Revista Brasileira de Ciências do Esporte Revista Brasileira de Ciências do Esporte
Revista Brasileira de Ciências do Esporte 2018;40:100-5 - Vol. 40 Núm.1 DOI: 10.1016/j.rbce.2018.01.003
Original article
A new age-based equation for predicting maximum heart rate in endurance-trained runners
Uma nova equação baseada em idade para predição da frequência cardíaca máxima em corredores aerobiamente treinados
Una nueva ecuación basada en la edad para pronosticar la frecuencia cardíaca máxima en corredores entrenados en resistencia
Fabiana Andrade Machadoa,b,c,?, , Ana Claudia Pelissari Kravchychyna,b, Cecília Segabinazi Pesericoa,b, Danilo Fernandes da Silvaa,b,c, Paulo Victor Mezzarobaa,b
a Universidade Estadual de Maringá (UEM), Programa de Pós-Graduação Associado em Educação Física, Maringá, PR, Brazil
b Universidade Estadual de Londrina (UEL), Maringá, PR, Brazil
c Universidade Estadual de Maringá (UEM), Departamento de Educação Física, Maringá, PR, Brazil
Recebido 10 Dezembro 2015, Aceitaram 12 Janeiro 2018

This study aimed to generate an age-based maximum heart rate (HRmax) equation for endurance-trained runners. Thirty-four male runners performed three tests on a motorized treadmill, starting at 8kmh-1 with increments of 1kmh-1 every 1, 2 or, 3min. HRmax was defined as the highest heart rate value recorded during each test. Post hoc analyses indicated that the HRmax derived from each test was significantly lower than the highest HRmax value, for each participant. HRmax predicted by “206-0.7×age” underestimated the highest HRmax by 8.6beatsmin-1. Thus, the generated age-based “218-0.8×age” equation should be used to predict HRmax in endurance-trained runners.


Esse estudo objetivou gerar uma equação de frequência cardíaca máxima (FCmax) baseada na idade para corredores aerobiamente treinados. Trinta e quatro corredores homens realizaram três testes incrementais em esteira motorizada, com início a 8km·h-1 e incrementos de 1km·h-1 a cada um, dois ou, três minutos. A FCmax foi definida como o valor mais alto de frequência cardíaca registrado em cada teste. As análises de post hoc indicaram que a FCmax de cada teste foi significativamente menor que o valor mais elevado de FCmax para cada participante. A FCmax predita pela equação “206-0,7 x idade” subestimou a mais alta FCmax em 8,6batimentos·min-1. Logo, a equação gerada baseada em idade “218-0,8 x idade” deveria ser usada para predizer a FCmax em corredores aerobiamente treinados.


El objetivo de este estudio fue generar una ecuación de la frecuencia cardíaca máxima (FCmáx) basada en la edad en corredores entrenados en resistencia. Treinta y cuatro corredores de sexo masculino realizaron 3 pruebas en una cinta ergométrica motorizada, comenzando con la velocidad de 8km/h-1 con incrementos de 1km/h-1 cada uno, 2 o 3min. La FCmáx fue definida como el valor de la frecuencia cardíaca más elevada registrada durante cada prueba. Los análisis posteriores indicaron que la FCmáx derivada de las pruebas fue considerablemente más baja que el valor más alto de la FCmáx de cada participante. La FCmáx pronosticada por la ecuación «206-0,7×edad», subestimada la más alta FCmás por 8,6lat/min-1. Así, la ecuación generada basada en la edad «218-0,8×edad» debería utilizarse para pronosticar la FCmáx en corredores entrenados en resistencia.

Cardiovascular system, Exercise test, Physical endurance, Running
Sistema cardiovascular, Teste físico, Resistência física, Corrida
Palabras clave
Sistema cardiovascular, Prueba física, Resistencia física, Carrera

Maximal heart rate (HRmax) is one of the most commonly used values in clinical medicine and physiology (Tanaka et al., 2001) where it is used as a criterion for assessing maximal effort during graded exercise testing (Duncan et al., 1997; Howley et al., 1995) and can be used to prescribe appropriate exercise intensity (ACSM, 2006; Robergs and Landwehr, 2002). For these purposes, exercise and fitness professionals often use age-based equations to estimate HRmax (Engels et al., 1998; Sporis et al., 2011; Tanaka et al., 2001). However, several equations are available to estimate this value (Engels et al., 1998; Gellish et al., 2007; Graettinger et al., 1995; Inbar et al., 1994; Jones et al., 1985; Lester et al., 1968; Miller et al., 1993; Schiller et al., 2001; Sheffield et al., 1978; Tanaka et al., 1997; Whaley et al., 1992), including meta-analyses of published equations (Londeree and Moeschberger, 1982; Tanaka et al., 2001) and most of them differ widely in their estimates of HRmax (Londeree and Moeschberger, 1982). Additionally, these equations were derived from different populations and testing protocols, which can also affect the determination of this value.

