**1. Introduction**

The benefits of regular physical exercise are widely debated and include reducing the risk of obesity [1] or cardiovascular diseases [2]. On the other hand, improperly performed training with excessive intensity may negatively affect the organism's homeostasis and increase the risk of injury [3].

**Citation:** Wiecha, S.; Kasiak, P.S.; Cie´sli ´nski, I.; Maciejczyk, M.; Mamcarz, A.; Sli ´ z, D. Modeling ˙ Physiological Predictors of Running Velocity for Endurance Athletes. *J. Clin. Med.* **2022**, *11*, 6688. https:// doi.org/10.3390/jcm11226688

Academic Editor: Birna Bjarnason-Wehrens

Received: 26 September 2022 Accepted: 9 November 2022 Published: 11 November 2022

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**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

The concept of anaerobic threshold (AT) is widely discussed in exercise physiology [4]. As envisioned by Karlman Wasserman, the AT linked the increase in blood lactate concentration ([La−]b), during a strenuous incremental cardiopulmonary exercise test (CPET), with an excess arterial CO2 accumulation and its further pulmonary output [5]. Above the AT, [La−]b increase leads to temporary acidosis. The endurance capacity of the whole system is usually sufficiently high to cope with the incoming state [6]. During steady-state exercise with intensity above the AT, an equilibrium in [La−]b appearance and its elimination is observed [6]. Thus, AT is a useful practical indicator to provide personalized training recommendations (with the aim of adjusting exercise intensity to set goals) and load monitoring [7].

The respiratory compensation point (RCP) is the intensity at which arterial CO2 begins to decrease during demanding activity due to breathing capacity [8]. Above the RCP, the intensity of acidic ion accumulation exceeds their systemic or respiratory elimination abilities and indicates reduced endurance capacity. This leads to an over-decrease in the serum pH during graded exercise. This threshold indicates how long a high-intensity effort can be sustained [9].

The velocity at the anaerobic threshold (VAT), at the respiratory compensation point (VRCP), and at its maximum (Vmax) play an essential position in the endurance performance assessment, both for professional and recreational endurance athletes (EA), as well as for the general population under clinical conditions [10,11].

These variables are the shift points of aerobic exercise to anaerobic metabolism and can be used as one of the parameters to evaluate the maximum endurance capacity [7]. Moreover, they closely positively correlate with exercise abilities [4]. They could be incorporated into the prescription for the advancement of training plans [7] or competition strategies [12] for special and narrow populations (e.g., EA), and in sports diagnostics whenever controlled running intensity is required (i.e., in clinical CPET) [7]. Furthermore, currently, these variables most closely correspond to the EA critical power sustainability [13,14].

Apart from Vmax, maximal aerobic speed (MAS), which is directly related to VO2max, is another important aspect of overall performance evaluation. However, as the aim of this research is to predict Vmax, we recommend that further studies should be performed to analyze the MAS.

Numerous parameters, such as heart rate (HR), oxygen uptake (VO2), or anthropometric data (i.e., height, age, and gender), are widely discussed in the development of multivariable prediction models that provide an increasingly more suitable alternatives to direct CPET measurements [15].

Several studies have attempted to develop and validate various non-invasive prediction equations for different sports performance measurements (i.e., for HR, VO2, and others) [11,15,16]. However, they were mostly conducted on general populations or on small athletic samples, and thus, they can only be extrapolated to a low degree [17]. In addition, their methodology is widely variable, and only a few of them fulfilled recommended TRIPOD guidelines [18]. Thus, the actual number of VAT, VRCP, and Vmax predictive models is limited, despite being significant measures of endurance capacity [19,20]. Moreover, although the variables influencing the running performance are well researched, the authors have not yet assessed how accurately they can be used to estimate running velocity by further including them in prediction models.

The aims of this study were: (1) to find the somatic and CPET variables that are the most responsible for running velocity, (2) to develop a prediction method for VAT, VRCP, and Vmax, in accordance with TRIPOD recommendations [18], (3) to internally validate the obtained formulae, (4) to assess the accuracy of the current machine-learning abilities to predict running velocity based on the primarily determined variables, and (5) to evaluate practical applications of such an approach in sports or clinical conditions based on actual knowledge regarding exercise physiology.
