Introduction

Blood lactate is a well-known important parameter for exercise physiology because the individual blood lactate threshold offers a simple and convenient way to distinguish trained from untrained subjects [1] and to see the improvement of the endurance level after certain training for a given subject. Although lactate determination in whole blood is a standard method in performance diagnostics, there is only limited knowledge about the dynamic distribution of lactate among different cells, tissues and compartments [e.g., plasma and red blood cells (RBCs)] during exercise. In this context blood represents the first great distribution space for lactate leaving the muscle. However, lactate is not homogeneously distributed in blood. The lactate concentration in plasma, denoted by [LA]plasma hereafter, could be much higher than the lactate concentration in RBC, denoted by [LA]RBC hereafter, especially during and after intensive exercises. For this reason, many researchers have studied [LA]plasma and [LA]RBC separately before, during and after certain exercises and lactate infusion. In one group of studies [2–9], [LA]RBC was calculated from the lactate concentrations in plasma and in whole blood. In the other group of studies [10, 11], [LA]RBC was measured. Although the exercise protocols used in these studies were different, some common features found in these studies could be summarized as follows: (1) In most of the studies, [LA]RBC was found to be significantly lower than [LA]plasma even in the resting state, resulting in a ratio [LA]RBC·[LA] −1plasma of about 0.5. (2) In the cases of a high increase rate of [LA]plasma caused by intensive exercise, the increase of [LA]RBC had a time delay compared to the increase of [LA]plasma, and therefore [LA]RBC reached the peak after [LA]plasma reached its peak. (3) After [LA]RBC reached its peak, [LA]RBC decreased when [LA]plasma was still higher than [LA]RBC, as shown by our own studies (cf. “Experimental studies”).

In order to explain the above findings, some kind of resistance against the influx associated with the plasma-to-RBC lactate gradient must be involved. Actually, some suggestions for the mechanism of the resistance/the delayed influx have been made in the literature, e.g., Donnan equilibrium [12] {the Donnan ratio ([An−]RBC·[An−] −1plasma ) under resting conditions is ~−10 mV (calculable from Cl− concentrations) at the inside of the red blood cell membrane, forcing diffusible anions out}, a barrier provided by the membranes of red blood cells [2] and the saturation of transporters [2, 3, 7, 11, 13]. However, none of these suggestions could give a detailed explanation and quantitative description for the entire dynamic process of lactate transport to RBCs during and after exercise.

The purpose of the present study is to develop and test a simple mathematical model that describes the dynamic processes of lactate transport into and out of RBCs. This model will be justified by the comparison between the model-predicted results and observations. The formulation of the model, as well as the check of the validity of the model, will be based on the analysis of the raw data of the time courses of [LA]RBC and [LA]plasma for all subjects. The previous results of the observed [LA]RBC and [LA]plasma found in the literature are not sufficient for our purpose because the raw data were usually not fully published. Therefore, the method of the present study will include two parts, our own experimental studies and mathematical model, which will be described in “Experimental studies” and “Mathematical model,” respectively. The results of the mathematical model and their comparison with observations will be presented in “Results of the model and the comparison with observations.” The advantages and the limitations of the present model as well as its potential applications to medicine and sports will be considered in “Discussion.” Finally, some conclusions will be drawn in “Conclusions.”

Methods

Experimental studies

Eleven male subjects (healthy sports students aged 24.3 ± 2.1 years) participated in these studies. All subjects gave written informed consent to contributing to the studies. The protocols used in these studies have been granted a license from the institutional office for conducting the human research described in the studies and are in line with the Declaration of Helsinki. The purpose of these studies was to see how different warm-up conditions influenced the lactate concentrations/distribution in RBCs and in plasma before and after a 30-s short-term maximal effort exercise on a cycle ergometer. But now we will only use the results of these experimental studies to formulate the ideas of the mathematical model and to check the validity of the model. Figure 1 shows the protocols of the three experimental studies, denoted by k = 1, 2, 3, respectively, hereafter. They differ from each other only in warm-up conditions. In the study k = 1 (moderate warm-up), there was a constant intensity (60% of individual VO2peak) of warming up for 12 min. VO2peak was previously determined by an incremental step test on a cycle ergometer (100 W, +40 W every 5 min). In study k = 2 (intensive warm-up), three 10-s periods of intensive warm-up were arranged during the 12 min at an intensity of 200% of individual VO2peak, as shown in Fig. 1. In the study k = 3, there was no warm-up at all. In all three studies, k = 1, 2, 3, the 30-s short-term maximal effort (“all-out”) exercise, as shown in Fig. 1, was carried out in an isokinetic mode on a cycle ergometer (Schoberer Rad Messtechnik SRM GmbH, Jülich, Germany) set to a cadence of 120 rpm. The thick arrows in Fig. 1 show the blood sampling times.

