Modeling Time To Exhaustion With Critical Power and Critical Metabolic Rate

Modeling time-to-exhaustion using the critical power / W' balance and critical metabolic rate / M' balance models

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🚴 What Is Fatigue and Why Is it So Difficult To Measure?

For much of the time that ‘fatigue science’ has been a field, the ‘catastrophic’ model of fatigue was used to describe what occurs when an athlete reaches the absolute limit of their physical performance. Proponents of this model assert that the body either runs out of key nutrients or is ‘poisoned’ due to metabolite accumulation in the working skeletal muscles.

However, as early as the dawn of the twentieth century, some individuals challenged these assumptions, one such example being Angelo Mosso, who stated, “At first sight[fatigue might appear an imperfection of our body, is on the contrary one of its most marvelous perfections. The fatigue increases more rapidly than the amount of work done saves us from the injury which lesser sensibility would involve for the organism”.

If we take a look through the lens provided by Angelo Mosso we can begin to see fatigue as an immensely complex derivative of a number of functions, behaviors, and psychological processes. As a result, exercise limitations involve a wide range of systems working together in harmony to maintain homeostasis.

While these descriptive views of fatigue and exercise limitations can be useful, they don’t improve practitioners’ ability to predict or manage fatigue in athletes. As a result, the link between fatigue and performance has always been elusive. However, in recent years compelling evidence has indicated that the relationship between fatigue and performance is enshrined in the concepts of critical power and critical metabolic rate.

Critical power represents the highest power output an athlete can sustain for an extended period without fatigue, while W' balance quantifies the anaerobic energy reserve that depletes and replenishes during varying intensities of exercise. Similarly, the critical metabolic rate indicates the highest sustainable rate of muscle oxygenation change, and M' balance tracks the total metabolic capacity in the context of muscle oxygenation.

In this article, you will learn how to apply the W' balance and M' balance models using real-world power output and muscle oxygenation data. By utilizing these models, we can gain insights into an athlete's performance, pacing strategies, and recovery capabilities.

🚴 Critical Power and The W' Balance Model

What Is Critical Power?

Critical power is mathematically defined as the power-asymptote of the hyperbolic relationship between power output and time to exhaustion. In essence, critical power describes the duration that an individual can sustain a fixed power output in the severe exercise intensity domain, and physiologically critical power represents the boundary between steady-state and non-steady-state exercise. As a result, critical power may provide a more meaningful fitness index over more well-known performance metrics such as VO2max or functional threshold power. The hyperbolic equation which describes the relationship between power output and exercise tolerance within the severe exercise intensity domain is as follows:

Time to Exhaustion = W’ / Power - Critical Power

This equation creates a two-parameter model where critical power represents the asymptote for power, and the W’ represents a finite amount of work that can be done above critical power, as demonstrated in the image below.

Taken together, these two parameters can be used to predict how long an individual can exercise at any intensity above their critical power output. Interestingly, the critical power model appears to apply across kingdoms, phylums, and classes of animal life as well as different forms of exercise, and individual muscles for a given athlete. These observations suggest a highly conserved and organized physiological process, and perhaps a unifying principle of bioenergetics.

How Is Critical Power Calculated?

There are currently two validated methods for determining critical power and the fixed amount of work that can be done above critical power, termed W’. Traditionally critical power and W’ were calculated after having an individual perform three to seven all-out work bouts where they hold a fixed power-output until failure. These test results are then plotted on a chart where the x,y variables represent time to failure and power for each trial. Critical power is then determined as the slope of the work-time relationship, whereas W’ is determined from the y-intercept. More recently, though, investigators have introduced a 3-minute all-out exercise test, known as the 3MT, that has enabled the determination of critical power and W’ from a single exercise bout. The idea behind the 3-minute all-out test is that when a subject exerts themselves fully and expends W’ wholly, their finishing power output equals their critical power.

In the code block below, I'll show you how to calculate critical power using an athlete's historic training data from multiple time to exhaustion trials (non-technical readers should feel free to skip the code blocks and read ahead. They are not required for understanding the physiology and exercise applications discussed in this article).

