP MAP−BMR+[E(lnT−lnT MAP)],T>T MAP (2) Peronnet and Thibault proposed that it can be assumed that the total anaerobic metabolic energy (A) can be used in its entirety for any race longer than 120 s to 150 s [11], but shorter thanT MAPin duration. BeyondT MAP, the anaerobic metabolic energy (S) gradually decays as the duration of the run continues to increase, wheref=−0.233 [21] S= ( A,T<T MAP A+A f(lnT−lnT MAP),T>T MAP (3) On the other hand, the actual average output power (Pv) required to run at a given speed (v) can be calculated by the formula proposed by di Prampero [22] Pv=BMR+3.86v+0.4BSA∗v 3 /BM+v 3 /D (4) whereBSArepresents body surface area,BMrepresents body mass (set at 70 kg for males and 50 kg for females, with correspondingBSAof 1.8m 2for males and 1.6m 2for females), andDdenotes running distance. Therefore, based on the energy balance between input and output and the relationship between speed, time, and distance, the P-T model can be concluded as ≤ S T ⊆ 1−e − T k 2 ⊇≥ + 1 T Z T 0 h BMR+B ≍ 1−e −T/k 1 ≡⟩ dt=BMR+3.86 D T +0.4BSA D 3 BM∗T 3 + D 2 T 3 (5)
Appl. Sci.2025,15, 10720 4 of 18 Consequently, based on these three parameters (MAP,EandA), a P-T model was constructed as shown in Equation (5). The ability of an athlete to use the anaerobic pathway to generate energy during a brief period of high-intensity activity is represented by total anaerobic energy (A). The higher the value of parameterA, the more explosive the athlete is in the anaerobic environment. The maximum amount of energy that an athlete can produce in an aerobic environment is referred to as maximum aerobic power (MAP). The magnitude ofMAPdirectly affects an athlete’s endurance capacity during prolonged exercise. In the P-T model, the degree of aerobic endurance is represented by the magnitude of aerobic energy expenditure (E). More specifically, a higher value of the parameterEindicates that the athlete has greater aerobic endurance and is more effective in maintaining a high level of power output throughout the competition. Furthermore, these three parameters interact differently depending on the race dura- tion relative toT MAP. WhenT<T MAP , the outcome is primarily determined by anaerobic metabolism, and a largerAenhances the athlete’s capability in short-distance, high-intensity efforts. WhenT>T MAP , aerobic metabolism predominates, with a largerMAPand a higherEimproving the performance of long-distance. By considering these effects and interactions, the P-T model bridges the gap between theory and practice. 2.2. The Minimum Power Consumption Model The Minimum Power (MP) Model was proposed by Emig et al. [13,14,23] to eliminate extraneous normalization parameters from the model through the lens of relative power. The model defines relative power (p) as the ratio of actual power output (P−P b ) to aerobic power reserve (Pm−P b ). By using velocity (v) as a parameter to quantify power, running economy (p(v)) is then derived. The relationship between power and velocity further simplifies the equation so that running economy can be expressed as p(v)=P(v)−P b/Pm−P b=v/vm (6) where the minimum velocity corresponding to the production of maximum aerobic power (Pm) is referred to as the crossover velocity (vm) [24]. AndP bdenotes the base power, and P(v)denotes the output power at an average speed (v). To construct a running performance
and velocity further simplifies the equation so that running economy can be expressed as p(v)=P(v)−P b/Pm−P b=v/vm (6) where the minimum velocity corresponding to the production of maximum aerobic power (Pm) is referred to as the crossover velocity (vm) [24]. AndP bdenotes the base power, and P(v)denotes the output power at an average speed (v). To construct a running performance prediction modelT(d), it is necessary to know the best time (T) a runner can run at a given power. The maximum average power (Pmax(T) ) that can be sustained for a durationTis used to quantify this information. In addition, the instantaneous power (PT(t)) consumed by the runner during a race of durationTis defined. Since during the actual race the runner will become fatigued, this will result in the instantaneous power being greater than the maximum average power, which needs to be compensated by supplementing with additional energy, which is known as supplemental power (P sub(T) ). Since different distances involve different energy mechanisms and differ- ent energy systems produce energy in different ways, different amounts of supplemental power need to be provided over different time frames. Therefore, to determine the supple- mentary power’s, a time scale (tc) should be introduced to distinguish between short and long distances. This means that the supplementary power can be expressed as P sub(T)= ( Ps,T<tc Ps tc T +P l T−tc T ,T>tc (7)
Appl. Sci.2025,15, 10720 5 of 18 To build the model, Emig et al. [13]. proposed a relation that states that the average of the instantaneous power is equal to the sum of the supplementary power and the maximum average power (Pmax(T)) over a durationT. This is expressed as Pmax(T)+P sub(T)= 1 T Z T o PT(t)dt (8) On this basis, they made a key conjecture that the instantaneous power utilized at a certain momenttis equal to the maximum average power utilized for the remaining time (T−t) [25]. It can be expressed as PT(t)=Pmax(T−t) (9) Through this relationship,Pmax(T)at different periods can be expressed as Pmax(T)= ( Pm−Pslog T tc ,T<tc Pm−P llog T tc ,T>tc (10) By solving this equation inversely and combining it with the relationship between power and speed, the expressionT(v)for the variation of time with speed can be obtained, based on which the expression is simplified