Fig. 1
figure 1

Protocols of the three experimental studies, denoted by k = 1, 2, 3, respectively. They differ from each other by different warm-up conditions. The thick arrows indicate the blood sampling times; −16, −4 and 0 mean “at rest,” “the end of warming up” and “1 min before the maximal exercise,” respectively. During the 12 min between −16 and −4, k = 1 means “moderate warm-up” of constant intensity at 60% of the individual VO2peak, k = 2 means “intensive warm-up” including three 10-s periods at 200% of the individual VO2peak and k = 3 means without warm-up

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For lactate determination in plasma and erythrocytes, 115 μl of blood was withdrawn from the earlobe and directly centrifuged for 1 min at 6,000 rpm with EBA20 (Hettich Zentrifugen, Tuttlingen, Germany) in order to interrupt the slow entrance of lactate into the erythrocytes and to obtain true in vivo [LA]plasma and [LA]RBC. It took <30 s to take the blood sample and to start centrifugation (since it needs 25 min at 25°C, which is ~5°C higher than the temperature in the laboratory during sampling) until equilibration [12]). Afterwards 20 μl of plasma or erythrocytes was analyzed respectively after discarding the buffy coat and the very lowest part of the sediment (however, there is always trapped plasma, as well as other cells in the cell pellet). All 20-μl samples were directly mixed with 1 ml of the EBIO plus system solution, which hemolyzes RBCs and stops further metabolism. All samples were measured with EBIO plus (EKF Diagnostic Sales, Magdeburg, Germany) (cf. [11]).

Figure 2a gives the observed [LA]RBC (solid lines with solid symbols) and [LA]plasma (dashed lines with hollow symbols), given in mean ± SD (standard deviation), under the three different warm-up conditions (k = 1, 2, 3), while Fig. 2b gives the observed lactate concentration ratio [LA]RBC·[LA] −1plasma as functions of time, also given in mean ± SD, for k = 1, 2, 3. In Fig. 2 and all the figures afterwards, the time point “1 min before the maximal exercise” is denoted by t = 0.

Fig. 2
figure 2

a Observed lactate concentrations in red blood cells (solid lines with solid symbols) and in plasma (dashed lines with hollow symbols), given in mean ± standard deviation, under the three different warm-up conditions (k = 1, 2, 3), where the unit of lactate concentration is millimole per liter, denoted by mmol l−1. b Observed lactate concentration ratio between red blood cells and plasma as functions of time, also given in mean ± standard deviation, for k = 1, 2, 3. k = 1, Moderate warm-up; k = 2, intensive warm-up; k = 3, without warm-up

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In order to avoid too many curves, we did not show the curves of [LA]RBC and [LA]plasma for individual subjects in Fig. 2. Actually, there would be 33 curves each for [LA]RBC and [LA]plasma (11 subjects, 3 warm-up conditions). Each curve in Fig. 2a, b is the mean ± SD over 11 curves for individual subjects.

For the convenience of the mathematical model, we will denote the 16 time points for blood sampling by t(−2), t(−1), t(0), t(1),…, t(13), where t(0) = is “1 min before the maximal exercise”. Thus, we have t(−2) = −16 min, t(−1) = −4 min, t(0) = 0, t(1) = 1.5 min, t(2) = 2.5 min,…, t(11) = 11.5 min, t(12) = 13.5 min, t(13) = 16.5 min.

Mathematical model

The mathematical model will describe the dynamic processes of lactate transport into and out of RBCs. Influx and efflux rate coefficients, representing lactate transfer from plasma to and from RBCs, will be calculated. The mathematical model we are going to present is based on the analysis of the results of experimental studies. Some important features of the observations shown in Fig. 2a, b can be summarized as follows: (1) Even for the resting state, [LA]RBC was lower than [LA]plasma. The ratio [LA]RBC·[LA] −1plasma was about 0.5 for the resting state (more precisely 0.51 as the mean ratio over all 11 subjects and all warm-up conditions for the resting state). (2) The supra-maximal exercise caused a more rapid increase for [LA]plasma than for [LA]RBC during and shortly after the exercise. (3) [LA]RBC reached its peak later than [LA]plasma reached its peak. (4) When [LA]RBC reached the peak, the ratio [LA]RBC·[LA] −1plasma roughly recovered to its beginning value (0.54 as the mean ratio over all subjects and all warm-up conditions at the peak of [LA]RBC). (5) After [LA]RBC reached the peak, it decreased, representing an outflow of lactate from RBC, in spite of the fact that [LA]plasma was still much higher than [LA]RBC. These experimental findings further confirmed the common features found by previous studies that we described in the introduction. The full raw data for all individual subjects of our own experimental studies are necessary for the development of the mathematical model and for the check of the validity of the model.