HRmax is a protocol-dependent physiological variable that is expected to be higher in long-duration tests as compared to short-duration tests (Bishop et al., 1998; Roffey et al., 2007). Nevertheless, some age-based HRmax equations were generated using short protocols (Graettinger et al., 1995; Inbar et al., 1994; Schiller et al., 2001; Tanaka et al., 1997), following the suggestion of Buchfuhrer et al. (1983) to bring the subject to the limit of tolerance in 10±2min. In fact, Buchfuhrer et al. (1983) suggested that short duration protocols were best to obtain the highest maximal oxygen uptake (VO2max) value, but not for attaining the highest HRmax value. This group also reported higher HRmax values in long (~26min) tests than in short (~11min) tests. These reports, therefore, suggest that age-based HRmax equations were generated using testing protocols that might not have been the most appropriate for attaining the highest value for HRmax. Additionally, one other shortcoming of the currently available age-based equations is that none of the previous studies derived a predictive equation by using the highest HRmax value obtained from two or more testing protocols, using the same individuals. Such a method could provide higher values for each subject and, consequently, higher values for the derived HRmax equation.

The incorrect prediction of HRmax may cause systematic errors in exercise prescriptions (Cleary et al., 2011). When HRmax is overestimated, the prescribed exercise intensity will be greater than needed to improve cardiovascular fitness. On the other hand, the underestimation of HRmax leads to lower stimulus that may not improve aerobic parameters. Thus, reducing the error associated with estimating HRmax would improve the accuracy of exercise prescriptions (Cleary et al., 2011), and improved estimates require the development and use of new equations designed for specific populations and modes of exercise (Robergs and Landwehr, 2002). Thus, the purpose of this study was threefold: (a) to compare three age-based HRmax equations derived for the same participants from each incremental test of different duration, (b) to derive an age-based HRmax equation for recreational endurance-trained runners from the highest HRmax value, for each participant, attained from the three incremental tests, and (c) to compare the generated equations with the well-known “206-0.7×age” equation proposed by Tanaka et al. (2001) for recreational endurance-trained individuals. We hypothesized that the “206-0.7×age” equation would underestimate the highest HRmax values obtained by the current protocol for recreational endurance-trained runners.

Materials and methodsParticipants

Thirty-four male, recreational, endurance-trained runners of regional and local level with a minimum of two years of training experience volunteered to take part in this study. The 10-km running times of the participants were between 35 and 60min, with a pace between 10 and 17kmh-1 (~45–75% of the world record). Prior to testing, all participants provided written informed consent and the local ethics committee approved the experimental protocol (#719/2010).

Experimental overview

In a counterbalanced order, participants who were habituated to running tests performed three continuous incremental exercise tests of different stage durations on a motorized treadmill (Super ATL; Inbrasport, Porto Alegre, Brazil), with the gradient set at 1%. The tests were performed over two weeks, with each test separated from the other by at least 48h. The three different stage duration protocols were as follows: (a) short stage duration of 1min (SS), (b) intermediate stage duration of 2min (IS), and (c) long stage duration of 3min (LS).

Exercise protocols

After a warm-up that consisted of walking at 6kmh-1 for 3min, each protocol started with an initial speed of 8kmh-1, followed by an increase of 1kmh-1 between each successive stage until volitional exhaustion. The speed was increased every 1min for the SS test, every 2min for the IS test, and every 3min for the LS test. Consistently across each trial, participants were strongly encouraged, verbally to invest maximum effort. All the three tests were performed at the same time of the day, under normal laboratory conditions (temperature=20–22°C and relative humidity=50–60%). Participants were instructed to report for testing well-rested, nourished, and hydrated, wearing lightweight comfortable clothing. Participants were also instructed to avoid eating 2h before the maximal exercise tests, to abstain from caffeine and alcohol, and to refrain from strenuous exercise for 24h before testing.

