Technical Explainer · NDA
Why we can do this — and why it's hard to copy.
A private walk-through of the seven scientific steps that turn a heart rate stream into a fitted metabolic model. We don't show the ODE form, the inversion algorithm, or the priors; we show enough that a sophisticated reader can verify the approach is principled and the discoveries non-trivial.
In brief. Heart rate carries the dynamics of an underlying metabolic system governed by three core parameters: endurance (E), capacity ceiling (Vmax), and cardiac ceiling (P). We recover them by Bayesian inversion of an ODE designed for identifiability. Once recovered, the same model simulates the entire metabolic state forward in time and supplies every downstream metric a platform partner needs.
Seven-step walkthrough
From raw heart-rate dynamics to fitted metabolic state, without exposing the protected implementation.
Heart rate is a window into metabolism — if you know how to look.
During exercise, heart rate is not just a function of intensity. Its shape in time encodes oxygen-demand kinetics, substrate switching, lactate accumulation, thermoregulation, and fatigue accrual. These are decades of established exercise physiology — the dynamics are known. The hard part is reading them simultaneously and individually, from one signal, in real time, on a moving athlete.
Population-level wearable models read intensity and stop. The underlying metabolic system, with its individual differences, stays invisible to them. The annotations below mark the regions a model that doesn't separate them collapses into a single number.
The metabolic system has three primary parameters.
After fitting against tens of thousands of sessions across running, cycling, and team sports, three parameters consistently and sufficiently explain individual variation:
- E (endurance). Aerobic efficiency relative to glycolytic — reflects oxidative capacity, fiber-type distribution, mitochondrial density, fat-oxidation reserve. Independent of absolute power.
- Vmax. Maximal sustainable speed or power, bounded by oxygen-utilization capacity at the cellular level. The absolute capacity ceiling.
- P (HRmax). Cardiac ceiling — autonomic limit, peak heart rate. Sets the cardiovascular operating envelope.
They are independent. An athlete with high Vmax can have low E; another with average Vmax can have exceptional E. Two-parameter models miss this variation. Four-plus parameter models lose identifiability — the data can't tell which combination produced the observed response. Three is the minimum sufficient parameterization, and finding it — through years of model development against validated cohorts — is the discovery.
Real participants from validation cohorts. Each occupies a distinct point in (E, Vmax, P) space; archetypes emerge but the volume is continuous.
We recover the parameters by Bayesian inversion — a property the ODE was designed for.
The technical claim that does most of the work in this whole story: TrueZone's ODE is parameter-identifiable from heart rate data alone. Different combinations of E, Vmax, and P produce distinguishably different HR responses to the same input, so given a sufficient HR trace, the parameters are recoverable.
This is not a trivial property. Most physiological ODEs in the literature are non-identifiable: distinct parameter sets produce indistinguishable outputs and any inversion converges to a flat valley rather than a peak. TrueZone's ODE was designed from the inside out for identifiability — that's where most of the modeling work went.
The inversion is Bayesian: maximum-a-posteriori parameter estimation under physiologically-informed priors. We don't show the priors or the inversion details, but the result is robust to noise, missing samples, and irregular session timing.
Two HR responses to the same intensity protocol, generated by two different parameter combinations. The shapes are visibly different — that distinguishability is what makes recovery possible.
Each session updates the posterior. The model has memory.
A single session is noisy. One workout might suggest E = 0.62, but that estimate carries uncertainty. The right way to handle this is Bayesian: each new session updates the posterior over (E, Vmax, P) for that participant, integrating prior estimates with new evidence.
The result: convergence in 3–10 sessions of routine activity, robust to outlier days (a cold, a poor sleep), and adaptive to longitudinal change as fitness improves or declines. Each participant's model is fully independent — no global retraining when the user base changes, no privacy pooling required. In production the posterior is the user's profile: lightweight, updates in seconds after each session, persists across the user's lifetime.
Posterior tightening over real validation data. Credible interval shrinks as evidence accumulates; the model adapts as fitness changes.
Because it's an ODE, we know the entire metabolic state — not just heart rate.