How should we explain the above-summarized features of observations found by our own and by previous studies? First, an inward force associated with the plasma-to-RBC [LA] gradient must exist. The rapid increase of [LA]plasma during and shortly after the exercise enlarged the plasma-to-RBC [LA] gradient and therefore induced the inward transport of lactate to RBC, resulting in the increase of [LA]RBC with a time delay. Second, the inward force alone cannot account for the ratio [LA]RBC·[LA] −1plasma at rest (about 0.5), and the outflow of lactate for the late stage of recovery after [LA]RBC reached the peak. Thus, in addition to the inward force associated with the plasma-to-RBC [LA] gradient, there must be an outward force against the inward force. Third, this outward force must not be a constant for the following reason. Let us compare the late recovery stage, say 11 min < t < 13 min in Fig. 2a, with the period during and shortly after the supra-maximal exercise, say 0 < t < 1.5 min. The difference [LA]plasma − [LA]RBC for the period 11 min < t < 13 min was even larger than for the period 0 < t < 1.5 min. However, in contrast to the influx of lactate for the period 0 < t < 1.5 min, there was an efflux of lactate for the period 11 min < t < 13 min. The only explanation would be that the outward force was not a constant; instead it increased as [LA]RBC increased. The simplest approximation would be to assume that the outward force is proportional to [LA]RBC. Thus, our mathematical model can be formulated as follows:

“The transport of lactate to RBCs during and after exercise is caused by an influx associated with the plasma-to-RBC [LA] gradient and an efflux that is assumed to have a rate proportional to the lactate concentration in RBC.”

If we simply denote [LA]RBC and [LA]plasma by x and y, respectively, this mathematical model can be expressed by the following differential equation:

$$ {\frac{{{\text{d}}x}}{{{\text{d}}t}}} = C_{1} \left( {y - x} \right) - C_{2} x $$
(1)

where C 1 and C 2 are called the influx rate coefficient and efflux rate coefficient [min−1], respectively. For the convenience of notations, we will use three indices, i, j and k, to distinguish different time points for sampling (i = −2 to 13), different subjects (j = 1–11) and different warm-up conditions (k = 1–3), respectively, where, e.g., t(−2) = −16 min is for the resting state, t(0) = 0 for pre and t(13) = 16.5 min for the last sampling. For given (j, k), C 1 and C 2 are constants, determined by the best fit to the observed [LA]RBC as a function of time for a given individual subject and a specific warm-up condition as follows.

Determinations of C1 and C2 as well as the model-predicted [LA]RBC were made as follows.

Now, let us denote the observed and the model-predicted [LA]RBC at t(i) by x(i) and \(\tilde{x}(i)\), respectively. The time range from t(0) to the moment for the last blood sampling t(13) contains the most measurements with a blood sampling frequency nearly one sample per minute, much higher than the sampling frequency in the time period from the resting state t(−2) to t(0). For this reason, we will call the time range t(0) to t(13) the major range and seek the best fit to the observed ones only for the major range. [An alternative fit for the entire range from t(−2) to t(13) and its disadvantages will be discussed later in “Discussions.”] Thus, the best fit means the minimization of the error

$$ \Updelta = \sum\limits_{i = 0}^{13} {\left[ {\tilde{x}\left( i \right) - x\left( i \right)} \right]}^{2} $$
(2)

where

$$ \tilde{x}(0) = x(0) $$
(3)
$$ \tilde{x}(i) = {x}(0) + C_{1} \int\limits_{t(0)}^{t(i)} {[y(t) - x(t)} ]{\text{d}}t - C_{2} \int\limits_{t(0)}^{t(i)} {x(t){\text{d}}t} \quad \left( {i = \, 1, \ldots,13} \right) $$
(4)

