Development of predictive mathematical models for physical performance parameters in sports and sports medicine

INTRODUCTION

The application of mathematical models to predict and evaluate physiological parameters in professional athletes is closely related to both advancements in sports science and analytical methods.

In the field of sports performance analysis and forecasting, modern statistical modeling techniques are increasingly finding application, driving the transformation of research methodologies worldwide. While sports science has traditionally relied on conventional statistical approaches, recent innovations have introduced more sophisticated tools such as machine learning algorithms and hierarchical modeling. These advanced techniques enable researchers to identify complex relationships within both medical and athletic data, leading to deeper insights into the predictors of performance decline and the optimization of training strategies [1–6].

Previous research in this domain has primarily focused on understanding individual and collective trends in general and sport-specific physical performance metrics. For instance, studies on running performance have explored three key areas: (1) physiological determinants of world-record achievements; (2) development of equivalent scoring and race outcome prediction systems; and (3) modeling individual physiological parameters in track and field athletes [7–9]. Despite these efforts, there is a lack of comprehensive knowledge-based models capable of integrating and comparing all the above aspects. This determines the relevance of developing versatile, personalized, and accessible mathematical models for predicting physical performance parameters in sports.

Recent achievements in the integration and analysis of big data have enhanced the accuracy of performance predictions. In one study, the researchers used an online database of the performance of British athletes (1954–2013) to propose a simplified model capturing key performance characteristics while maintaining empirical validity. This model demonstrated a remarkably low mean prediction error for specific athletic outcomes, marking an important step toward unifying performance understanding through data analytics [10][11].

A review of literature on predictive mathematical models in sports medicine reveals that most studies focus either on injury risk prediction across various sports or on estimating the probability of achieving target performance outcomes [12–15]. However, no studies examining potential correlations between diagnosed medical conditions in world-class athletes and their predicted impact on physical performance decline have been found.

With advances in neural networks and machine learning, researchers — including international collaborations — are now combining efforts to study the interplay of physiological and psychological factors affecting athletic performance. The goal is to optimize training regimens through predictive models capable of accounting for the dynamic nature of sports performance and its relationship with athletes’ physiological processes.

The present study is aimed at developing methods for analyzing clinical data from comprehensive medical examinations (CME) of athletes.

MATERIALS AND METHODS

This study analyzed and mathematically processed anonymized clinical and instrumental examination data derived from medical records of world-class athletes who underwent comprehensive medical examinations (CME) in the Burnasyan Federal Medical Biophysical Center in 2019–2023. The dataset included clinical information from 6222 athletes (3792 males and 2430 females, mean age 23.3 ± 5.1 years).

The data were stratified by sex and sport categories: cyclic sports (1376 athletes, including 861 males and 515 females); complex coordination sports (1342 athletes, including 761 males and 581 females); team sports (1618 athletes, including 980 males and 638 females); and combat sports (1886 athletes, including 1190 males and 696 females).

The article presents data on male athletes from cyclic (n = 861) and team sports (n = 980). These cohorts were selected as those capable of providing the most representative and homogeneous samples for robust statistical analysis and reliable mathematical modeling. Data from other sports categories were utilized for preliminary screening and selection purposes.

During model development, the following inclusion criteria were applied:

• parameters had to reflect the key indicators of physical condition and functional capacity (morphofunctional status, aerobic/anaerobic capabilities, etc.);
• data were collected within homogeneous athlete groups (matched by sex, age, sports category, and skill level);
• measurements were obtained using standardized protocols (with uniform CME procedures for exercise testing).

The exclusion criteria comprised incomplete, erroneous, or anomalous parameters that could distort modeling results.

Thus, the following baseline parameters for mathematical modeling were used: age (years); weight (kg); height (cm); oxygen consumption at AT (anaerobic threshold) level (VO_2 AT, mL/min/kg); peak oxygen consumption during exercise testing (VO_2 peak, mL/min/kg); respiratory exchange ratio (R, relative units); resting heart rate (HR_rest, bpm); heart rate at aerobic threshold level (HR_AerT, bpm); heart rate at AT level (HR_AT, bpm); peak exercise heart rate (HR_peak, bpm); heart rate at 3 min of recovery (HR_3min, bpm); power output at anaerobic threshold level (Pwr_AT, W); peak power output during testing (Pwr_peak, W); relative power output at anaerobic threshold level (Pwr_AT/weight, W/kg); relative peak power output during testing (Pwr_peak/weight, W/kg). In each sports category, the statistical dataset of the studied parameters was combinatorially grouped by qualitative presence (1)/absence (0) of pathological conditions diagnosed by the following medical specialists: gastroenterologist, dermatovenerologist, cardiologist, neurologist, otolaryngologist, ophthalmologist, dentist, orthopedic traumatologist, and endocrinologist.

