# A Novel Approach of Modelling and Predicting Track Cycling
## 2 Materials and Methods  
### Physiological Model  
Time trial efforts, particularly the 1 km, are usually executed in an all-out fashion to complete exhaustion [1; 2]. This strategy is considered to lead to the best result possible due to the optimal utilisation of physiological capacity [19]. The mechanical power exerted by the athlete equals the maximal power attainable for the present pedal rate and exhaustion state at all times. The maximal force and maximal power an athlete can generate at a given cadence can be described by linear force-velocity (\(F/v\)) and parabolic power-velocity (\(P/v\)) profiles, where \(F\) denotes the mean tangential pedal force per one revolution, \(P\) is the power output, and \(v\) corresponds to the movement velocity of the cranks or pedals. As the pedalling rate \(PR\) is proportional to the tangential speed of motion \(v\) at the pedal, the \(F/v\) and \(P/v\) profiles are usually based on cadence in cycling. The linear function:  
\[F(v)=a\cdot PR+b \tag{1}\]  
approximates the relationship between mean pedal force \(F\) and the movement velocity \(PR\) in the absence of fatigue [20; 21]. Constant \(a<0\) specifies the decline in mean pedal force with increasing cadence and \(b\) the individual maximum force \(F_{max}\). The relationship between power (\(P\)) and the pedalling rate is then parabolic:  
\[P(v)=a\cdot PR^{2}+b\cdot PR \tag{2}\]  
From these functions, parameters such as the current maximum force (\(F_{max}=b\)), maximum pedalling rate (\(PR_{max}=-b/a\)), maximum power output (\(P_{max}=-b^{2}/4a\)) and related optimal pedalling rate (\(PR_{opt}=-b/2a\)) can be derived [20; 22]. These profiles have been used to describe a (track) cyclist's fatigue-free neuromuscular performance by determining the maximal resistance an athlete can overcome at a certain cadence and the maximal power output at that cadence [22; 23; 24]. Previous research results revealed that elite track sprinters leave this fatigue-free \(F/v\) profile after less than 3 s in maximal sprints [25]. An approximately synchronous and uniform reduction in force development and velocity of contraction is reported with increasing fatigue in muscles [26; 27; 28; 29]. These fatigue-induced changes can be described by a parallel shift in the \(F/v\) profile towards the origin [29; 30]. Power output declines in an approximately exponential fashion as a result of increasing fatigue when performing maximal exercise to exhaustion [1; 2]. In accordance, we found that the time course of the shift in the \(F/v\) profile with fatigue is also exponential. By adding a time-domain and incorporating the duration of exercise, two-dimensional \(F/v\) profiles form a three-dimensional model. This method is limited to sprints with a maximum duration of \(\approx\)60 s:  
\[F(v,t)=F(v)\cdot\mathrm{e}^{-t/\tau} \tag{3}\]  
where \(F(v)\) is the fatigue-free \(F/v\) profile, and \(\tau\) denotes the time constant of its decline due to fatigue. With the parallel shift in the \(F/v\) profile, characteristic parameters of the fatigue-free profiles also decrease [3]. This enables a simplification from three dimensions to two dimensions by eliminating the dependence on cadence. The temporal behaviour of the \(F/v\) or \(P/v\) profile can, therefore, be characterised by the variation of a single parameter, such as the maximum mean pedal force:  
\[F_{max}(t)=F_{max}\cdot\mathrm{e}^{-t/\tau} \tag{4}\]  
with \(F_{max}\) derived from the fatigue-free \(F/v\) profile. Equation (4), analogue to Equation (3), returns the state of the \(F/v\) profile at time \(t\) and allows for the calculation of the maximum average pedal force and maximum power output of an athlete at a given cadence at that time. In our recent publication [31], we demonstrate the application of this method by calculating optimal cadences to maximise mean power output in maximal sprints in the laboratory and on the cycling track.