Equation 4 is simply the integration of Eq. 1. We denote the two integrals in Eq. 4 by I 1(i) and I 2(i), respectively:

$$ I_{1} \left( i \right) = \int\limits_{t\left( 0 \right)}^{t\left( i \right)} {\left[ {y\left( t \right) - x\left( t \right)} \right]} {\text{d}}t\quad I_{2} \left( i \right) = \int\limits_{t\left( 0 \right)}^{t\left( i \right)} {x\left( t \right){\text{d}}t} $$
(5)

Now Eq. 4 becomes

$$ \tilde{x}\left( i \right) = x\left( 0 \right) + C_{1} I_{1} \left( i \right) - C_{2} I_{2} \left( i \right)\quad \left( {i \, = \, 1, \ldots,13} \right) \, $$
(6)

The error is then given by

$$ \Updelta = \sum\limits_{i = 1}^{13} {\left[ {\tilde{x}\left( i \right) - x\left( i \right)} \right]^{2} = \sum\limits_{i = 1}^{13} {\left[ {x\left( 0 \right) + I_{1} \left( i \right)C_{1} - I_{2} \left( i \right)C_{2} - x\left( i \right)} \right]}^{2} } $$
(7)

The minimization of Δ requires that

$$ {\frac{\partial \Updelta }{{\partial C_{1} }}} = 2\sum\limits_{i = 1}^{13} {I_{1} (i)[x(0) + I_{1} } (i)C_{1} - I_{2} (i)C_{2} - x(i)] = 0 $$
(8)
$$ {\frac{\partial \Updelta }{{\partial C_{2} }}} = - 2\sum\limits_{i = 1}^{13} {I_{2} (i)[x(0) + I_{1} (i)C_{1} - I_{2} (i)C_{2} - x(i)] = 0} $$
(9)

Thus, we get two linear equations for C 1 and C 2 as follows:

$$ \left[ {\begin{array}{*{20}c} {a_{11} } & {a_{12} } \\ {a_{21} } & {a_{22} } \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {C_{1} } \\ {C_{2} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {b_{1} } \\ {b_{2} } \\ \end{array} } \right] $$
(10)

where

$$ \left[ {\begin{array}{*{20}c} {a_{11} } & {a_{12} } \\ {a_{21} } & {a_{22} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {\sum\limits_{i = 1}^{13} {I_{1}^{2} (i)} } & { - \sum\limits_{i = 1}^{13} {I_{1} (i)I_{2} (i)} } \\ { - \sum\limits_{i = 1}^{13} {I_{1} (i)I_{2} (i)} } & {\sum\limits_{i = 1}^{13} {I_{2}^{2} (i)} } \\ \end{array} } \right] $$
(11)

and

$$ \left[ {\begin{array}{*{20}c} {b_{1} } \\ {b_{2} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {\sum\limits_{i = 1}^{13} {I_{1} (i)[x(i) - x(0)]} } \\ { - \sum\limits_{i = 1}^{13} {I_{2} (i)[x(i) - x(0)]} } \\ \end{array} } \right] $$
(12)

I1(i) and I2(i) will be calculated by the linear interpolations of x(t) and y(t) in any interval [t(i), t(i + 1)]. Thus, we have

$$ I_{1} \left( i \right) = \sum\limits_{j = 1}^{i} {\frac{1}{2}} \left[ {d\left( {j - 1} \right) + d\left( j \right)} \right]\left[ {t\left( j \right) - t\left( {j - 1} \right)} \right]\quad \left( {i = \, 1, \ldots,13} \right) $$
(13)

where d = y − x and

$$ I_{2} (i) = \sum\limits_{j = 1}^{i} {\frac{1}{2}} [x(j - 1) + x(j)][t(j) - t(j - 1)]\quad \left( {i = \, 1, \ldots,13} \right) $$
(14)

Once I 1(i) and I 2(i) (i = 1,…,13) have been calculated by Eqs. 13 and 14, respectively, the coefficients a 11, a 12, a 21, a 22, b 1 and b 2 are obtained by the substitution of I 1(i) and I 2(i) into Eqs. 11 and 12. Thus, C 1 and C 2 as solutions of Eq. 10 are given by

$$ C_{1} = D_{1} /D\quad C_{2} = D_{2} /D $$
(15)

where D 1, D 2, D 3 are the following determinants:

$$ D = \left| {\begin{array}{*{20}c} {a_{11} } & {a_{12} } \\ {a_{21} } & {a_{22} } \\ \end{array} } \right| {D_1}= \left| {\begin{array}{*{20}c} {b_{1} } & {a_{12} } \\ {b_{2} }&{a_{22} }\\ \end{array} } \right| {D_2}= \left| {\begin{array}{*{20}c} {a_{11} } & {b_{1} } \\ {a_{21} }& {b_{2} }\\ \end{array} } \right| $$
(16)

Once C 1 and C 2 have been determined by Eq. 15, the model-predicted [LA]RBC can be determined by Eq. 6. The justification of the model will be given by the comparison of various features of the model-predicted [LA]RBC with those from observations as presented in the next section.

