Estimating carbohydrate and fat oxidation from heart rate: a practical model¶

This document describes a model for estimating the oxidation rate of carbohydrate (CHO) and fat (FAT) during exercise from a single, non-invasively measurable variable: heart rate (HR). The physiology and mathematics of the model are presented together with their scientific references, deriving it step by step from the objective to its final form.

1. Objective¶

We want to know, at any instant \(t\) during an activity, the oxidation rate of carbohydrate and fat:

and, from these instantaneous rates, the cumulative total over the whole activity (from \(0\) to the total duration \(T\)):

The rest of the document introduces the necessary background concepts (§2), and then answers, in order, two questions: how \(CHO(t)\)/\(FAT(t)\) is calculated (§3-7), and how this instantaneous rate relates to a cumulative total (§8).

2. Background concepts¶

Before getting into the model, it is worth clearly defining the physiological variables that will be used throughout.

2.1. Oxygen consumption (\(VO_2\)) and CO2 production (\(VCO_2\))¶

\(VO_2\) is the volume of oxygen the body consumes per unit of time. It is measured in litres/minute (absolute value) or in ml·kg⁻¹·min⁻¹ (relative to body weight, useful for comparing people of different sizes). \(VCO_2\) is, analogously, the volume of carbon dioxide produced per unit of time. Both are the basis of indirect calorimetry: since oxidizing fat or oxidizing carbohydrate consumes O2 and produces CO2 in different proportions (due to the different chemical composition of both molecules), measuring \(VO_2\) and \(VCO_2\) allows deducing which fuel is being burned, without having to measure it directly.

2.2. Maximal oxygen consumption (\(VO_{2max}\))¶

\(VO_{2max}\) is the highest \(VO_2\) value a person can reach during maximal effort — the classic reference measure of cardiorespiratory fitness (how much blood/oxygen the cardiovascular and muscular system can pump and use per unit of time). It is used as the upper end of the scale in this model (§4):

Bassett DR Jr, Howley ET. Limiting factors for maximum oxygen uptake and determinants of endurance performance. Medicine & Science in Sports & Exercise. 2000;32(1):70-84.

2.3. Respiratory Exchange Ratio (RER)¶

A single number, typically between \(\sim 0.7\) and \(\sim 1.0\) at rest/submaximal exercise, that indicates the mix of substrates being oxidized: \(RER=0.7\) corresponds to almost exclusively fat oxidation; \(RER=1.0\), to almost exclusively carbohydrate oxidation. It is the central variable of the model, since it relates directly to the percentage of energy coming from each substrate (§5).

2.4. Relative heart rate: %HRmax and Heart Rate Reserve (%HRR)¶

Absolute heart rate (HR, beats/min) is not comparable between people — it must be expressed relatively. Two common ways:

• %HRmax, the simplest, but physiologically less precise:

• %HRR (heart rate reserve), which relativizes HR against the full available range (from rest to maximum), not only against the maximum:

This is the one used by this model (§4.1). The conceptual origin of the "reserve method" is found in:

Karvonen MJ, Kentala E, Mustala O. The effects of training on heart rate; a longitudinal study. Annales Medicinae Experimentalis et Biologiae Fenniae. 1957;35(3):307-315.

2.5. Relative oxygen consumption and VO2 Reserve (%VO2R)¶

Analogously to §2.4, \(VO_2\) can be expressed relative to its own range, between the resting value and \(VO_{2max}\) (§2.2):

This model estimates \(\%VO_2R\) from \(\%HRR\) (§2.4), thanks to the empirical equivalence between the two, discussed in detail in §4.2.

3. The starting point: the Jeukendrup & Wallis equations¶

The relationship between oxygen consumption, CO2 production, and grams of substrate oxidized per minute has been known in exercise physiology since the classic non-protein (i.e., ignoring the small contribution of protein oxidation) indirect calorimetry equations of:

Frayn KN. Calculation of substrate oxidation rates in vivo from gaseous exchange. Journal of Applied Physiology. 1983;55(2):628-634.