Physiological variables

Before testing, participants were familiarized with the 6–20 scale by Borg (1982), which was used to measure the rating of perceived exertion (RPE) during the last 15s of each stage and at exhaustion. The highest RPE value was adopted as the peak RPE (RPEpeak). Heart rate (HR) was recorded every five seconds throughout the tests (Polar RS800sd, Kempele, Finland) and HRmax was defined as the highest HR value recorded during the test (Bentley and McNaughton, 2003; Schiller et al., 2001; Tanaka et al., 2001). Neither respiratory gases nor blood lactate were monitored during the tests concerning that such interventions would affect the performance of the participants (Schabort et al., 1998). Earlobe capillary blood samples (25µL) was collected into a glass tube at the end of the tests and at the third, fifth, and seventh minute of passive recovery, sitting in a comfortable chair. From these samples, blood lactate concentration was subsequently determined by electroenzymatic methods using an automated analyzer (YSI 2300 STAT, Ohio, USA). Peak blood lactate concentration (LApeak) was defined for each participant as the highest post-exercise blood lactate concentration value. The maximal effort was deemed to be achieved if the incremental test met two of the following criteria: 1) LApeak higher than 8mmolL-1, 2) HRmax within ±10beatsmin-1 of endurance-trained age-predicted HRmax using the age-based “206-0.7×age” equation (Tanaka et al., 2001) and 3) RPEpeak greater than 18 in the 6–20 Borg scale (Howley et al., 1995).

Statistical analyses

Data are presented as mean±SD and were analyzed using the SPSS 17.0 software. The Shapiro–Wilk test was used to check the normality of the data distribution. HRmax values were compared using repeated measures ANOVA with Bonferroni post hoc test. The sphericity assumption was checked by Mauchly's test of sphericity. A simple linear regression was employed to determine the age-based HRmax equation from the SS (HRmaxSS), IS (HRmaxIS) and LS (HRmaxLS) tests. Additionally, an age-based HRmax equation was derived from the highest HRmax value attained for each participant from the SS, IS, and LS tests (HRmaxSIL). The relationship between HRmax and age was examined using the Pearson's correlation coefficient (r), the coefficient of determination (R2) and the standard error of estimate (SEE). Statistical significance was set at p<0.05.


The characteristics of the participants (mean±SD) are presented in Table 1.

Table 1.

Characteristics of the participants and training.

Variables  Mean±SD 
Age (years)  40.1±12.8 
Body mass (kg)  67.7±7.2 
Height (cm)  173.3±7.0 
Body mass index (kgm-222.5±1.8 
Training frequency (dayswk-14.9±1.5 
Training distance (kmwk-158.5±31.7 


Four participants did not meet at least two criteria of maximal effort in at least one test. The tests from these participants were not included in the results. The summary of the physiological parameters of the remaining thirty participants (mean±SD) is given in Table 2. Analysis of variance revealed a significant effect of the stage duration on the LApeak (p=0.001), percentage of endurance-trained age-predicted HRmax (p=0.003), time to exhaustion (p<0.001) and HRmax (p=0.003). The RPEpeak did not significantly differ among the three protocols (p=0.36). Post hoc analyses indicated that the HRmaxSIL was significantly higher than the HRmax derived from the SS, IS or LS tests (p<0.001).

Table 2.

Values of maximum rating of perceived exertion (RPEpeak), peak blood lactate concentration (LApeak), percentage of age-predicted maximum heart rate (%APMHR), and maximal heart rate (HRmax) during treadmill incremental protocols with stage length of 1min (SS), 2min (IS) and 3min (LS).

Protocol  RPEpeak (6–20 Borg scale)  LApeak (mmolL-1%APMHR (%)  Time to exhaustion (min)  HRmax (beatsmin-1
SS  19.8±0.6  9.4±2.2  102.3±4.0  11.1±1.6  181.8±12.1 
IS  19.8±0.4  9.2±1.9  104.0±4.3a  19.4±3.0a  184.8±12.7a 
LS  19.9±0.4  7.9±2.2a  103.0±4.1a  26.7±4.7a,b  183.1±12.9 
SIL  20.0±0.0  10.3±2.0a,b,c  104.8±4.1a,b,c  –  186.3±12.7a,b,c 

Values are means±SD, n=30. SIL, highest values attained for each participant from the SS, IS and LS protocols.


p<0.05 compared with the SS protocol.


p<0.05 compared with the IS protocol.


p<0.05 compared with the LS protocol.