Results of the model and the comparison with observations

For all 33 cases (11 subjects j = 1–11 and 3 warm-up conditions k = 1, 2, 3), the model-predicted [LA]RBC, as calculated by the procedure described in “Mathematical model,” turns out to be in reasonable agreement with the observed [LA]RBC. In order to avoid too many figures, in Fig. 3, we only show the comparison between the model-predicted and the observed individual [LA]RBC for subject 7 as one example. Figure 4 shows the comparison of the mean ± SD of [LA]RBC over all subjects between model predictions and observations for (a) k = 1, (b) k = 2 and (c) k = 3 respectively. Again, we see reasonable agreement between model predictions and observations.

Fig. 3
figure 3

Observed (solid lines with solid symbols) and model-predicted (dashed lines with hollow symbols) lactate concentrations in red blood cells under conditions k = 1, 2, 3 for subject 7. k = 1, Moderate warm-up; k = 2, intensive warm-up; k = 3, without warm-up

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Fig. 4
figure 4

Observed (solid lines with solid symbols) and model-predicted (dashed lines with hollow symbols) lactate concentrations in red blood cells, given in mean ± standard deviation, for (a) k = 1, (b) k = 2 and (c) k = 3. k = 1, Moderate warm-up; k = 2, intensive warm-up; k = 3, without warm-up

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As a statistical comparison, we show the correlation between the observed and the model-predicted values of [LA]RBC for all the subjects, under all three warm-up conditions at all time points in the major range in Fig. 5. It shows very good and very significant correlation (r = 0.9773, p = 0.0000), and the best linear fit (the thin solid line) is very close to the thick solid line, which means the equality of the model predictions and the observations.

Fig. 5
figure 5

Correlation between the observed and the model-predicted [LA]RBC for all 11 subjects, under all three warm-up conditions, and at all 14 time points for blood sampling [t(0) to t(13)]. Thus, the graph includes 11 × 3 × 14 = 462 circles where each circle represents the observed and the model-predicted values of [LA]RBC for a given subject under a given warm-up condition at a given moment for blood sampling. The thin solid line shows the best linear fit, which is close to the thick solid line, which means the equality of the observations and the model predictions. The very good and very significant correlation is shown by r = 0.9773 and p = 0.0000

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The influx rate coefficients C 1 and the efflux rate coefficients C 2 for all subjects (j = 1–11) and all warm-up conditions (k = 1, 2, 3) are given in Table 1.

Table 1 C1 (influx rate coefficient) and C2 (efflux rate coefficient) for different subjects (j = 1–11) and different warm-up conditions (k = 1–3)
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Statistical analysis (repeated measure ANOVA) is made to compare C 1 and C 2 under different warm-up conditions, k = 1, 2, 3. The results are shown in Fig. 6. C 2 is significantly smaller than C 1 (p = 0.000000), while the differences made by different k are not significant (p = 0.19).

Fig. 6
figure 6

Repeated measure ANOVA for C 1 (influx rate coefficient) and C 2 (efflux rate coefficient) under the three warm-up conditions, k = 1, 2 and 3, where the symbols stand for the means over the 11 subjects and the vertical bars give the 95% confidence ranges. C 2 is significantly smaller than C 1 (p = 0.000000), while the differences made by different warm-up conditions are not significant (p = 0.19). k = 1, Moderate warm-up; k = 2, intensive warm-up; k = 3, without warm-up

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As a further check of the validity of the model, let us consider the ratio [LA]RBC·[LA] −1plasma . The observed approximate equality between the ratios at the peak of [LA]RBC and at rest can be easily explained by the present model. Actually, for both resting state and the peak of [LA]RBC, the time derivative of [LA]RBC, i.e., dx/dt on the left of Eq. 1, should be zero. Therefore, the right side of Eq. 1 must also be zero at these two moments. Thus, for both resting state and the peak of [LA]RBC, the model-predicted ratio should be the same, given by

$$ \frac{x}{y} = {\frac{1}{{1 + C_{2} /C_{1} }}}. $$
(17)

This is directly derived by letting the right side of Eq. 1 be zero. In Fig. 7, we give the comparison between the model-predicted ratios [LA]RBC·[LA] −1plasma at the peak of [LA]RBC, based on Eq. 17, and those from observations for all subjects j = 1–11 and for (a) k = 1, (b) k = 2 and (c) k = 3, respectively. For all 33 cases (11 subjects and 3 warm-up conditions), the comparison shows excellent agreement. This gives further confirmation of the validity of the model.