This model specifically uses the modification of these equations for exercise (Frayn derived them with rest in mind, assuming glucose as the CHO substrate; during exercise the predominant substrate is glycogen, which slightly changes the coefficients):

Jeukendrup AE, Wallis GA. Measurement of substrate oxidation during exercise by means of gas exchange measurements. International Journal of Sports Medicine. 2005;26(Suppl 1):S28-S37.

What do we need to use this equation? The two continuous variables defined in §2.1: \(VO_2(t)\) and \(VCO_2(t)\). Neither can be measured without a laboratory gas analyzer, so indirect methods of estimating them are needed (§4-7).

3.1. Rewriting the equation in terms of RER¶

Instead of estimating \(VO_2\) and \(VCO_2\) separately, it is more practical to estimate \(VO_2\) and the respiratory exchange ratio \(RER\) (§2.3). Substituting \(VCO_2(t) = RER(t)\cdot VO_2(t)\):

Now two independent estimates are needed: \(VO_2(t)\) (§4) and \(RER(t)\) (§5-6).

4. Estimating \(VO_2(t)\) from heart rate¶

4.1. Heart rate reserve (%HRR)¶

(definition introduced in §2.4).

4.2. From %HRR to %VO2 reserve¶

The estimate is based on the empirical equivalence \(\%HRR \approx \%VO_2R\) (§2.5; \(VO_2\) reserve, not \(\%VO_{2max}\), which is less precise):

Swain DP, Leutholtz BC. Heart rate reserve is equivalent to %VO2 reserve, not to %VO2max. Medicine & Science in Sports & Exercise. 1997;29(3):410-414.

where \(VO_{2,rest} = 3.5\ \text{ml}\cdot\text{kg}^{-1}\cdot\text{min}^{-1}\) (1 MET, the standard resting oxygen consumption).

4.3. From relative to absolute¶

Model limitations: \(\%HRR\approx\%VO_2R\) is a statistical approximation (population average, with individual variability). The result depends directly on the quality of \(HR_{rest}\), \(HR_{max}\) and \(VO_{2max}\) as model inputs — in practice, these three values are themselves often estimates too (not direct laboratory measurements), which introduces an additional source of accumulated uncertainty in the final result.

5. Estimating \(RER(t)\): metabolic thresholds¶

Real RER is not constant: it rises non-linearly with intensity, especially around the metabolic thresholds (\(LT_1\), aerobic threshold; \(LT_2\), anaerobic/lactate threshold). This general behaviour is documented in:

Faude O, Kindermann W, Meyer T. Lactate threshold concepts: how valid are they? Sports Medicine. 2009;39(6):469-490.

\(LT_1\) is physiologically defined as the point where blood lactate starts to rise above resting level. At this point, a well-trained endurance athlete is still burning mostly fat (metabolism remains highly aerobically efficient), not a 50%/50% mix. The point where the energy contribution is exactly 50% fat / 50% CHO is a related but different concept, known in the literature as the "crossover point":

Brooks GA, Mercier J. Balance of carbohydrate and lipid utilization during exercise: the "crossover" concept. Journal of Applied Physiology. 1994;76(6):2253-2261.

According to the classic non-protein respiratory quotient table (Lusk 1924; updated by Péronnet & Massicotte 1991, already cited in §3), the 50%/50% point corresponds to \(RER\approx 0.85\), while at \(LT_1\) (a little earlier, at lower intensity) RER tends to be slightly lower — consistent with fat still predominating slightly at that point. \(LT_2\), in turn, marks the maximal lactate steady state: the limit intensity at which the body can still clear lactate at the same rate it is being produced; at this point RER is typically between \(0.95\) and \(1.00\) (there is still a small residual fat oxidation, typically \(6\)-\(10\%\) of energy). Above \(LT_2\), RER can exceed \(1.00\) (up to \(1.05\)-\(1.10\) in near-maximal efforts), due to the buffering effect of blood bicarbonate on accumulated lactic acid, which releases additional CO2 through the lungs.

