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A physiological heart-rate model for energy expenditure across work, exercise, and recovery: development and in-sample evaluation against indirect calorimetry

This study developed and evaluated an in-sample physiological heart-rate model that accurately estimates total energy expenditure across diverse activity states (work, exercise, and recovery) using only heart rate and protocol-derived parameters, achieving high agreement with indirect calorimetry despite notable errors in specific conditions like stair descent and recovery.

Original authors: Agnar Steinarsson, Jón Hlöðver Friðriksson, Gréta Jakobsdóttir, Guillaume Bonnier, Christopher Moufawad El Achkar, Erlingur Jóhannsson, Alia Lemkaddem

Published 2026-10-26

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CC BY 4.0

Original authors: Agnar Steinarsson, Jón Hlöðver Friðriksson, Gréta Jakobsdóttir, Guillaume Bonnier, Christopher Moufawad El Achkar, Erlingur Jóhannsson, Alia Lemkaddem

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ⚕️ This is an AI-generated explanation of a preprint that has not been peer-reviewed. It is not medical advice. Do not make health decisions based on this content. Read full disclaimer

Every day, our bodies burn fuel to keep us moving, thinking, and breathing. Scientists call this energy expenditure, and knowing exactly how much we burn is crucial for managing weight, training for sports, or simply understanding our health. For decades, the gold standard for measuring this has been a method called indirect calorimetry, where a person wears a mask that analyzes the oxygen they breathe in and the carbon dioxide they breathe out. It is accurate, but it is also bulky, expensive, and impossible to wear while living a normal life. To fill this gap, wearable technology has turned to heart rate. The logic is simple: as our muscles work harder, our heart beats faster to deliver oxygen. By tracking the pulse, devices try to guess how much energy is being burned. However, this relationship is not a straight line. A heart rate of 120 beats per minute might mean something very different when you are walking up a hill compared to when you are lifting heavy weights or sitting still after a sprint. The heart responds to stress in complex ways that do not always match the actual fuel consumption, especially during non-rhythmic movements or recovery.

A team of researchers set out to build a new way to calculate this energy burn using only heart rate, without relying on general guesses about a person's age, gender, or height. They developed a model that treats the heart rate not just as a number, but as a signal of the body's specific physiological state. To test it, they brought twenty healthy adults into a laboratory for a single, long session that mimicked a varied day. The participants spent an hour doing office work, followed by a circuit of exercises that included cycling, lifting weights, and climbing stairs, and finally, they ran on a treadmill until they were exhausted, followed by twenty minutes of sitting and recovering. Throughout this entire time, the researchers measured the true energy burned using the gold-standard gas analysis mask, while simultaneously recording the heart rate data to see if their new model could match the real numbers.

The results showed that the new model was remarkably accurate when looking at the total energy burned over the entire session. On average, the model's estimate was almost identical to the true measurement, missing by less than one kilocalorie across the whole day. This suggests that by using specific physiological markers—like a person's maximum heart rate and the speed at which their breathing changes during exercise—the model can create a personalized map of how that specific body burns fuel. It managed to do this without needing to know the person's age or gender, which are usually required by standard formulas. The model worked best during rhythmic activities like running and walking, where the heart rate and energy burn are tightly linked. In these moments, the estimates were very close to the truth, often within a few percentage points.

However, the story becomes more complicated when the activity changes. The model struggled most with short, intense bursts of resistance exercise and the period immediately after stopping. During the recovery phase, when the heart rate remained high even though the body was slowing down, the model tended to overestimate the energy burned. Similarly, during eccentric movements, such as walking down stairs where muscles lengthen under tension, the model had difficulty predicting the low energy cost of the action. These errors occurred because the heart rate can stay elevated or rise for reasons other than immediate fuel consumption, such as the body's effort to cool down or the specific mechanics of the muscle. While the model successfully avoided the need for demographic guesses, the individual errors for these specific states were still significant, showing that a single heart rate number cannot perfectly explain every type of movement.

Ultimately, the study demonstrates that a heart-rate-based approach can provide a highly accurate picture of total daily energy expenditure without needing to know a person's age or gender, provided the model uses individualized physiological quantities derived from the specific session being analyzed. The findings suggest that the key to better tracking lies not in adding more sensors, but in understanding the specific context of the movement. If a device can recognize whether a person is running, lifting, or recovering, it can apply the right rules to the heart rate data to get a much clearer picture. While the current version of this model is a promising step forward, the researchers note that it still needs to be tested in the real world, outside the controlled environment of a laboratory, to see if it holds up when people are living their lives without a researcher telling them exactly what they are doing.

Technical Summary: A Physiological Heart-Rate Model for Energy Expenditure

Problem Statement
Accurate estimation of energy expenditure (EE) across the heterogeneous activities of daily life remains a significant challenge for wearable technology. While accelerometry is the standard, it performs inconsistently during static, resistance, and non-rhythmic exercises where metabolic demand is high but movement is low. Conversely, heart-rate (HR)-based methods offer a complementary signal reflecting cardiovascular metabolic demand but often rely on population-based equations (incorporating age, sex, and height) that introduce individual error, particularly at low intensities or during recovery. Existing flex-heart-rate approaches require individual laboratory calibration, limiting their practicality. There is a need for a model that utilizes individualized physiological parameters to estimate EE across diverse states—sedentary work, exercise, and recovery—without relying on demographic population equations.

