The maximum heart rate (MHR) is the highest heart rate achieved during a graded exercise performed at maximum effort, and it can be measured in the laboratory or in the field. MHR determines the upper limit of cardiovascular function (Mahon et al., 2010[18]). MHR is a physiological measure that is frequently used as a benchmark for maximal exercise intensity to avoid errors when defining training loads (Antonacci et al., 2007[3]). This is owing to the established linear relationship between heart rate (Zanuto et al., 2020[31]) and oxygen consumption (VO₂) in activities with progressive intensity (da Silva et al., 2021[7]). According to this relationship, percent of MHR corresponds to a percentage of VO₂max at a particular effort intensity. Therefore, we can supervise the HR in % MHR and control the intensity of a given physical activity, thus personalized the physical training (Antonacci et al., 2007[3]).

Coaches and physical trainers in soccer frequently base training HR intensities on MHR, and that in cases where direct measurement of MHR is not possible, an equation based on age can be used to predict MHR. FOX's equation (MHR = 220 - age) (Fox et al., 1971[10]) and TANAKA's equation (MHR= 208 - 0.7 x age) (Tanaka et al., 2001[29]) are two widely used equations in sports practices. The validity of these equations has been rigorously tested in large samples of adults who are healthy, sedentary, athletes and persons with health complications. In contrast, few investigations in children and adolescents have been conducted (Nikolaidis, 2015[20]). In a given population, MHR values demonstrate a large inter-individual variation. This may make estimated MHR using linear equations more inaccurate. It is strongly influenced by a person's physiology and lifestyle (Sarzynski et al., 2013[28]).

Reference percentile curves are widely used in medical practice as a screening tool during the important period of childhood. They identify abnormal subjects, in the sense that their value of a given measure is located in one or the other tail of the reference distribution (Borghi et al., 2006[4]). An important aspect of this process is drawing the curves in percentiles, which are generally presented as the frequency distribution of values based on gender and age. The graphs' common pattern is shown as a series of smooth curves drawn through the points of each age and transformed into a normal distribution (Cole, 2012[6]; Ghouili et al., 2018[12]).

During the last decades, several countries have established their references or models of physical and physiological parameters in the fields of health and physical activity of trained or untrained subjects (Eytan et al., 2017[9]; Lesinski et al., 2020[15]; Santos et al., 2014[27]). For example, a recent study showed that athletes practicing endurance sports have similar MHR values as those engaged in speed and strength activities, and both have lower values than untrained subjects. This highlights the need to review popular prediction equations in sport fields (Nikolaidis, 2015[20]). Therefore, the objective of the present study is to develop smoothed reference percentile curves of MHR and resting heart rate (RHR) based on age (11-18 years), as well as a predicted equation specifically for Tunisian young soccer players.

The descriptive characteristics of the sample are presented in Table 1(Tab. 1).

The sample was distributed over the eight age groups as follows: 12.23 % (11 years), 10.61 % (12 years), 13.61 % (13 years), 13.48 % (14 years), 13.11 % (15 years), 12.86 % (16 years), 10.99 % (17 years) and 13.11 % (18 years). There were significant differences between all successive age groups in terms of height, body mass, and trainability. However, the differences were particularly noticeable for a certain age group in other variables. For BF %, there were significant differences between the 11/12 years (p<0.001), 13/14 years (p<0.0001), and 16/17 years (p<0.01) age groups. For MHR, there were significant differences between the 11/12 years (p<0.001), 14/15 Years (p<0.01), and 17/18 years (p<0.001) age groups. For RHR, there was a significant difference only between the 11/12 years (p<0.001) age group (Table 1(Tab. 1)).

The aim of this study was to develop reference curves of MHR and RHR and produce an estimated equation for children and adolescents' soccer players.

The 3^rd, 50^th, and 95^th percentile values of MHR showed a continuous decrease with age, through variation of decline of around 1.67 ± 0.30 bpm, 1.37 ± 0.72 bpm, and 1.53 ± 0.62 bpm per year, respectively, resulting in an overall decline rate of -4.88 %, -5.61 %, and -4.89 % between the ages of 11 and 18 years. Tanaka et al. (2001[29]) have demonstrated that the decline in MHR is not related to physical activity level or gender, but rather primarily due to the reduction in intrinsic heart rate. A constant relationship between the variation of heart rate and the intrinsic contractile function of the myocardium has been found in previous studies by Jose and Taylor (1969[13]) and Opthof (2000[22]).

Smoothed percentile values for RHR are also inversely associated with age. Indeed, the values for the 3^rd, 50^th, and 97^th percentiles vary by 1.52 ± 0.72 bpm, 1.46 ± 0.75 bpm, and 1.01 ± 0.62 bpm per year, respectively, with a total percentage decrease of -17.01 %, -13.84 %, and -8.38 % from 11 to 18 years of age. In the same context, Zanuto et al. (2020[31]) found, through simple regression analysis, that RHR was associated with age (β = -1.61; 95 % CI: -2.11, -1.16) and total physical activity (β = -0.51; 95 % CI: -0.91, -0.11) parameters in boys aged 10 to 17 years.