5.1. Interpolation model¶

Since lactate or gases cannot be measured without laboratory equipment, this curve can be approximated with a piecewise linear interpolation, based on four HR reference points: rest, an onset point (\(I_{onset}\), §6), \(LT_1\), \(LT_2\), and \(HR_{max}\):

with the reference values chosen for this model:

Model limitations: this interpolation assumes that \(RER(t)\) responds instantaneously to HR. Physiologically, at the onset of exercise, fat mobilization and oxidation take a few minutes to fully activate (typically \(10\)-\(20\) min), so real RER rises a bit faster at the start than this purely HR-based model predicts. This delay is shortened if the athlete has done a warm-up or recent prior physical activity, since fatty-acid mobilization and blood flow to adipose tissue are already partially activated:

Andersson Hall U, Edin F, Pedersen A, Madsen K. Whole-body fat oxidation increases more by prior exercise than overnight fasting in elite endurance athletes. Applied Physiology, Nutrition, and Metabolism. 2016;41(4):430-437.

6. Estimating the model's parameters¶

The model needs four HR reference points (\(I_{onset}\), \(LT_1\), \(LT_2\), \(HR_{max}\), §5.1) and \(VO_{2max}\) (§2.2). Often, an athlete who has undergone specific testing (a lactate threshold test, a maximal exercise test with gas analyzer) already knows the real values of \(LT_1\), \(LT_2\), \(HR_{max}\) and \(VO_{2max}\) directly, in which case these are simply used as-is — no approximation is needed. When this direct data is not available, the parameters must be estimated indirectly; there are several reasonable ways to do this, but they are outside the scope of this document (it will depend on what indirect information is available in each specific case).

Special case — \(LT_1\) as a fraction of \(LT_2\): when \(LT_1\) is not known directly but (an estimate of) \(LT_2\) is, it is common to approximate it as a fraction of \(LT_2\), since \(LT_1\) tends to sit below \(LT_2\) (§5). This fraction depends on training level: in the general population the two thresholds tend to sit further apart, while in an endurance athlete with a good aerobic base they are closer together. This model assumes the latter profile (consistent with the RER reference values chosen in §5.1):

Special case — \(I_{onset}\) (onset of exercise): this point differs from the other three because it cannot be measured directly, not even with a laboratory test — there is no test that reveals "the point where RER starts rising gradually before \(LT_1\)", because it is not a discrete physiological event but a smoothing device of the model itself (§5.1: no biological system switches metabolic regime with a sharp step, so the model introduces this gradual transition instead of an abrupt change of slope exactly at \(LT_1\)). This is why \(I_{onset}\) is more arbitrary than the other parameters: even an athlete with all their laboratory data known would not have a "real" value for this point. The following relationship is used:

where the specific \(0.70\) does not come from any scientific article — it is a design choice of the model to make the transition gradual, not the value of a measurement.

7. Complete point-in-time model¶

Putting §3-6 together: given HR at an instant \(t\) and the personal parameters (weight, resting HR, max HR, \(LT_2\), \(VO_{2max}\)), the model calculates, in this order:

where \(RER(t)\) is obtained in parallel from HR, by piecewise interpolation (§5-6).

This is the point-in-time model: given an isolated instant \(t\) (and only the HR at that instant), it calculates an oxidation rate — not a total, nor anything that depends on what happened before. The result, \(CHO(t)\) and \(FAT(t)\), is always expressed in g/min (§3), regardless of how often this calculation is used in practice: the model does not know or care whether it is evaluated once per second, once per minute, or a single isolated time — it always answers the same question ("at what rate would CHO/FAT be oxidized if HR stayed constant at this value"), in the same unit.

