Stochastic optimization for the calculation of the time dependency of the physiological demand during exercise and recovery

Abstract

The stochastic optimization method ALOPEX IV is successfully applied to the problem of estimating the time dependency of the physiological demand in response to exercise. This is a fundamental and unsolved problem in the area of exercise physiology, where the lack of appropriate tools and techniques forces the assumption and the use of a constant demand during exercise. By the use of an appropriate partition of the physiological time series and by means of stochastic optimization, the time dependency of the physiological demand during heavy intensity exercise and its subsequent recovery is, for the first time, revealed.

Introduction

The version IV of the stochastic optimization method ALOPEX (ALgorithm Of Pattern EXtraction) has been proven to be the fastest and easiest in its implementation [1], [2], [3] when compared to other, previous versions of the same method (versions ALOPEX I, ALOPEX II [3], [4], [5] and ALOPEX III [3], [6], [7]). All ALOPEX methods, however, are shown to be very powerful, especially in optimization problems of many variables in real time. They are fast, effective and very easy to implement. The main advantage of ALOPEX methods is that no knowledge of the dynamics of the system or of the functional dependence of the cost function on the control variables, is required.

To give a brief introduction to ALOPEX IV stochastic optimization, let us assume that the aim is to find the values of the N control variables $x_1,x_2,\dots ,x_N$ that maximize function $f(x_1,x_2,\dots ,x_N)=f(\overrightarrow{x})$ (the cost function). The method of ALOPEX IV works as follows: if ${x_i}^{(k)}$ is the value of the ith control variable after the kth iteration and $f^{(k)}({x_1}^{(k)},{x_2}^{(k)},\dots ,{x_N}^{(k)})$ is the value of the control function after the kth iteration, then, the value of the ith control variable in the next $(k+1)$th iteration is

$$x_i^{(k+1)}=x_i^{(k)}+c\mathrm{\Delta }{x_i}^{(k)}\frac{\mathrm{\Delta }{f}^{(k)}}{|\mathrm{\Delta }{f}^{(k-1)}|}+g_i^{(k)},i=1,\dots ,N$$

where

$$\mathrm{\Delta }{x}_i^{(k)}\equiv {x_i}^{(k)}-{x_i}^{(k-1)},$$

$$\mathrm{\Delta }{f}^{(k)}\equiv f^{(k)}-f^{(k-1)},$$

and

$$\mathrm{\Delta }{f}^{(k-1)}\equiv f^{(k-1)}-f^{(k-2)}.$$

A proof of convergence of ALOPEX IV, in the absence of noise, can be found in [8]. For more detailed discussion on ALOPEX IV see [3], [8].

Let us denote a physiological variable as $s(v,t)$ (referring to either the heart rate, or the rate of change of Oxygen uptake), assuming that the current value of s depends on the intensity of the particular exercise (velocity v) and the time passed after the beginning of the exercise, t. The understanding of $s(v,t)$ in response to exercise, is a fundamental area of exercise physiology and has many applications in sport, see [9], [10], [11], [12], [13] as well as medicine and health in general [14].

The values of $s(v,t)$ always lie within the physiological limits of that particular variable, i.e. $s_{\mathit{min}}⩽s(v,t)⩽s_{\mathit{max}}$ (for example there is a maximum rate that the heart is able to sustain, as well as a minimum rate, which corresponds to absolute resting values). In the present study we focus on exercises of constant intensity (v constant), as this simplest case is currently the main area of research in exercise physiology. The term ‘on-transient’ kinetics refers to the change in the values of $s(v,t)$ in response to an increase in the intensity of the exercise, while the term ‘off-transient’ refers to the kinetics of $s(v,t)$ as the body recovers to a new lower value of s, following a decrease in the exercise intensity.