All quantitative parameters in the modeling are presented as: X_М — sex (0 — female, 1 — male); X_age — age; X_weight — weight; Х_height — height; X_gastro — pathological condition by gastroenterologist (0 — no, 1 — yes); X_derm — pathological condition by dermatovenerologist (0 — no, 1 — yes); X_cardio — pathological condition by cardiologist (0 — no, 1 — yes); X_neuro — pathological condition by neurologist (0 — no, 1 — yes); X_oto — pathological condition by otolaryngologist (0 — no, 1 — yes); X_ophth — pathological condition by ophthalmologist (0 — no, 1 — yes); X_dent — pathological condition by dentist (0 — no, 1 — yes); X_trauma — pathological condition by orthopedic traumatologist (0 — no, 1 — yes); X_endo — pathological condition by endocrinologist (0 — no, 1 — yes); Х_VO2 AT — V(O₂) at AT level; X_VO2 peak — V(O₂) at peak; X_R — respiratory coefficient; X_HR rest — HR at rest; Х_HR AerT — HR at AerT; X_HR AT — HR at AT; Х_HR peak — HR at peak; Х_HR 3min — HR at 3-min recovery; X_PWR AT — power at AT; X_PWR peak — power at peak; X_{PWR AT/weight} — power at AT divided by weight; X_{PWR peak/weight} — power at peak divided by weight.

Statistical analysis was performed using the StatTech v. 4.6.0 software (StatTech, Russia). The linear regression method was used to examine the dataset structure and establish relationships between its parameters. Mathematical models were developed to describe dependencies between the group of quantitative physical performance indicators and the presence/absence of pathological conditions identified by physicians during in-depth medical examinations, for each sport category and sex. Regression equation coefficients were determined using the least squares method with Cramer’s formulas.

RESULTS AND DISCUSSION

During data preparation for linear regression modeling, the relevant CME results of athletes were compiled to ensure the required precision. Incomplete, erroneous, or anomalous values were excluded, along with non-informative features. Table 1 presents the general characteristics of male athletes from cyclic and team sports included in the final sample.

Among the exercise testing parameters characterizing the overall physical performance (items 4–15 in Table 1), the most physiologically relevant indicators for sports medicine applications are those of gas exchange: VO_2 AT, VO_2 peak, respiratory exchange ratio, as well as the direct measure of achieved power output at anaerobic threshold.

The models presented below describe general relationships (and interdependencies) between the selected physical performance metrics (Y value in the formula) and all other parameters, including the presence/absence of pathological conditions during examination (X values in the formula).

The observed relationship for oxygen consumption at anaerobic threshold VO_2 AT (1 — cyclic sports, 2 — team sports) is described by the following linear regression equations:

Y_VO2 AT = –5.313 – 0.424 × X_neuro – 0.369 × X_dent + 0.921 × X_endo + 0.058 × X_weight + 0.479 × X_VO2 peak – 0.024 × X _HR rest + 0.029 × X_HR AT + 0.038 × X_PWR AT – 0.047 × X_PWR peak + 5.924 × X_{PWR AT/weight}, (1)

Y_VO2 AT = –0.301 + 0.572 × X_М + 0.576 × X_VO2 peak – 0.022 × X_HR rest + 0.072 × X_HR AT – 0.033 × X_HR peak – 0.012 × X_HR 3min – 0.004 × X_PWR peak + 7.671 × X_{PWR AT/weight} – 3.725 × X_{PWR peak/weight}. (2)

The obtained regression models for oxygen consumption at anaerobic threshold (VO_2 AT) demonstrated the following characteristics:

1. The multiple correlation coefficient was R_xy= 0.965 for cyclic sports and R_xy= 0.948 for team sports, indicating a highly strong relationship according to the Chaddock scale;
2. The coefficient of multiple determination was R²≈ (0.965)²= 0.931 for cyclic sports and R² ≈ (0.948)² = 0.899 for team sports. These models can predict VO_2 AT values with high accuracy: they explain 93.1% of observed variance in cyclic sports and 89.9% in team sports. The models were statistically significant (p < 0.001).

After accounting for interdependencies among physical performance parameters, negative associations were found between VO_2 AT and neurological/dental pathological conditions, while a positive association was observed with endocrine disorders.

The relationship for peak oxygen consumption (VO_2 peak) (3 — cyclic sports, 4 — team sports) is described by the following linear regression equation:

Y_VO2 peak = 5.920 + 0.737 × X_М – 1.417 × X_endo + 0.865 × X_VO2 AT – 0.050 × X_HR AT + 0.036 × X_HR peak – 6.026 × X_{PWR AT/weight} + 6.668 × X_{PWR peak/weight}, (3)

Y_VO2 peak = 5.743 + 5.743 × X_М – 0.091 × X_age + 0.823 × X_VO2 AT – 6.022 × X_R + + 0.023 × X_HR rest – 0.076 × X_HR AT + 0.048 × X_HR peak – 6.090 × X_{PWR AT/weight} + + 8.022 × X_{PWR peak/weight}. (4)

The key characteristics of the developed models:

1. The multiple correlation coefficient was R_xy= 0.933 for cyclic sports and R_xy= 0.919 for team sports, indicating a highly strong association according to the Chaddock scale.
2. The coefficient of determination reached R²≈ (0.933)²= 0.871 for cyclic sports and R² ≈ (0.919)² = 0.844 for team sports.

