W’_bal model to monitor real time work capacity balance during exercise.

Exercise intensity is classified as severe, heavy, or moderate [1], [2] based on blood lactate levels [3] or maximum oxygen uptake (VO_2max) [4], [5]. This classification is derived from the energy systems used by the human body, i.e. ATP-PCr, anaerobic glycolysis, and aerobic glycolysis [6]. Energy expenditure in the severe exercise intensity domain has been modelled using the parameters Critical Power (CP) and Work Capacity (W’). Please refer to the article in [7] to understand how CP is related to exercise intensity.

What are CP and W’?

Critical Power, CP: Critical power is a represents a power output above which muscle metabolic homeostasis cannot be attained [8]–[10]. In layman terms, it is a theoretical power output that can be maintained for a very long time without getting tired.

Work Capacity, W’: W’ (pronounced double-U-prime) represents the finite amount of work that can be done above CP [9], [10].

The Mathematics

The two terms CP and W’ are the power-asymptote and curvature constant of the hyperbolic relationship between power and time-to-exhaustion. The hyperbolic equation was derived from Monod and Scherrer’s [11] seminal work. The equation is given by,

where, P is power in Watts, CP is critical power in Watts, W’ is the work capacity in Joules, and t is the time to exhaustion in seconds. The figure below shows the hyperbolic relationship with CP as the power asymptote and W’ as the curvature constant.

How to determine CP and W’?

There are several ways to determine CP and W’, but two of them are most popular: (i) The constant work rate (CWR) protocol, derived from Monod and Scherrer [11] (ii) The 3-minute all-out test (3MT) proposed by Vanhatalo and colleagues [12]

The CWR protocol: The athlete, post warmup, is made to pedal at 3-6 constant powers based on their VO_2max till exhaustion (indicated by a drop in preferred cadence). The power versus time data points are plotted and the hyperbolic relation is fitted to the data to determine CP and W’. Some researchers prefer to use the same hyperbolic equation as a linear regression by plotting power versus reciprocal of time. Curve fitting from both methods is shown in the figure below.

The 3MT: The athlete, post warmup, is made to go on an all-out sprint for 3 minutes. The athlete should be at their maximal effort at every second of the three minutes. CP is calculated as the average power output of the last 30 seconds of the test, and the W’ is calculated as the area below the curve above CP. The figure below shows a schematic of the 3MT.

Modelling expenditure and recovery of energy, i.e. W’

Expenditure of W’ is modelled by the hyperbolic equation discussed above. The amount of W’ remaining is given by subtracting the energy above CP from the amount of W’ remaining in the previous instance of time. This is illustrated below:

where,

where, W'_bal is the W' balance at any time during exercise, W'_exp is the amount of W' expended, W'₀ is W' at time t=0, and t_e(i) and t_e(i-1) are time data that capture the duration of exercise above CP, i.e. duration of expenditure. The available recovery of W’ models in literature have been proposed by Skiba and colleagues [13]–[16]. Here, the latest model presented by Skiba and colleagues [15], referred to as SK2, is considered due the robustness of its mathematical derivation from first principles. The original model from Skiba and colleagues [13] has some mathematical and applicability issues [7], [17]–[19]. The SK2 model combines the expenditure and recovery of W’ in a biconditional model given by,

where, D_CP is the recovery power below CP and t_r is the time spent below CP, i.e. duration of recovery. The P > CP condition is same as the hyperbolic model for expenditure of W'. The P < CP condition models the recovery of W'.

Note: It is important to note that the SK2 model also has its drawbacks with respect to applicability to all athletes. Please refer to published articles [7], [17]–[19] for the criticisms of Skiba’s models.

The python code

The python program reads the workout file saved as an excel with time data on column 1 and power data on column 2. Then it asks the user to input their CP and W’. After the CP and W’ is entered by the user, the python program computes the W’_bal at the end of the workout and writes the W’_bal data points back to the excel file on column 3. Additionally, the python program plots Power, CP, and W’_bal versus time and saves the plot as an image in the location where the python program would be running. Furthermore, the python program can analyse multiple files at the same time with the plot function disabled.

Further reading:

The hyperbolic model of CP and W’ can also be applied to running. In running, CP is replaced by Critical Speed or Critical Velocity, CS or CV, and W’ is replaced by distance capacity, D’. Recently, the hyperbolic model was used to simulate the sub-2 hour marathon by Hoogkamer and colleagues [20].

References:

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11. H. Monod and J. Scherrer, “The work capacity of a synergic muscular group,” Ergonomics, vol. 8, no. 3, pp. 329–338, 1965.
12. A. Vanhatalo, J. H. Doust, and M. Burnley, “Determination of critical power using a 3-min all-out cycling test,” Med. Sci. Sports Exerc., vol. 39, no. 3, pp. 548–555, 2007.
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14. P. F. Skiba, D. Clarke, A. Vanhatalo, and A. M. Jones, “Validation of a novel intermittent W’ model for cycling using field data.,” Int. J. Sports Physiol. Perform., vol. 9, no. 6, pp. 900–904, 2014.
15. P. F. Skiba, J. Fulford, D. C. Clarke, A. Vanhatalo, and A. M. Jones, “Intramuscular determinants of the ability to recover work capacity above critical power,” Eur. J. Appl. Physiol., vol. 115, no. 4, pp. 703–713, 2015.
16. P. F. Skiba, S. Jackman, D. Clarke, A. Vanhatalo, and A. M. Jones, “Effect of work and recovery durations on W’ reconstitution during intermittent exercise,” Med. Sci. Sports Exerc., vol. 46, no. 7, pp. 1433–1440, 2014.
17. J. C. Bartram, D. Thewlis, D. T. Martin, and K. I. Norton, “Predicting Critical Power in Elite Cyclists: Questioning the Validity of the 3-Minute All-Out Test,” Int. J. Sports Physiol. Perform., vol. 12, no. 6, pp. 783–787, 2017.
18. K. Caen, J. G. Bourgois, G. Bourgois, T. Van der Stede, K. Vermeire, and J. Boone, “The Reconstitution of W′ Depends on Both Work and Recovery Characteristics,” Med. Sci. Sports Exerc., vol. 51, no. 8, pp. 1745–1751, 2019.
19. V. S. M. Sreedhara, F. Ashtiani, G. M. Mocko, A. Vahidi, and R. E. Hutchison, Modeling the Recovery of W’ in the Moderate to Heavy Exercise Intensity Domain, vol. Publish Ah, no. June. 2020.
20. W. Hoogkamer, K. L. Snyder, and C. J. Arellano, “Modeling the Benefits of Cooperative Drafting : Is There an Optimal Strategy to Facilitate a Sub‑2‑Hour Marathon Performance?,” Sport. Med., vol. 48, no. 12, pp. 2859–2867, 2018.