Note — You can find the code for this article along with sample datasets in my GitHub Repository, which you can find HERE.

import numpy as np
import pandas as pd
from scipy.optimize import curve_fit
import matplotlib.pyplot as plt

data = pd.DataFrame({'Power (watts)': [278, 349, 455, 435,  725], 'Duration (sec)': [3533, 1790, 600, 300, 120]})

def power_duration_curve(t, CP, W_prime):
  power = CP + W_prime / t
  return power

power = data['Power (watts)'].values
duration = data['Duration (sec)'].values

popt, pcov = curve_fit(power_duration_curve, duration, power)
CP, W_prime = popt

print(f"Estimated Critical Power (CP): {CP:.1f} Watts")
print(f"Estimated W': {W_prime:.1f} Joules")

Which, produces the following output:

• Estimated Critical Power (CP): 306.8 Watts

• Estimated W': 50664.4 Joules

After calculating critical power with the code block above, we can plot the athlete's power-duration curve using the following code:

plt.scatter(duration, power, label='Athlete data', color='blue')
t_fit = np.linspace(min(duration), max(duration), 100)
P_fit = power_duration_curve(t_fit, CP, W_prime)
plt.plot(t_fit, P_fit, label='Fitted Model', color='red')
plt.xlabel('Duration (seconds)')
plt.ylabel('Power (watts)')
plt.legend()
plt.show()

Which produces the following chart:

Now, as previously mentioned, the following formula can be used to calculate an athlete's time to exhaustion when exercising at a fixed power output when both their critical power and W' are known: Time to Exhaustion = W’ / Power - Critical Power. For example, if an athlete's critical power is 306 watts, their W' is 50664 joules we can use the following formula to see how long they can sustain a fixed power output of 500 watts: Time to exhaustion = 50664 / 500 - 306 = 261 seconds

However, a major limitation of the formula above is that it's only applicable during fixed power output work bouts and does not account for scenarios where power output varies, such as during interval training or racing where power can fluctuate by a large degree moment to moment. In these scenarios, it's necessary to go beyond the traditional critical power model.

Beyond Critical Power: The W' Balance Model

Whereas the traditional critical power model allows you to predicting the time to exhaustion when power is held constant, the W' balance model allows us to account for varying power outputs and dynamic work/rest scenarios. This model tracks the depletion and recovery of W' over time, providing a more accurate prediction of an athlete's fatigue status during activities with changing intensities. For example, when an athlete's power output is greater than their critical power, W' is depleted (the rate of depletion is proportional to how much power exceeds critical power). Alternativley, when an athlete's power ouput is below their critical power, W' is restored and the rate of recovery is a function of the difference between their critical power and current power output and their recovery constant, Tau.

Thus, the W' balance model continuously updates the remaining W' based on the power output and CP over time. In the code block below I'll show you how to apply the W' balance model to a real athlete's power data, recorded during a ramp incremental exercise test (note, this athlete's estimated critical power is ~306 watts and their estimated W' is ~50664 joules, as calculated in the last section):

# initialize the W' balance model
W_bal = np.zeros(len(workout_data))
W_bal[0] = W_prime
tau_recovery =250  # adjust time constant based on experimental data

# function to update W' balance
for i in range(1, len(workout_data)):
    dt = workout_data['time'].iloc[i] - workout_data['time'].iloc[i - 1]
    power = workout_data['power'].iloc[i]

    if power > CP: # depletion of W'
        W_bal[i] = W_bal[i - 1] - (power - CP) * dt
    else: # recovery of W'
        W_bal[i] = W_bal[i - 1] + (W_prime - W_bal[i - 1]) * (1 - np.exp(-dt / tau_recovery))

    # ensure W' balance does not go below 0 or above W'
    W_bal[i] = max(0, min(W_prime, W_bal[i]))

# add W' balance to workout data
workout_data['W_bal'] = W_bal

We can then plot the athlete's time series power data with their modeling W' balance, as demonstrated below:

fig, ax1 = plt.subplots(figsize=(10, 6))
ax1.plot(workout_data['time'], workout_data['W_bal'], label="W' Balance", color='red')
ax1.set_xlabel('Time (seconds)')
ax1.set_ylabel("W' Balance (Joules)", color='red')
ax1.tick_params(axis='y', labelcolor='red')
ax2 = ax1.twinx()
ax2.plot(workout_data['time'], workout_data['power'], label='Power Output', color='blue')
ax2.set_ylabel('Power Output (Watts)', color='blue')
ax2.tick_params(axis='y', labelcolor='blue')
lines, labels = ax1.get_legend_handles_labels()
lines2, labels2 = ax2.get_legend_handles_labels()
ax1.legend(lines + lines2, labels + labels2, loc='best')
plt.show()

Which produces the following chart:

As you can see, the W' balance model is a powerful tool for evaluating an athlete's pacing strategy and energy utilization during a race or a training session. By plotting the power output and W' balance over time, we can gain insights into how effectively the athlete managed their energy reserves. For example, when W' balance hits zero at the end of a race or test, it indicates that the athlete has fully utilized their available W' and paced the effort perfectly. This is often a sign of optimal pacing strategy. Alternativley, if the athlete finishes the race with a positive W' balance, it suggests that they had some unused energy reserve. This might indicate conservative pacing, where the athlete could have potentially exerted more power and achieved a faster time.

