Week 3 - Velocity-Duration, Power-Duration, and Critical Power Concepts

Velocity-Duration and Power-Duration Foundations

• A. V. Hill established the curvilinear relationship between average velocity and exercise duration across world record events in men's and women's swimming, running, and men's rowing.

• Kennerly (also cited as Kennelly or McKenley) demonstrated that plotting the natural logarithm of distance against log time produces a linear relationship, highlighting a scale-invariant power law function.

• Modern data demonstrates a near-perfect fit ($R^2 \approx 1$) for power law functions between $1000\,m$ and marathon distances, but the relationship deteriorates at extreme short ($60\,m$-$100\,m$) and long ($24\text{-hour}$ ultramarathon) distances due to physiological discrepancies across intensity domains.

The Critical Power and Critical Speed Framework

• Wilkie (1985) along with Monod and Scherrer (also referred to as Mono and Shera) defined Critical Power ($CP$) and Critical Speed ($CS$) as the asymptote of the power-duration or velocity-duration curve.

• Work capacity above $CP$ or $CS$ is defined by the curvature constant $W'$ (work prime, in joules or $kJ$) or $D'$ (distance prime, in meters), historically termed anaerobic work capacity.

• Work performed at or below $CP$/$CS$ does not utilize $W'$/$D'$, whereas exercise above $CP$/$CS$ depletes this fixed energetic reserve until exhaustion occurs.

• Research by Mark Burnley confirmed that the $CP$/$CS$ relationship applies across diverse species (including horses, painted ghost crab, fiddler crab, lungless salamander, and desert iguana under cool vs. warm conditions) and modalities (cycling, isometric knee extension at $14\%$ to $40\%$ of MVC, shuttle running).

Mathematical Modeling and Testing Protocols

• Multi-Visit Protocols: Uses $3$ to $5$ laboratory visits for time trials to exhaustion ($T_{\text{lim}}$) evaluated via three primary mathematical models:

  • Hyperbolic model: $T_{\text{lim}} = \frac{D'}{\text{Speed} - CS}$.

  • Linear distance-time model: Plotted as distance over time, where the slope represents $CS$ and the y-intercept represents $D'$.

  • Linear inverse-time ($\frac{1}{t}$) model: Plotted as speed against $\frac{1}{t}$, where the slope represents $D'$ and the y-intercept represents $CS$.

• 3-Minute All-Out Test ($3\text{-min AOT}$):

  • Requires continuous $100\%$ maximal effort for $180\,\text{seconds}$ without pacing to fully deplete $W'$/$D'$.

  • Power output plateaus during the final $30\,\text{seconds}$, which approximates $CP$

  • Correlates tightly with multi-visit testing, provided athletes receive non-time-based verbal encouragement and complete familiarization to prevent pacing.

  • Field-based testing developed by Kramer and colleagues adapts this protocol using maximal shuttle running.

Metabolic Domains and Physiological Determinants

• $CP$ and $CS$ delineate the domain boundary between heavy-intensity and severe-intensity exercise (whereas lactate threshold delineates moderate-intensity from heavy-intensity exercise).

• Severe Intensity Domain ($>\!CP$ / $>\!CS$):

  • Oxygen uptake ($\text{VO}_2$) fails to achieve a steady state and increases toward $\text{VO}_2\text{ max}$.

  • Blood lactate continually accumulates without stabilizing.

  • Causes rapid depletion of intramuscular phosphocreatine ($\text{PCr}$), marked acidosis, and elevated inorganic phosphate.

• Moderate and Heavy Domains ($\le\!CP$ / $\le\!CS$):

  • Achieve a metabolic steady state for $\text{VO}_2$, blood lactate, and $\text{PCr}$ (with a $\sim 60\%$ drop in $\text{PCr}$ during heavy exercise before stabilization).

  • Fatigue during prolonged exercise below $CP$ is driven primarily by muscle glycogen depletion rather than high-energy phosphate depletion.

• $CP$ is strongly associated with muscle oxidative capacity, exhibiting strong correlations with the percentage of type 1 muscle fibers ($r \approx 0.7$) and type 1 fiber capillary contacts.

Applied Uses: Profiling, Prediction, and Intermittent Exercise

• Athlete Profiling and Performance Prediction:

  • Elite marathoners (such as Eliud Kipchoge / Elliot Kupchogo / Elliot Kimchogue and Antonio Pinto, as studied by Andy Jones and Annie Van Hatland / Anni Van Hartelow) run personal bests at an average mean speed of $96\%$ of their $CS$

  • Mathematical modeling predicts maximal performance capability ($T_{\text{lim}}$), such as a theoretical maximum of $15\,\text{min}\,13\,\text{s}$ ($913\,\text{seconds}$) for a $5000\,m$ run.

• Intermittent Work and $W'$ Balance Modeling:

  • $W'$/$D'$ acts as a rechargeable energetic store; working above $CP$/$CS$ depletes the balance, while working below $CP$/$CS$ reconstitutes it.

  • Phil Skiba (or Phils schema) and Christian Vassallo (from Michael Wardroam / Ward-Hopkins group) developed integral and differential $W'$ balance models ($W'_{\text{bal}}$) to track energy expenditure in real time during variable-paced sports and protocols like the $30\text{-}15$ Intermittent Fitness Test.

  • Depletion rate depends on work interval intensity and duration; recovery rate depends on how far recovery intensity is below $CP$/$CS$ and recovery interval duration (e.g., changes observed across $5\,s$ to $20\,s$ vs. $30\,s$ intervals).

• Training Adaptations:

  • Training responses are better explained by changes in $CP$ and $W'$ than by changes in $\text{VO}_2\text{ max}$.

  • Endurance training increases $CP$/$CS$ but frequently reduces $W'$/$D'$.

  • Resistance and high-intensity strength/power training (Sawyer and colleagues; Geyser and Wilson / Geyzer and Wilson) can enhance $W'$/$D'$ without altering $CP$.