The models demonstrate high predictive accuracy for VO_2 peak values, explaining 87.1% of the observed variance in cyclic sports and 84.4% in team sports. All models showed statistical significance (p < 0.001).

After analyzing interdependencies among physical performance parameters, a negative association between VO_2 peak and endocrine disorders was observed.

The relationship between the respiratory exchange ratio (R) (5 — cyclic sports, 6 — team sports) and quantitative factors is described by the following linear regression equation:

Y_R = 1.436 – 0.013 × X_М + 0.013 × X_dent – 0.002 × X_height – 0.002 × X_weight – 0.0001 × X_HR rest + 0.0001 × X_HR AerT – 0.001 × X_HR AT + 0.001 × X_HR peak + 0.001 × X_HR 3min + 0.001 × X_PWR AT – 0.167 × X_{PWR AT/weight} + 0.107 × X_{PWR peak/weight}, (5)

Y_R = 0.388 + 0.388 × X_М + 0.012 × X_ophth + 0.009 × X_dent + 0.002 × X_weight – 0.002 × X_VO2peak – 0.001 × X_HR AT + 0.000 × X _HR 3min + 0.001 × X_HR peak – 0.001 × X_PWR AT + 0.092 × X_{PWR peak/weight}. (6)

The characteristics of the developed models are as follows:

1. For cyclic sports, the multiple correlation coefficient was R_xy= 0.830, while for team sports it was R_xy= 0.783, indicating a strong association according to the Chaddock scale.
2. The coefficient of determination was R²≈ (0.830)²= 0.689 for cyclic sports and R² ≈ (0.783)² = 0.613 for team sports.

The models demonstrate moderately high predictive accuracy for the respiratory exchange ratio (R), explaining 68.9% of the observed variance in cyclic sports and 61.3% in team sports. All models were statistically significant (p < 0.001).

After controlling for interdependencies among physical performance parameters, positive associations were identified between the respiratory exchange ratio (R) and ophthalmological/dental pathological conditions.

The relationship between power output at anaerobic threshold (Pwr_AT) (7 — cyclic sports, 8 — team sports) and quantitative factors is described by the following linear regression equation:

Y_PWR AT = –62.319 + 0.115 × X_height + 0.274 × X_weight + 0.167 × X_VO2 AT + 17.519 × X_R – 0.092 × X_HR AerT +  0.153 × X_HR AT – 0.069 × X_HR peak + 0.724 × X_PWR peak + 74.556 × X_{PWR AT/weight} – 55.694 × X_{PWR peak/weight}, (7)

Y_PWR AT = –49.907 + 1.485 × X_neuro + 1.185 × X_офт – 0.102 × X_age + 0.089 × X_height + + 0.434 × X_weight + 0.154 × X_HR AT – 0.159 × X_HR peak + 76.528 × X_{PWR AT/weight} + 0.652 × X_PWR peak – 48.533 × X_{PWR peak/weight}. (8)

The characteristics of the developed models are as follows:

1. The multiple correlation coefficient was R_xy= 0.996 for cyclic sports and R_xy= 0.994 for team sports, indicating an exceptionally strong association according to the Chaddock scale.
2. The coefficient of determination was R²≈ (0.996)²= 0.993 for cyclic sports and R² ≈ (0.994)² = 0.988 for team sports. The models demonstrate extremely high predictive accuracy for power output at anaerobic threshold (Pwr_AT), explaining 99.3% of the observed variance in cyclic sports and 98.8% in team sports. All models were statistically significant (p < 0.001).

After controlling for interdependencies among physical performance parameters, positive associations were identified between Pwr_AT and neurological/ophthalmological pathological conditions.

In a similar manner, clinical data from 6222 athletes (accounting for gender and sport category differences) were processed. After developing 12 separate models for each sports category, significant relationships (both positive and negative) between the studied parameters and the presence/absence of diagnosed pathological conditions were quantified.

The total number of significant associations between physical performance parameters and the presence/absence of documented pathological conditions across all studied athlete groups reached 46 (Table 2).

Out of 46 established dependencies, the strongest correlations were found with ophthalmologist-diagnosed conditions (8 confirmed relationships) and dentist-diagnosed conditions (11 confirmed relationships). The developed mathematical models for predicting physical performance parameters confirmed a strong relationship between the probability of achieving planned training results or performance in major competitions and the presence of specialist-diagnosed conditions identified during the comprehensive medical examinations of athletes.

CONCLUSION

In this study, effective predictive mathematical models of physical performance parameters, accounting for the presence/absence of diagnosed conditions, have been developed using clinical data from comprehensive medical examinations of world-class athletes and linear regression methods. The developed models can be used to carry out a comprehensive assessment of athletes’ functional status, thus facilitating a more accurate prediction of physical performance levels and optimization of professional activities while minimizing risks of overtraining and injuries.

Thus, the study results contribute to the development of sports medicine and provide a scientific basis for decision making in the field of athlete preparation and medical support. This will subsequently lead to changes in sports physicians’ approaches to interpreting the results of comprehensive medical examinations.