🚴 Critical Metabolic Rate and The M' Balance Model

Critical Metabolic Rate (CMR) and Critical Power (CP) are concepts used to model an athlete's performance and endurance capabilities. While they share some similarities, they are applied to different physiological parameters and provide insights into different aspects of athletic performance. For example, critical metabolic rate (CMR) is applied to muscle oxygenation rates of change (ΔSmO2), and CMR represents the highest sustainable rate of muscle oxygenation change (%/s) an athlete can maintain over a prolonged period without leading to an unsustainable drop in muscle oxygenation levels.

In the code block below, I'll show you how to calculate an athlete's critical metabolic rate using historic training data from multiple time to exhaustion trials collected with the NNOXX wearable device:

data = pd.DataFrame({'De-oxy rate': [-0.019, -0.035, -0.048, -0.068, -0.22], 'Duration': [7200, 4800, 3600, 2400, 1200]})

def critical_metabolic_rate(t, CMR, M_prime):
    return CMR + M_prime / t

deoxy_rate = data['De-oxy rate'].values
duration = data['Duration'].values
 
popt, pcov = curve_fit(critical_metabolic_rate, duration, deoxy_rate)
CMR, M_prime = popt

print(f"Estimated Critical Metabolic Rate (CMR): {CMR:.5f} %/s")
print(f"Estimated M': {M_prime:.5f}")

Which produces the following output:

• Estimated Critical Metabolic Rate (CMR): 0.03111 %/s

• Estimated M': -290.96203

Ordinarily critical metabolic rate is negative and approximates zero, but in some cases we may see a positive value. Additionally, M' is negative since it represents the area between the deoxy-duration curve and the y-intercept (i.e, critical metabolic rate), as demonstrated in the image below. However, we will invert the sign on M' when modeling M' balance as you'll see in the next code block.

Now, we'll use our calculated critical metabolic rate and M' as inputs for our M' balance model, which is an adaptation of the W' balance model used in the context of critical power. The M' balance model applies the same principles to muscle oxygenation data, focusing on the rate of change of muscle oxygen saturation (ΔSmO2) instead of power output, as demonstrated in the code block below:

CMR = CMR
M_prime = M_prime * - 1 # Invert sign

# initialize M' balance model
M_bal = np.zeros(len(workout_data))
M_bal[0] = M_prime
tau_recovery = 300  # Example recovery time constant, adjust as needed

# update M' balance
for i in range(1, len(workout_data)):
    dt = workout_data['time'].iloc[i] - workout_data['time'].iloc[i - 1]
    d_rate = workout_data['SmO2_accel_rolling_mean'].iloc[i]
    if d_rate < CMR: # Depletion of M'
        M_bal[i] = M_bal[i - 1] - (CMR - d_rate) * dt
    else: # restoration of M'
        M_bal[i] = M_bal[i - 1] + (M_prime - M_bal[i - 1]) * (1 - np.exp(-dt / tau_recovery))

    # Ensure M' balance does not go below 0 or above M'

# add M' balance to workout data
workout_data['M_bal'] = M_bal

# create plots
fig, ax1 = plt.subplots(figsize=(10, 6))
ax1.plot(workout_data['time'], workout_data['M_bal'], label="M' Balance", color='red')
ax1.set_xlabel('Time (seconds)')
ax1.set_ylabel("M' Balance", color='red')
ax1.tick_params(axis='y', labelcolor='red')
ax2 = ax1.twinx()
ax2.plot(workout_data['time'], workout_data['SmO2_rolling_mean'], label='Δ SmO2 Rate', color='blue')
ax2.set_ylabel('Muscle oxygenation (%)', color='blue')
ax2.tick_params(axis='y', labelcolor='blue')
lines, labels = ax1.get_legend_handles_labels()
lines2, labels2 = ax2.get_legend_handles_labels()
ax1.legend(lines + lines2, labels + labels2, loc='best')
plt.show()

Which produces the following output:

In the image above, you can see an athlete's muscle oxygenation data during a ~4-hour bike ride in the mountains. Throughout the ride, their M' progressively decreases, with brief periods of restoration when they lighten the pace, although it never fully depletes. With sufficient prior data, it's possible to retroactively analyze an athlete's muscle oxygenation data using the M' balance model. This analysis helps determine how close they pushed themselves to their physiological limit. Additionally, this information can be used for real-time pacing guidance, optimizing performance during training and racing. By understanding the athlete's muscle oxygenation dynamics, we can tailor strategies to enhance endurance, delay fatigue, and improve overall performance.

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