## Introduction

### Premises of workload analysis

One of the fundamental reasons that athletes are monitored is to measure their progress in response to their training.1 2 Individual responses to training stress may vary,3 4 and an appropriate follow-up method could aid in identifying these.5 The monitoring of training also plays an important role in injury prevention.6 In particular, previous research has often focused on the relationship between training load and injury risk.7–10

Banister11 was one of the first people to introduce the notions of fitness and fatigue that correspond to the positive and negative adaptations from training. The largest difference between these adaptations is obtained when athletes reach their highest performance level, that is, when the negative consequences of their training (injury, illness, fatigue and over-reaching) are limited and the positive ones are optimised.12

Foster proposed the rating of perceived exertion (RPE) session assessment, an index that can be adapted to a large number of activities.12–14 The training load is subjectively measured through the RPE in order to relatively quantify the burden imposed on athletes. This index can be computed by multiplying the perceived intensity of the training session by its duration.15 Correlations between RPE and other intensity measurements, such as heart rate (r=0.89) or plasma lactate concentration (r=0.86), were demonstrated.16 Several other methods have been proposed to quantify workload: the *work endurance recovery* takes into account objective and subjective parameters17 18 and estimates the level of fatigue induced by exercise using a cumulative work:endurance limit ratio associated with the natural logarithm of the work:recovery ratio.

### Acute:chronic workload ratio (ACWR) and its limitations

More recently, Gabbett19 has updated the ratio proposed by Allen and Coggan.20 They suggested comparing the current week of training to the previous four. The aim of their ratio was to ensure that the workload was kept within a ‘high-load, low-risk’ range of values. When the ratio (last week/previous four) was too low (<0.8) or too high (≥1.5), the risk of subsequent muscular and non-contact injury was shown to initially increase.19

The ACWR is based on specific sport and injury data.21–25 However, many criticisms have emerged concerning both the structure of the ratio and its interpretations. Despite some beliefs about training load, injury and performance that ACWR and its derivative have generated, some methodological issues appear.24 The first problem is the requirement to wait 4 weeks before any ratio can be computed. Second, each time missing data appear in the collecting process, it leads to spurious values in the ratio until another 4-week period allows for stabilisation. This type of problem can be addressed with imputation methods,26 but it complicates the procedure.

Menaspà's paper, although presented in an editorial format (ie, the lowest level of evidence), puts forward another limit of the ACWR.27 First, the averaging of the 4 weeks does not allow for variations within a given period and can only show the general trends in the training load while masking potential peaks and troughs in the load. For example, different workload levels (chronic and acute) can lead to identical ratios. Moreover, computing the average load does not take into account stimuli that may occur in the meantime, such as, the effect of a training intensity peak that decreases over time.28 The chronic load calculated with a moving average gives as much weight to a training session performed the day before as to one that took place 4 weeks before.

On top of the aforementioned limitations, Lolli *et al*
29 highlighted the problem of dependence between acute and chronic loads: when calculating the ACWR, the acute load also constitutes a substantial part of the chronic one. This inadequate mathematical coupling between the two variables,30 also called ‘connecting a part to the whole’,31 raises the possibility that athlete monitoring may be compromised by spurious correlations. The proposed solution has been to exclude acute periods in the calculation of chronic load.29 However, this is an opinion piece, in that the strength of this paper remains limited.

More recently Gabbett *et al*
32 have shown in elite cricket fast bowlers that the use of coupled and uncoupled ACWRs produces the same injury likelihoods. Although findings do not imply that injuries can be predicted from a single training variable, no evidence was found of the rejection of ACWRs coupled in a real practical context.32

ACWR is a useful method to analyse training load. However, like all tools, it has its limitations, which we have tried to address through our theoretical study.33

### Exponentially weighted moving average (EWMA) and its limitation

Williams *et al*
34 proposed EWMA to calculate a load ratio. Authors shared concerns regarding the use of moving averages to compute ‘acute’ and ‘chronic’ loads in the ACWR as these measurements do not account for the declining nature of fitness and fatigue effects over time, nor do they accurately represent variations about how loads cumulate.

The EWMA35 alleviates some of these problems by assigning a decreasing weight to older load values. Specifically, the EWMA for a given day is calculated by

where is a value between 0 and 1 that represents the level of workload decrease. It is defined as

where *N* represents the selected time constant of the decrease, generally 7 and 28 days, respectively, for acute and chronic loads. The time frames of 1 and 4 weeks are frequently used in the periodisation strategies used by many team sports, although other time constants may be more appropriate in different contexts.

A first limitation of the EWMA lies in the complexity of this recursive equation, which may complicate the interpretation, implementation and computation of the coefficients. Moreover, the way each workload is weighted only depends on the number (N) of days considered in the calculation.

In terms of a long-term follow-up, the EWMA weight coefficients of loads tend to be equivalent to the ACWR and very small (eg, with 100 days, the most recent load accounts for just 2/101 of the total average). Thus, in this context, the EWMA value merely approximates the unweighted average load over N days and decreases the importance of recent workloads in favour of historical cumulated ones.

Therefore, EWMA is more consistent and accurate than the ACWR with a small value for N and a rolling average. However, in this context, both the EWMA and ACWR become very sensitive to missing data and need a period of initialisation that cannot be computed.

On the other hand, the impact of acute load differs greatly according to various sports disciplines. Currently, none of the presented methods provide a parameter able to adjust the decreasing influence of load, depending on sport context.

Using what has already been proposed in the literature and taking into account the different limitations outlined earlier, the purpose of this study was to propose a new way to compute cumulative training loads.