Fig. 1 presents the measured HRmax (i.e., the highest HRmax values attained for each participant from the SS, IS, and LS tests) and the age-based HRmax equations from the SS (HRmaxSS), IS (HRmaxIS) and LS (HRmaxLS) tests. Further, it presents the proposed HRmaxSIL equation from the measured HRmax and the “206-0.7×age” meta-analysis equation proposed by Tanaka et al. (2001). In developing the HRmaxSIL equation from the three protocols, the IS protocol contributed with 56.7% of the HRmax values. The LS and SS protocols contributed with 33.3% and 10.0%, respectively. The correlation coefficient, coefficient of determination and standard error of estimate of the generated “2180.8×age” equation were r=-0.81; R2=0.66, and SEE=7.5beatsmin-1. The relative SEE (i.e., SEE expressed as a percentage of the mean of the outcome measure) was 4.1%. The HRmax predicted by the “206-0.7×age” equation was 177.7±9.2beatsmin-1 and, on average, according to repeated measures ANOVA, underestimated the measured HRmax by 8.6beatsmin-1 in the present population.

Figure 1.

Comparison between age-based maximal heart rate equations derived from incremental protocols with stage length of 1min (HRmaxSS), 2min (HRmaxIS) and 3min (HRmaxLS). It also presents the proposed HRmaxSIL equation from the highest maximal heart rate value attained for each participant from the three tests of different stage duration and the meta-analysis equation “206-0.7×age” for endurance-trained individuals.


The main purpose of the present study was to derive an age-based HRmax equation for recreational endurance-trained runners. The main finding was that the generated “218-0.8×age” equation should be used to predict HRmax in endurance-trained runners rather than the well-known “206-0.7×age” meta-analysis equation by Tanaka et al. (2001). In fact, the “206-0.7×age” equation, on average, underestimated the highest HRmax values by 8.6beatsmin-1.

The incorrect prediction of HRmax may cause systematic errors in exercise prescriptions (Tanaka et al., 2001). When the HRmax is overestimated, the prescribed exercise intensity will be greater than needed to improve cardiovascular ?tness. On the other hand, the underestimation of HRmax leads to lower stimulus that possibly will not improve aerobic parameters. Reducing the error of estimating HRmax would improve the accuracy of exercise prescription (Cleary et al., 2011). Consequently, several studies have been conducted to generate new age-based HRmax equations to estimate the HRmax more accurately (Engels et al., 1998; Gellish et al., 2007; Graettinger et al., 1995; Inbar et al., 1994; Jones et al., 1985; Lester et al., 1968; Londeree and Moeschberger, 1982; Miller et al., 1993; Schiller et al., 2001; Sheffield et al., 1978; Tanaka et al., 1997, 2001; Whaley et al., 1992). Nevertheless, most of them are too general or unspecific for the population of endurance-trained runners.

Tanaka et al. (2001) presented one of the most usual equations to predict HRmax (Cleary et al., 2011; Franckowiak et al., 2011). They proposed the general age-based “208-0.7×age” equation, independent of gender and habitual physical activity status, which was generated by a meta-analytic study that included 351 studies involving 18.712 individuals. This general equation is valid for heterogeneous groups within a wide age range including children and adolescents (Machado and Denadai, 2011). They also proposed the equation “206-0.7×age” for the specific population of endurance-trained individuals. Nevertheless, in contrast to some studies (Camarda et al., 2008; Cleary et al., 2011; Silva et al., 2007), we found that the specific “206-0.7×age” equation underestimated the actual HRmax in endurance-trained runners.

Camarda et al. (2008), for example, analyzed the data from 2047 sedentary individuals (age, 12–69 years), and reported that the HRmax obtained by the “208-0.7×age” equation (182.7±8.2beatsmin-1) overestimated the measured HRmax (180.8±13.8beatsmin-1). Similarly, Cleary et al. (2011) evaluated 96 active subjects ranging from 18 to 33 years old. They found that the general “208-0.7 × age” equation overestimated the measured HRmax (192.6±1.9 and 190.1±7.9beatsmin-1, respectively). Silva et al. (2007) assessed 93 elderly women (age, 67±5 years) during the modified Bruce protocol and found that the measured HRmax (145.5±12.5beatsmin-1) was highly overestimated by the general “208-0.7×age” equation (161.0±3.9beatsmin-1). On the other hand, we found that the specific “206-0.7×age” equation for endurance-trained runners underestimated the actual HRmax for the sample of this study. Supporting our results, Sporis et al. (2011), assessed 509 members of the Croatian Armed Forces (age, 29.1±5.5 years) and concluded that the general “208-0.7×age” equation underestimates the actual HRmax value by 4.2beatsmin-1 in military personnel. Similarly, Nes et al. (2013) examined the relationship between HRmax and age in 3320 healthy individuals within a wide age range (19–89 years) and reported that the “208-0.7×age” equation underestimated the measured HRmax by 6.2beatsmin-1 even using a individualized protocol that brought the subjects to exhaustion within 8–12min.

