The kinetics of lactate production and removal during whole-body exercise

Background: Based on a literature review, the current study aimed to construct mathematical models of lactate production and removal in both muscles and blood during steady state and at varying intensities during whole-body exercise. In order to experimentally test the models in dynamic situations, a cross-country skier performed laboratory tests while treadmill roller skiing, from where work rate, aerobic power and blood lactate concentration were measured. A two-compartment simulation model for blood lactate production and removal was constructed.

Results: The simulated and experimental data differed less than 0.5 mmol/L both during steady state and varying sub-maximal intensities. However, the simulation model for lactate removal after high exercise intensities seems to require further examination.

Conclusions: Overall, the simulation models of lactate production and removal provide useful insight into the parameters that affect blood lactate response, and specifically how blood lactate concentration during practical training and testing in dynamical situations should be interpreted.

Blood lactate is one of the most frequently measured values in endurance testing. It is a concentration. It rises when lactate is produced faster than it is removed, and it falls when removal is faster. In this model, all production and all removal happen in the muscles and other tissues. The blood only exchanges lactate with those tissues, but the blood is where the sample is taken.

The paper turns this into a small set of differential equations with a single input, the exercise intensity. It then compares the model with measurements on one elite cross-country skier.

The model rests on four assumptions:

1. Production follows aerobic demand. Glycolysis supplies pyruvate in proportion to the aerobic power the exercise needs. The mitochondria oxidise part of it, up to a limit. The rest becomes lactate.
2. Removal needs spare aerobic capacity. Lactate is oxidised or turned back into glycogen. This removal rises with aerobic power, but drops to zero when aerobic power reaches its maximum, Q_max.
3. Removal has a ceiling. More lactate speeds up removal, but only up to a maximum rate. Above a certain intensity, production exceeds that maximum and lactate can no longer level off. That intensity is the model's threshold.
4. Blood lags behind muscle. Lactate reaches the blood only by crossing from the muscle compartment. This is why blood lactate keeps rising for a while after a hard effort stops.

On a treadmill, the mechanical work rate is the power needed against gravity and rolling friction: W = m g v(\mu\cos\alpha + \sin\alpha), with friction coefficient μ. The paper converts this into a virtual steady-state aerobic power: the aerobic power the work would need once everything has settled, even if that is more than the athlete can deliver.

Here Q_b = 80 J/s is the resting metabolic power, Q_{ul} = 111 J/s is the cost of moving arms and legs without load, and c_2 = 5.8. The factor \bar A(\alpha) makes uphill skiing slightly cheaper per watt of work. Actual aerobic power cannot change instantly. It follows the virtual value with a delay of 30 seconds, and it cannot exceed Q_{\max} = 1886 J/s:

Figure 1. Aerobic power against work rate⛶ Fullscreen

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Glycolysis produces pyruvate at a rate P. The mitochondria take up part of it, \alpha_0\tanh(\beta_0 P)/\beta_0, an amount that levels off at high rates. The rest appears as lactate. Removal, by oxidation and by conversion back to glycogen, is the product of three factors: one that grows with the concentration and levels off, one that grows with aerobic power, and the spare aerobic capacity Q_{\max} - Q_a.

Pyruvate appearance is proportional to the virtual aerobic power. It follows that power with a 10 s delay (Eqs. 4 and 5). For the function D, the paper reuses the shape of the appearance term:

Appearance can be written as R_a = p_0\,D(\tilde{\bar Q}_a), so production and removal share the factor D. In a steady state, D cancels out. Move the slider to see at which intensities removal wins and at which production wins.

At constant aerobic power, setting production equal to removal gives a closed formula (Eq. 8). It depends on only two combinations of parameters: the ratio p_0\chi/d_0 and \chi. Because removal has a ceiling, \tanh(\chi C)/\chi \le 1/\chi, the steady-state lactate goes to infinity at a finite intensity. The paper identifies this intensity with the maximal lactate steady state (MLSS, Eq. 9).

Figure 3. Steady-state lactate⛶ Fullscreen

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A model with a single lactate pool (Eq. 6) cannot explain why blood lactate keeps rising after exercise stops. The paper therefore splits the pool (0.18 L/kg, 14 L for this skier) into two compartments. Lactate is produced and removed in the muscle compartment. The blood compartment only exchanges lactate with it:

Here V_b = 4 L and V_m = 10 L, and k_{1a} = 0.05 L/s. The blood has no removal of its own. Both compartments start at 0.5 mmol/L. Figure 4 runs the model at three constant speeds on a 0.05 incline.

Figure 4. Lactate at three constant speeds⛶ Fullscreen

Drag horizontally to zoom. Double-click to reset. Hover for values.

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The real test of the model is a 20-minute protocol with changing speed and incline (Figure 2). Gas exchange was measured throughout and blood was sampled every few minutes, followed by recovery at 2.2 m/s. Hovering over one chart shows all four at the same moment.

Figure 2. Speed and incline⛶ Fullscreen

Figure 5. Work rate and aerobic power⛶ Fullscreen

Figure 6. Anaerobic power from lactate accumulation (Eq. 10)⛶ Fullscreen

Figure 7. Muscle and blood lactate⛶ Fullscreen

Drag horizontally to zoom; the four charts zoom together. Double-click to reset.

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The third chart is the paper's Eq. 10. It converts the rate at which lactate accumulates into energy, using di Prampero and Ferretti's 3 mL O₂/kg per mmol/L and about 20 J per mL of O₂:

Negative values during recovery mean lactate is removed faster than it is produced. The published Figure 6 matches this calculation from about two minutes on.

In the first trial, simulated blood lactate fell much faster after exercise than the measured values. The authors then limited the flux from blood back into muscle, C_m - C_b \to \max(-0.1,\ C_m - C_b), with the value chosen by eye. With this cap in place, blood can lose lactate at most at a fixed rate, k_{1a}\,c/V_b. That is why the blood curve in Figure 7 falls along a straight line.

This has a consequence the paper does not discuss. While the cap is active, the speed at which blood lactate falls no longer depends on what the athlete does during recovery. Change the recovery intensity with the cap on and off to see this.

The figures of the paper are the reference. All of them were traced from the PDF at pixel level, and every curve on this page is compared with its published counterpart. The table gives the distance between the two: the average distance from each traced point of a published curve to the curve on this page, measured in the original figure and expressed in the units of its y axis. Smaller is closer. The last rows compare the model with the skier's measured blood lactate.

What the figures show

Where the text leaves room for interpretation, the figures can decide. These findings are also shown as comments next to the relevant text.

What the paper leaves open

The model code runs in your browser and can be read directly: model.js. The traced figures (digitized.json) and every number quoted on this page (forensics.json) can be downloaded.

The equations are solved with the classic fourth-order Runge-Kutta method and a fixed step of 0.5 s. The protocol is interpolated linearly between its corner points. A finer step changes the results by less than 0.003 mmol/L.

All parameters belong to one elite skier and were tuned on the same data they are compared with. The authors note that they depend on the type of exercise, the fitness of the athlete and glycogen stores, and that they have to be determined for each individual. Nothing on this page is a prediction for anyone else.

Figures 1 to 7 are reproduced from Moxnes and Sandbakk (2012), an open access article published under the Creative Commons Attribution License 2.0.