Estimating Non-Linear Power Degradation in Elite Athletes

By the Forwod Research & Engineering Team | February 14, 2026

In our previous discussions, we established the mathematical architecture required to calculate absolute mechanical Work—bridging the gap between the conservative force of gravity acting on a barbell and the non-linear fluid dynamics of an ergometer.

However, calculating the Work (Joules) or instantaneous Power (Watts) of a single movement represents only a fraction of the metrological equation. The human body is not an internal combustion engine; it does not possess an infinite battery, nor does it output power linearly.

Human performance is governed by distinct, overlapping bioenergetic pathways. An elite athlete might generate $1,500 \text{ Watts}$ during a maximal-effort, 3-second heavy deadlift, relying entirely on the highly volatile adenosine triphosphate-phosphocreatine (ATP-PCr) system. Yet, that same athlete may only sustain $250 \text{ Watts}$ over a 40-minute rowing bout as their body shifts into aerobic oxidative metabolism.

To definitively quantify an athlete’s overarching "Work Capacity," analytical models cannot isolate a single 1-Repetition Maximum (1RM) lift or a singular $5\text{km}$ run. They must continuously and mathematically map how the human nervous system exhausts its power reserves across all physiological time domains.

Historically, tracking this bioenergetic degradation has been the Achilles' heel of sports informatics.

The Infinite Power Hallucination

For decades, sports scientists and endurance coaches have relied on the classic two-parameter Critical Power (CP) model, originally pioneered by Monod and Scherrer in 1965.

The CP model was designed to identify an athlete’s theoretical sustainable aerobic threshold (Critical Power, or $CP$) and their finite anaerobic battery ($W'$). For any given time ($t$) in seconds, the relationship between Power ($P$) and time is expressed algebraically as:

Clinical physiologists have long acknowledged that the traditional CP equation possesses strict physiological boundaries, generally limiting its validity to continuous exercise lasting between two and fifteen minutes. Because the formula attempts to divide by a microscopic number as $t \to 0$, it mathematically predicts infinite power for short durations.

While academics respect these boundaries in a lab, the modern mixed-modal athlete routinely operates outside them in the real world—transitioning fluidly from 3-second maximal ATP-PCr outputs to 40-minute oxidative efforts. If an analytical model attempts to holistically map this athlete without respecting the finite depletion and reconstitution kinetics of the phosphocreatine system, it inherently hallucinates limitless kinetic energy.

Conversely, at the long-duration extreme, the model draws a horizontal asymptote, predicting that the athlete can maintain their Critical Power threshold forever. It completely ignores the biological reality of central nervous system exhaustion and ultra-endurance systemic fatigue.

The Omni-Domain Resolution

To accurately quantify elite athletes who transition seamlessly across energy systems, the computational framework must mathematically cap short-duration peak power and apply mathematical penalties for long-duration fatigue.

To resolve this, the Forwod Calculation Engine adapted a sophisticated Omni-Domain Power-Duration (OmPD) architecture. Instead of a simplistic linear asymptote, our engine evaluates a massive historical array of an athlete's Maximum Mean Power (MMP) and fits it to a complex non-linear equation.

The core of this non-linear prediction cap can be expressed as:

By introducing $P_{max}$ (Maximum Instantaneous Power) into the exponential decay function, the equation establishes an absolute physiological ceiling. It physically prevents the algorithm from hallucinating infinite power at $t \to 0$, gracefully mapping the violent depletion of the ATP-PCr system. For time intervals that stretch into ultra-endurance territories, the engine subsequently applies an aggressive logarithmic penalty to model ultimate systemic metabolic decay.

Constraining Computational Calculus

Rendering this equation in an academic paper is straightforward. Translating it into a computational environment that evaluates millions of historical data points is an exercise in extremely volatile calculus.

Extracting these parameters ($P_{max}$, $CP$, $W'$) from raw, chaotic human data requires heavy non-linear least-squares optimization algorithms. But mathematical optimizers do not intuitively understand human biology. If a curve-fitting algorithm is fed a raw dataset and left completely unbounded, it will frequently fail to converge or get trapped in local mathematical minima. Unconstrained, an algorithm might hallucinate that an athlete possesses a negative endurance decay rate—mathematically implying they magically gain power the longer they run.

To make our engine computationally stable and biologically valid, we engineered a proprietary matrix of dynamic heuristics directly into our analytical pipelines. Before our engine runs the heavy calculus, it programmatically intercepts the telemetry and cages the optimizer within strict physiological boundaries. By mathematically anchoring the algorithmic starting points, we force the computation to obey the absolute limits of human physiology.

A Moving Target

Predicting human bioenergetics is arguably the most dynamic field in sports science. At Forwod, we treat this architecture as a moving target. We are continually analyzing the biological bounds of systemic fatigue, tracking how different morphologies degrade over time, and aggressively refining the calculus that governs our engine in the background. As we accumulate deeper datasets, our ability to map the exact intersection of mathematical optimization and human decay becomes exponentially sharper.

Selected Clinical Context & Further Reading

• Monod, H., & Scherrer, J. (1965). The work capacity of a synergic muscular group. Ergonomics.
• Puchowicz, M. J., Baker, J., & Clarke, D. C. (2020). Development and field validation of an omni-domain power-duration model. Journal of Sports Sciences.
• Gastin, P. B. (2001). Energy system interaction and relative contribution during maximal exercise. Sports Medicine.
• Motulsky, H., & Christopoulos, A. (2004). Fitting models to biological data using linear and nonlinear regression: A practical guide to curve fitting. Oxford University Press.