by the introduction of two new variables, the endurance indices for long and short distances,γs=Ps/(Pm−P b) , andγ l=P l/(Pm−P b) . Thus, the time that running can be sustained under a given power condition is expressed as T(v)= ( tcexp h vm−v γ lvm i ,v<vm tcexp[ vm−v γsvm ],v>vm (11) Using the relationship between speed, distance, and time (v=d/t ), we may estimate the duration of a run at a specific distance (T(d)), that is T(d)= − d γ lvm 1 W −1 h − d dcγ l e −1/γ l i,d>dc − d γsvm 1 W −1 h − d dcγs e −1/γs i,d<dc (12) The equation is expressed as the real branchW−1(z) of the Lambert W-function [26], defined as the inverse of the functionw→we w . This branch provides an analytical solution to equations of the formy=xe x , which naturally arise in the derivation ofT(d). Thus, the selection ofW−1(z)is a natural choice for obtaining a closed-form solution. The expressionT(d)denotes the Minimum Power Consumption (MP) model, which is based on four key parameters: crossover velocity (vm), crossover time (tc), long distance endurance index (γ l), and short distance endurance index (γs). As a
equations of the formy=xe x , which naturally arise in the derivation ofT(d). Thus, the selection ofW−1(z)is a natural choice for obtaining a closed-form solution. The expressionT(d)denotes the Minimum Power Consumption (MP) model, which is based on four key parameters: crossover velocity (vm), crossover time (tc), long distance endurance index (γ l), and short distance endurance index (γs). As a model of important physiological parameters, the model provides a systematic framework for a comprehen- sive assessment of an athlete’s competitive level. Among them, parametervmis a key index to measure the aerobic metabolic capacity and endurance level of athletes; param- etertcrefers to the time for which an athlete is able to sustain maximal aerobic power, reflecting the aerobic endurance as well as the ability to maintain high-intensity output during the competition. In addition, parametersγ landγsare used to characterise athletes’ endurance at different distances. The higherγsand lowerγ lcorrelate with better en- durance and running performance. The model specifies long-distance running endurance asE l=exp(1−p/γ l) , whereas short-distance running endurance asEs=exp(1−p/γs) . E lmainly measures the stability of athletes’ performance under prolonged, high-continuous loading conditions; whereasEsfocuses more on the athletes’ ability to produce output
Appl. Sci.2025,15, 10720 6 of 18 during short, high-intensity exercise, reflecting in particular their anaerobic metabolism level and explosive power. Combining these four parameters, the generalized running model can comprehen- sively reflect the physiological characteristics of athletes in various distance running events. The model provides a scientific and systematic assessment tool for understanding key physiological mechanisms such as endurance and aerobic capacity, helps to analyze physi- ological differences between individuals and supports the development of personalized training plans. 2.3. Data Resource In this study, we used two main data sources: (1)World Running Records (WRR): The dataset is derived directly from the authoritative IAAF Handbook [27], which provides us with running world records for nine different distances ranging from the 1000-m sprint to the full marathon, with records for males ranging from 1918 to 2023 and for females ranging from 1984 to 2023. This choice attempts to fully cover a wide range of results in terms of short-distance explosive power and long-distance endurance, in addition to guaranteeing the validity and authenticity of the data. (2) the average runner, we have selected data from the BRR database from a widely used online resource (http://www.thepowerof10.info/ 28]. We focus on the analysis of five representative race distances—5 km (5 K), 10 km (10 K), 10 miles (10 M), half marathon (HM) and full marathon (Mar). Based on this, we select for runners who had raced at all five distances to construct an exhaustive dataset of 2079 UK runners. The dataset is relatively gender balanced, with 38.3% female and 61.7% male. 2.4. Calculating Model Parameters The P-T model and the MP model rely on a set of independent parameters that are used to characterize physiological information about an individual. These parameters can be estimated by fitting a given set of exercise performance data derived from exercise performed at maximum intensity. To calculate the model parameters, a nonlinear least squares method [29] was used to fit the WRR data for a range of distances between 1000 m and 10,000 m to minimize the difference between actual race times (T(d)) and predicted times ( . T(d) ).
fitting a given set of exercise performance data derived from exercise performed at maximum intensity. To calculate the model parameters, a nonlinear least squares method [29] was used to fit the WRR data for a range of distances between 1000 m and 10,000 m to minimize the difference between actual race times (T(d)) and predicted times ( . T(d) ). Specifically, the objective of fitting using the least squares method was to minimize the residual sum of squares (RSS). RSS=min ⊂ ∑ . T(d)−T(d) 2 ⊃ (13) To analyze the distributional properties of the parameters in more detail and to assess their validity in characterizing individual endurance levels and aerobic capacity, we collected personal bests from BRR dataset at four different race distances (5 K, 10 K, 10 M, and HM), and fit the data to the distances using a unified nonlinear least squares method [29].
Description
The study evaluates physiological models for running performance using real-world data.