It has been observed that the on transient kinetics of $s(v,t)$ are slightly different for different constant exercise intensities. For moderate intensities, $s(v,t)$ follows an exponential-like rise [15], [16] until it reaches a steady state. For very heavy intensity exercise the final steady state value is the maximum value $s_{\mathit{max}}$; there is, however, a delay (a slowing down of the kinetics) in reaching this value. For severe intensity exercise the values of $s(v,t)$ rise very steeply and approximately exponentially, until they are limited by the maximum value $s_{\mathit{max}}$, which then becomes the steady state for the remaining time the exercise can be carried out at. Note that if the exercise is too severe then the subject may have to stop due to fatigue before they reach the $s_{\mathit{max}}$.

Let us assume the velocity and time-dependent function $D(v,t)⩾0$ that describes the physiological demand for that particular exercise (see also [8], [17], [18]). It is generally assumed that, for constant intensity exercises where the demand is such that $D(v,t)\ll s_{\mathit{max}}$, the value of the demand is constant and equal to the steady state value that $s(v,t)$ finally reaches [19] (the asymptote of the physiological time series). This time-independent value $D(v)$ depends on the intensity of the particular exercise can easily be obtained from the time series of $s(v,t)$.

For severe or very high intensity exercise there is $D(v,t)⩾s_{\mathit{max}}$. In the literature it is commonly assumed that, even in these cases, the demand is only a function of v and does not depend on time [26]. However this assumption can be shown to be an approximation which is not valid in general [8]. Indeed, for higher values of demand, where $D(v,t)>s_{\mathit{max}}$, the function of demand is probably also a function of time.

It is known that the energy cost of accelerating from a particular value of velocity to another value of velocity is greater, due to inertia, than the energy cost of remaining at that velocity. It is also known that the efficiency of muscle decreases with time, during constant intensity exercise [20], [21], [22], [23], [24]. For prolonged exercise and at a given constant intensity the energy cost increases due to a loss of efficiency during the exercise. Efficiency has also been shown to be dependent on exercise intensity, with the mechanical efficiency being lower in exhaustive exercise than in sub-maximal exercise [19], [25], [26]. The on-transient demand should be therefore considered to be a function of time, both in the initial and also the final stages of a bout of heavy intensity exercise.

In the case of recovery (off-transient), when the intensity of the exercise is reduced to a new constant level, there is also a time dependency in the demand. It has been observed that in the initial stages of recovery following a very heavy work load, high values of $s(v,t)$ [27] persist for some seconds before they rapidly drop (see also [28]). The reduction in the heart rate then begins to decrease as we approach the new recovery demand. There is then an ultra-slow reduction in the heart rate depending on the severity and duration of the previous exercise; this can last for several hours [9]. It is known that the time to reach resting level of heart rate or oxygen uptake can be very long following very intense exercise, in fact it may not reach resting levels the same day of the exercise [9], [26]. These phenomena suggest that the demand remains elevated for a period of time during recovery.

A previous study by the authors [8] showed that the on-transient heart rate demand (and also the oxygen demand [29]) can be modelled, using the model of Stirling et al. [17], [18], as a constant, only for sufficiently low exercise intensities and for the off-transients following sufficiently low on-transient exercise intensities. The same study showed, however, that these assumptions break down for higher exercise intensities; a model curve assuming constant, time-independent demand could not be optimally fit to the time series of $s(v,t)$ for heavy intensity exercise [8]. This result justified the expectation for the demand stated in the paragraphs above.

The demand, therefore, during exercise and recovery for physiological variables such as the heart rate and oxygen uptake is, in general, a function of time and not a constant, as is usually assumed. The present study applies ALOPEX IV stochastic optimization to the problem of optimizing the fit of the model [17], [18] to physiological time series data (see also [8]) with the aim of revealing, through the process of optimization, the time dependent nature of the physiological demand during exercise and recovery.

For the purposes of our study heart rate time series data corresponding to heavy intensity exercise and its subsequent recovery is used. The data consists of two sets of exercise and recovery for two different heavy exercise intensities. This data is the same data that was used in [8] to show that the assumption of a constant demand during heavy intensity exercise and the recovery that follows is incorrect.

Our method assumes a partition of the physiological time series in respect to time and calculates a value of the demand for each subset of the data (see Section 3).