Methodology
The study developed and evaluated a Physiological Heart-Rate Model (PHM) using a single continuous physiological signal: heart rate. The study involved 20 healthy adults (11 men, 9 women; mean age 39.5 ± 7.3 years) who completed a single-session protocol comprising three phases:

Work: 50–60 minutes of seated/standing office activity.
Exercise: A 15-minute non-rhythmic circuit including cycling, resistance exercises (bicep curls, push-ups, squats, planks), and stair ascent/descent.
CPET + Recovery: An incremental treadmill test to volitional exhaustion followed by 20 minutes of seated recovery.

Reference and Model Inputs:

Reference EE: Measured breath-by-breath via indirect calorimetry (COSMED K5) and calculated using the Weir equation.
Model Inputs: The PHM utilized HR as the sole continuous signal, supplemented by activity-state labels (work, walking, exercise, recovery). Individualized parameters were derived from the session itself:
- Maximal Heart Rate ( $HR_{max}$ ) and speed at the second ventilatory threshold ( $vVT2$ ) from the CPET.
- An endurance parameter ( $E$ ) fitted from the recovery phase heart-rate response.
- Body mass.
Model Architecture: The model estimates EE per second as a resting baseline plus a heart-rate-driven term. It defines a heart-rate threshold interval ( $P_i$ ) and a resting/response anchor ( $P_c$ ). A basal metabolic rate (BMR) is derived from an activity factor ( $AF$ ) rather than demographic equations. The instantaneous activity factor ( $\alpha(t)$ ) transitions between low-intensity (basal) and high-intensity values via a logistic function, with adjustments for cardiovascular drift above the second ventilatory threshold.

Key Results
The evaluation was an in-sample assessment, meaning coefficients were calibrated on the same cohort used for testing.

Total Session Agreement: The PHM demonstrated high agreement with indirect calorimetry for total session energy.
- Mean EE: 465.9 kcal (PHM) vs. 466.8 kcal (Reference).
- Bland-Altman Bias: -0.9 kcal (-0.2%).
- 95% Limits of Agreement: -33.6 to +31.7 kcal.
- Lin's Concordance Correlation Coefficient: 0.98.
- Root Mean Square Error (RMSE): 16.3 kcal (3.5%).
Performance by Activity State:
- Rhythmic Locomotion: Highest accuracy was observed during running (Bias: -3.1%, MAD: 3.5%) and walking (Bias: -5.9%, MAD: 10.8%).
- Resistance and Eccentric Exercise: Errors were larger and varied by mode. Weighted exercise showed a +18.1% bias (MAD 21.2%), while stair descent showed the largest relative error at +35.3% (MAD 35.3%). Push-ups and squats were underestimated.
- Sedentary Work: The model estimated sedentary work with a mean bias of +0.6% and a Mean Absolute Deviation (MAD) of 9.2%. Individual standard deviation was 11.2%, resulting in 95% limits of approximately ±22%.
- Recovery: Recovery EE was systematically overestimated by 15.7%.
Demographic Independence: The model successfully estimated light sedentary and recovery EE without using age-, sex-, or height-based population equations. However, the individual precision for sedentary work was comparable to that of conventional population-based equations.

Significance and Claims
The authors claim that the PHM demonstrates the feasibility of estimating energy expenditure across diverse activity states using individualized physiological quantities ( $HR_{max}$ , $vVT2$ , $E$ , body mass) rather than demographic proxies. The model successfully reproduced total session energy expenditure with high concordance, suggesting that individualized heart-rate physiology can serve as a robust basis for EE estimation.

However, the paper maintains a modest stance regarding its current limitations:

In-Sample Nature: The reported performance is "apparent in-sample" because coefficients were fine-tuned on the evaluation cohort. External validation with coefficients fixed a priori is required to confirm generalizability.
Activity Context: The model relies on broad activity-state labels (supplied manually in this study). The authors suggest that future wearable implementations could improve accuracy by using accelerometry to automatically detect specific modes (e.g., distinguishing stair descent from running) to select appropriate HR-metabolic mappings.
Recovery and Eccentric Errors: The systematic overestimation during recovery and eccentric activities highlights that the coupling between heart rate and metabolic demand is weaker in these states, a known physiological challenge.
Future Requirements: The authors explicitly state that the study does not yet validate demographic-free estimation for free-living conditions. Future work must include independent cohorts, field-estimated inputs, automatic activity-state detection, and comparison against doubly labeled water.

In conclusion, the study presents a promising physiological framework that achieves high aggregate accuracy without demographic inputs, but emphasizes that individual error in specific states (particularly recovery and resistance) and the need for external validation remain critical areas for development.

Technical Summary: A Physiological Heart-Rate Model for Energy Expenditure

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