The question of how to link this rate (g/min) to real measurements taken at any other frequency (per second, per minute...) without needing to convert the rate itself is precisely what the discretization in §8 solves: instead of changing the units of \(CHO(t)\)/\(FAT(t)\), the elapsed interval \(\Delta t\) is expressed in minutes (the same unit as the rate), whatever the actual interval between samples.

8. From instantaneous rate to cumulative total: the integral and its discretization¶

8.1. Continuous formulation¶

As stated in §1, what really matters is the cumulative total, not just the rate at one instant:

This integral cannot be solved analytically because \(CHO(t)\) depends on \(HR(t)\), which is an arbitrary function (whatever the person's heart actually does during the activity) — not a closed-form mathematical formula.

8.2. Discretization: Riemann sum¶

With data available only in discrete samples (\(t_0=0, t_1, t_2, \ldots, t_n\)), the simplest numerical approximation of an integral is the Riemann sum: hold the rate constant from the last sample to the current one and multiply it by the elapsed interval:

This is a first-order (rectangular) numerical integration, valid when \(\Delta t\) is much smaller than the speed at which the real physiological state changes. If \(\Delta t_i\) were large, the Riemann sum would apply the instantaneous rate at the moment of the sample retroactively over the whole interval — a real integration error, which grows with the size of the gap.

8.3. Instantaneous rate vs. smoothed rate¶

The cumulative total (§8.2) is, by definition, an exact sum with no smoothing — it gives the best possible estimate of how many grams have been oxidized in total. But as a point-in-time reference value for decisions during the effort (e.g., fueling), instantaneous \(CHO(t)\) and the cumulative total give two very different kinds of information:

• The point-in-time calculation \(CHO(t)\) answers "at what rate am I oxidizing CHO right now?" — it is very sensitive to momentary HR fluctuations (a short spike, a bend in the road, a scare), and therefore noisy to interpret at a glance, but it is the only one that reacts instantaneously to a real intensity change.
• The cumulative total answers "how much have I oxidized in total so far?" — useful for the session's final balance, but says nothing about the current rate.

There is a middle ground between the two: a moving average of \(CHO(t)\), which answers "at what rate have I been oxidizing CHO lately?" — more stable than the instantaneous rate (because it integrates several recent samples), but still sensitive to sustained real intensity changes (unlike the cumulative total, which dilutes them more and more as time passes).

A common way of computing this moving average, when samples arrive at not-necessarily-regular intervals, is the exponential moving average (EMA), with a time constant \(\tau\) that determines how much "weight" recent samples get relative to older ones:

Hunter JS. The exponentially weighted moving average. Journal of Quality Technology. 1986;18(4):203-210.

The larger \(\tau\) is, the more \(CHO_{avg}\) resembles the cumulative total (very stable, not very reactive); the smaller, the more it resembles the instantaneous rate (very reactive, noisier). The specific choice of \(\tau\) is a tradeoff between stability and reactivity, not a physiological property of the model.

9. Carbohydrate intake guidelines during exercise (fueling)¶

Practical recommendations exist for carbohydrate intake during endurance exercise, which depend on the duration of the effort (and not directly on body weight, since the limiting factor is intestinal absorption capacity —SGLT1/ GLUT5 transporters— rather than body mass):

Jeukendrup AE. A step towards personalized sports nutrition: carbohydrate intake during exercise. Sports Medicine. 2014;44(Suppl 1):S25-S33. DOI: 10.1007/s40279-014-0148-z

This source broadly establishes: - Short efforts (\(<45\)-\(60\) min): no significant CHO intake needed. - Efforts of 1 to 2-3 hours: \(\sim 30\)-\(60\) g/h. - Long efforts (\(>2.5\)-\(3\)h): up to \(\sim 90\) g/h, provided multiple-transportable carbohydrates are used (glucose + fructose blends), since a single carbohydrate source cannot be absorbed at such a high rate.

10. Reference summary table¶