First of all, let us introduce the mathematical structure of FFM. In the original form, fitness and fatigue components are described by first order differential equations. That is, they are convolution based features in which each training input—expressed as a discrete function \(\omega \left( i \right)\)—is convolved with an exponential transfer function. Hence, the equation of the model is given by the basic level of performance \(p^{*}\)—a model intercept—and the difference between the two features, with

$$\hat{p}\left( n \right) = p^{*} + k_{1} \mathop \sum \limits_{i = 1}^{n - 1} \omega \left( i \right)e^{{ - \frac{1}{{\tau_{1} }}\left( {n - i} \right)}} - k_{2 } \mathop \sum \limits_{i = 1}^{n - 1} \omega \left( i \right)e^{{ - \frac{1}{{\tau_{2} }}\left( {n - i} \right)}} ,\quad n \in {\mathbb{N}}.$$

(1)

Here, \(\hat{p}\left( n \right)\) is a modelled performance, \(k_{1}\) and \(k_{2 }\) denote two gain terms and \(\tau_{1}\), \(\tau_{2}\) denote two time constants for fitness and fatigue impulse responses, respectively. In this form, the model is commonly described as a linear time-invariant system. However, some alternatives motivated by relevant physiological assumptions make the features' parameters varying over time, being dependent on the accumulation of training input [4, 5, 7]. This results in time-variant systems that should better represent the true responses to training.

Yet, the use of FFMs for the purpose of modelling complex phenomena such as athletic performance might be in some ways unsuitable. In the following, we simply decompose the FFMs’ framework into three levels and briefly highlight conceptual issues responsible for errors in prediction.

### The Input: Quantification of Training

The first step of any training effect modelling using FFMs requires to quantify the training itself. Mainly used for modelling the training effects on performance in individual and endurance sports, a few methods for quantifying the training dose exist. Hence, the aforementioned discrete function \(\omega \left( i \right)\) can take various expressions. One physiology based on the product of training duration and its exponentially weighted physiological response (e.g. heart rate changes) is termed Training Impulse (TRIMP) [3]. Some other methods commonly rely either on products of volume and intensity parameters, being physiology-based (e.g. using heart-rate variations) [18] or not [19]. When exercise intensity cannot be objectively measured, the session TL is usually estimated using an ex post rating of perceived exertion multiplied by the session's duration [20, 21]. Exercise intensity can also be measured in arbitrary units, especially in cases of technical sport disciplines [22]. On this basis, the training sessions are the only cause of adaptations. That means training responses are independent of any other external factors to training, yet known to impact athletic performance but not accounted for in the model (e.g. environmental factors, nutritional and psychological status). Hence, two identical training sessions that occur at different training stages would induce similar adaptations and responses. Besides, various training sessions (e.g. a low intensity and prolonged exercise, and high intensity and short exercise) may result in similar TL estimates and so Fitness and Fatigue states, despite specific responses and adaptations to exercise exist [23, 24]. For example, two resistance training sessions (a low intensity, high volume and a high intensity, low volume) may lead to similar TL indexes according to the product of exercise volume and intensity [19]. Finally, athletes usually practice endurance and resistance training, and other disciplines to enhance performance.

Since FFMs are sensitive to the nature of the model input [25], a consistent training quantification method that is not biased by the type of training is required across training sessions.

Taking this stand, a univariate configuration of FFMs reduces the space of dimensions around adaptations to training into one single dimension, solely characterised by the training quantification. This is at the expense of all relevant information that can be captured and that may explain a part of athletic performance, even if the training quantification is objectively well estimated.

It also questions training quantification based on arbitrary methods, which might bring "noise" into the modelling in cases where there is an inexact appreciation of the exercise demand by the coach.

### The Function: A Physiological Approximation

Attempting to model athletic performance upon a mathematical representation of physiological principles is obviously commendable. However, it implies being confident in the model itself, leaving no room for vague theoretical approximations. Among the aforementioned variants of the original FFM, improving model complexity (e.g. by adding components in the model) does not guarantee the best model performances [26], even though such models are supposed to represent the physiological responses better. Therefore, the pertinence of adding antagonistic components to the most basic structure (i.e. only based on the fitness component) and more generally, the theoretical hypothesis behind FFMs might be questioned. However, some authors have proposed refinements and extensions of the two-components FFM formulation (see Eq. 1) in light of physiological responses to exercise. On one side, non-linear modifications of the mathematical structure allow model saturation effects [7, 27], describing over-training phenomena. Otherwise, state kinetics were adjusted in order to better represent physiological mechanisms (e.g. tissue remodelling, myosin ATPase activity) [28, 29] through delayed [8], growth and decay kinetics in response to exercise [9]. However, these modifications remain to be more broadly tested in ecological conditions with real data.

### The Output: The Performance

Finally, FFMs attempted to model either an athletic performance during a competitive season, a physical ability that relates to an athletic performance (e.g. mean power or velocity sustained over shorter distances than ones performed during competitions) [5, 30] or a physiological state [31, 32]. In general, choosing the appropriate model output has a strong implication in the modelling process. Modelling changes in physical ability instead of a full discipline-specific performance may allow for repeating less invasive and better controlled testing all along a training process. However, whatever form the output takes (i.e. an athletic performance or a physiological state), its multifaceted nature involves factors that are not considered in the univariate model. Therefore, the training history merely characterised by training loads may only explain a part of changes in the output, somehow resulting in a lack of model performances.

To summarise, FFMs’ ability to predict changes in athletic performance is greatly impacted by univariate modelling issues and a simplification of human physiological adaptations to exercise and training. Moreover, considering only the training loads responsible for changes in athletic performance implies neglect of all related confounding variables that influence both independent and dependent variables, causing spurious associations between input and output of the model.