An important factor to develop an accurate age-based HRmax equation for a specific population is the homogeneity of the sample. This study involved 34 recreational endurance-trained runners. The small number of participants was the main limitation of this study. However, the difficulty in obtaining a homogeneous group of participants willing to perform three running tests in laboratory until exhaustion was a factor that restricted the sample size. In contrast, the sample of this study ranged widely in age, contributing to the generation of the equation. Another factor that influences the accuracy of a generated age-based HRmax equation is the duration of the incremental test. Roffey et al. (2007) showed that the HRmax was significantly higher during long (25±4min) incremental tests than it was in shorter tests (10±2min). Moreover, Bishop et al. (1998) found significantly higher values for HRmax when a stage duration of 3min was used, rather than 1min, during incremental tests. Thus, the design of the incremental tests might have affected most of the available age-based HRmax equations.

In this study, the age-based equation generated from IS protocol was the most similar to the HRmaxSIL equation (Fig. 1) and contributed with about 60% of the data for its generation. Additionally, not only the HRmaxSIL but also all the three equations generated from just one test (i.e., HRmaxSS, HRmaxIS and HRmaxLS) presented higher values for HRmax than the “206-0.7×age” equation. Further, the “206-0.7×age” equation, on average, underestimated the measured HRmax values by 8.6beatsmin-1 in the recreational endurance-trained population. Thus, the proposed “218-0.8×age” equation, generated from three incremental tests of different durations, seems to be more appropriate for predicting HRmax in this population of runners.

The majority of age-based univariate equation for predicting HRmax have large errors between 7 and 11beatsmin-1 (Robergs and Landwehr, 2002). Further, according to these authors, equations need to be developed that are population specific. The error of the proposed “218-0.8×age” equation for the specific population of recreational endurance-trained runners was relatively low, and in accordance with previous studies. Nes et al. (2013), for example, proposed recently the “211-0.64×age” equation to predict the HRmax in healthy individuals and reported a higher, but also in accordance with previous studies, error (i.e., SEE=10.8beatsmin-1; r=-0.60). Nevertheless, due to the inherent error associated with the estimation of HRmax, using the proposed “218-0.8×age” equation, approximately 67% of the population should fall within ±8beatsmin-1 of the actual HRmax. Further, for 5% of the population, the actual HRmax could differ by more than 15beatsmin-1. Therefore, direct measurements of HRmax should be used whenever possible.

Although this study adds important value to the literature as well as present great practical application, it also has a limitation. Due to the specific characteristics of the sample (recreational endurance-trained runners), our sample size is small for this type of study. However, as mentioned, the equation was built to provide a great practical application for this specific population.


In summary, the results from the present study suggest that the proposed “218-0.8×age” equation should be used to predict HRmax in recreational endurance-trained runners rather than the well-known “206-0.7 × age” equation proposed by Tanaka et al. (2001). Additionally, the “206-0.7×age” equation, on average, underestimated the measured HRmax values by 8.6beatsmin-1 in the recreational endurance-trained population. Researchers, coaches, and practitioners are recommended to use this formula to determine HRmax more accurately in endurance-trained runners. Because the results of this study are limited to this specific population, further research is warranted to verify whether these results apply to other populations that have different performance levels.

Ethical statement

Comite de Ética e Pesquisa em Seres Humanos, Universidade Estadual de Maringá (#1.262.502/2015).


This work was supported by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior – CAPES/Brazil and Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq/Brazil).

Conflicts of interest

The authors declare no conflicts of interest.

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Corresponding author. (Fabiana Andrade Machado
Copyright © 2018. Colégio Brasileiro de Ciências do Esporte
Revista Brasileira de Ciências do Esporte 2018;40:100-5 - Vol. 40 Núm.1 DOI: 10.1016/j.rbce.2018.01.003