In a meta-analysis review, Pieles and Stuart (2020[25]) showed that training-related changes in adolescent athletes include increased cardiac output, as well as increased size and thickness of the ventricular cavity, with a decreased RHR. Training three days per week for 12 weeks with high-intensity physical activity for 15 minutes each day resulted in significant improvements in systolic blood pressure and RHR in school children.

Zanuto et al. (2020[31]) described that the decrease in resting heart rate under the effects of physical activity may provide three action mechanisms: (1) neural mechanism, which regulates the necessary stimuli through the action of muscle afferent nerves (chemoreceptors and mechanoreceptors) based on exercise intensity, (2) baroreflex mechanism, which acts as a behavioral regulator with every beat, and (3) central mechanism, which controls changes in sympathetic and parasympathetic efferent activity during exercise (Zanuto et al., 2020[31]). In other words, moderate to vigorous physical activity promotes health benefits by producing physiological changes in the cardiovascular, pulmonary, and metabolic systems (Nystoriak and Bhatnagar, 2018[21]).

Several variables, including anthropometric parameters, age, trainability, and RHR variables, were examined to identify the best predictor of MHR. Our analysis revealed that age is the most significant predictor, with an inverse correlation of -0.56 (p<0.001) with MHR. Previous studies have shown varying degrees of correlation between MHR and age across different age groups of athletes. For example, a correlation coefficient of -0.41 (p<0.001) was reported for soccer players aged 11-36 years (Nikolaidis, 2015[20]), while a weaker correlation (-0.28; p<0.001) was found for athletes aged 9 to 18. In contrast, studies covering larger age ranges reported much stronger correlations, such as -0.60 for age groups of 14-77 years (Whaley et al., 1992[30]) and 19-89 years (Nes et al., 2013[19]). In our study, we developed two new prediction equations for MHR in young Tunisian soccer players, based on age and BMI as independent variables. The first model, which uses only age as a predictor, had R² of 0.317 with SEE of 5.22. The second model, which includes both age and BMI, had an R² of 0.324 with SEE of 5.20. Since the two models had similar R² values and SEE, we chose the simpler model that includes only age as the predictor for MHR.

After analyzing the difference between observed MHR values of young Tunisian soccer players and the predicted values using our model 1, as well as the BOX's equation and TANAKA's equation, we observed biases and standard deviations of -0.003 ± 5.22, -3.17 ± 5.37, and +4.33 ± 5.57, respectively. The errors that exceed the limits of agreement can reach up to 3.6 % for our model 1, 4.8 % for the BOX's equation, and 5.4 % for TANAKA's equation. Our findings align with previous research by Gellish et al. (2007[11]) and Pereira Rodriguez et al. (2016[24]), indicating that the BOX's equation underestimates MHR values, while TANAKA's equation overestimates them. However, some studies, such as those by Nikolaidis (2015[20]) and Papadopoulou et al. (2019[23]), have reported conflicting results, showing that BOX's equation overestimates and TANAKA's equation underestimates MHR in adolescents. According to Almeida et al. (2010[1]), predictive equations become valid when they are applied to populations with similar characteristics to those of the sample for which the equation was developed (Machado and Denadai, 2011[16]). TANAKA's prediction equation was developed using a meta-analysis approach that extracted MHR values from 18,712 healthy adult subjects of 492 different groups in 351 studies (Tanaka et al., 2001[29]). The BOX model was based on an estimate derived from the observation of a linear best fit of a series of raw and averaged data compiled in 1971 (Fox et al., 1971[10]), and research since 1971 has revealed errors in the estimation of MHR, rendering the BOX's prediction equation inaccurate (Robergs and Landwehr, 2002[26]).

Numerous authors have attempted to develop alternative formulas for predicting MHR, but the resulting equations may not be applicable to the general population due to various factors, such as the sample size, age group, and type of stress test (Magrì et al., 2022[17]).

Our study has several strengths. (1) We developed reference values for MHR and RHR that allow for ranking players according to their aerobic capacity compared to those of subjects of the same age and from the same population. (2) We included several parameters, such as age, stature, body mass, BMI, percentage of body fat, trainability, and RHR, as possible predictors for MHR in young Tunisian footballers. (3) We developed first prediction equation for MHR during the adolescent period. However, our study also has some limitations. For instance, the equation we developed only has a 30 % predictive capacity and is applicable to a little age group of 11-18 years. Therefore, future studies should expand the age group by combining the period of adolescence with that of young adult footballers. Additionally, longitudinal studies would be beneficial to better specify the tendencies of the curves of references over time.

Based on these results, it is suggested that the reference curves developed from the Tunisian children and adolescent sample can be used to classify RHR and MHR of each player and to monitor their changes over time. In the context of testing and training, practitioners are advised to use the prediction equation developed in this study, which is more accurate than BOX's and TANAKA's equations. Finally, the reliability of the tools developed in this study needs to be tested by Tunisian coaches and physical trainers.