A regression gives one number per timestep. An ODE gives a dynamical state vector — multiple coupled physiological variables evolving together. Once parameters are fitted, the same ODE simulates the system forward: oxygen demand, substrate split (fat vs carbohydrate), lactate accumulation, fatigue, recovery kinetics, time-to-exhaustion at current intensity.
All of these are derived from one model in one simulation, mutually consistent. They don't contradict each other because they share the underlying physics. This is what enables real-time outputs other systems can't compute: a substrate-aware pacing engine, a runway forecast at current intensity, a difficulty estimator that responds to internal state rather than a stopwatch. None of these are post-hoc heuristics — they're first-class outputs of the simulation.
One consequence is the aerobic + glycolytic flux partition. The state vector decomposes total metabolic flux into oxidative (aerobic) and substrate-level (glycolytic) ATP production continuously. Indirect calorimetry sees only the aerobic share via VO₂; a blood-lactate test sees one moment's glycolytic accumulation. HR responds to the integrated demand — oxygen delivery, metabolite clearance, sympathetic drive, buffering — and therefore tracks the total. TrueZone reads the partition back from the HR shape, which is why HR-based energy expenditure stays accurate at intensities where lab calorimetry under-counts the glycolytic contribution.
One workout. One ODE simulation. HR (top), substrate balance (middle), and fatigue with runway (bottom) — all from the same model, all coupled.
Three parameters, every metric that matters.
From a single fitted (E, Vmax, P) per participant, we derive every cardiovascular and metabolic metric of interest — aerobic thresholds, individualized HR and power zones, race predictions, glycemic-validated metabolic-fitness indices, calorimetry-validated energy expenditure, substrate balance, recovery kinetics, real-time time-to-exhaustion. Each derivation is physiologically grounded and shares the underlying parameters, so they're internally consistent. Replacing a dozen separate population models with one individualized one is the architectural advantage.
Every downstream metric a partner platform needs, derived from three core parameters. Each derivation is physically grounded; outputs share the underlying model and so cannot contradict each other.
Every output anchored to a clinical reference.
We don't ask partners to take any of this on faith. Each derived biomarker is validated against the same gold-standard reference a clinical investigator would use: CPET for VO₂max, indirect calorimetry for energy expenditure, glycemic response markers for metabolic fitness. Two clinical studies (METFIT at University of Iceland, FITSILVER with CSEM Switzerland — 70 participants, three papers submitted), plus five independent real-world datasets covering 815 participants and 28,516 sessions. One head-to-head against Apple's hybrid ODE + neural network (PHM): TrueZone matches accuracy with three parameters and no global training data.
Output · Clinical reference · Result
| Output | Validated against | Result |
|---|---|---|
| MFI (Metabolic Fitness Index) | Glycemic response markers | R² = 0.99 |
| VO₂max | Gold-standard CPET | MAE 2.1 ml/kg/min; no significant difference (p > 0.05) |
| Energy expenditure | Indirect calorimetry (metabolic cart) | 5.8% MAE — rest, walking, running, recovery |
| Heart rate prediction (real-time) | Held-out HR time series across 5 datasets | 4–6 bpm median MAE — matches Apple's hybrid ODE + neural network |
| E, Vmax, P (model parameters) | Bayesian per-participant fit | Convergence in 3–10 sessions of routine activity |
| Cardiac thresholds (VT1, VT2, LT) | Geometric derivation from fitted model | Replaces single max-effort lab test |
What this means for integration
A structured-feature service over heart rate.
TrueZone runs server-side via the C++ SDK or REST API. Submit raw heart rate and movement; receive validated, individualized, internally-consistent biomarkers and real-time outputs. The lightweight engine for second-by-second outputs runs on-device. Each user's model is fully independent — no global retraining, no privacy pooling. One granted patent and one pending across US and Europe; protection runway to 2045.
What this page deliberately does not show: the ODE form, the parameter identifiability proof, the inversion algorithm, the priors, the closed-form derivations for each downstream metric, the production fitting code. Under NDA we go meaningfully deeper in conversation. This page is the floor — what we can show in writing. The ceiling is a working session, where we can prove together how far the model goes on your data.