Fig. 7
figure 7

Comparison between the observed and the model-predicted ratio [LA]RBC·[LA] −1plasma at the peak of [LA]RBC for different subjects (j = 1–11) under different warm-up conditions, (a) k = 1, (b) k = 2 and (c) k = 3. k = 1, Moderate warm-up; k = 2, intensive warm-up; k = 3, without warm-up

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Discussion

Now we discuss the advantages and the limitations of the present model, the physical and chemical processes involved in the influx and efflux, the physiological meanings of the influx rate coefficient C 1 and the efflux rate coefficient C 2, as well as the potential applications of the model to medicine and sports.

The advantages and the limitations of the model

Simple as it is, the present model can quantitatively reproduce the basic features of the time courses of the observed lactate concentrations in RBC. The close agreement between the model predictions and observations confirms the main influencing factors for the transport of lactate to RBCs suggested by the model. Namely, in addition to the influx associated with the plasma-to-RBC [LA] gradient, there is an efflux that has a rate approximately proportional to the lactate concentration in RBCs. Although the existence of some kind of efflux has been mentioned in previous research, the present study is the first mathematical model that has been completed for the quantitative description of both the influx and the efflux to calculate the detailed results of the model, to compare the model-predicted results with the corresponding observations in detail and therefore to confirm the validity of the model. The validity of the model does not depend on the specific mechanism for the efflux. In other words, the present model is valid no matter what the specific mechanism for the efflux turns out to be. However, the only way to really validate the model/efflux rate coefficient would be to label a pool of intracellular lactate and directly measure efflux rates at different extracellular lactate concentrations.

Nevertheless, the present model is still a semi-empirical model because the mechanism of the efflux is still not entirely clear (cf. the next subsection) and also because the influx rate coefficient C 1 and the efflux rate coefficient C 2 are determined by the best fit to observations.

It is important to note that C 1 and C 2 are constants for a given subject during a given protocol, but C 1 and C 2 vary for a given subject between different protocols. They vary with different conditions/protocols in order to have the best fit. The present model can be applied well to all the three warm-up conditions, k = 1, 2 and 3. We could even think of it in another way. Namely, the data of k = 1 are used for the development of the model, while the data of k = 2 and 3 are used to further check the validity of the model. However, all three conditions used 30-s short-term maximal effort exercises. Therefore, future research needs to determine how well the present model would apply to conditions with longer term exercises. In this case, such an acute setting has been chosen because only minor changes or even no changes in the ratio [LA]RBC·[LA] −1plasma occur during steady-state exercise [6] or incremental step tests (even when the work load was increased by 30 W every minute) [5, 8]. For the development of the model, major changes in the ratio are much more interesting than steady-state conditions. Our purpose was to fit the model with big changes in the lactate distribution.

On the physical and the chemical processes involved in the influx and the efflux

The terms influx and efflux used in the model are certainly just simplifications of the real physical and chemical processes involved in the lactate transport to RBCs. Actually, only a small part (roughly 4–5%) of the inward transport of lactate is caused by physical diffusion across the membrane of RBCs. The major part (80–90%) of the transport is through the transporter MCT-1 [14–16]. In this context it has to be considered that the lactate transport by MCTs is a facilitated but passive transport mainly driven by the concentration gradient. Therefore, the velocity of reaching equilibrium depends on the initial difference and on the permeability. Since MCT-1 always carries the lactate anion LA− together with the hydrogen cation H+ [10, 17], not only the gradient of [LA−], but also the gradient of [H+] would influence the direction of transport of the couple {LA−, H+}, where the bracket [] stands for the concentration. At rest, the effects of the plasma-to-RBC [LA−] gradient and RBC-to-plasma [H+] gradient may balance each other. During and shortly after exercise, the rapid increase of [LA−]plasma is associated with a greater relative increase of [H+] in plasma than in RBCs, resulting in the increase of the plasma-to-RBC [LA−] gradient and the decrease of the RBC-to-plasma [H+] gradient [3], both favoring the influx of the couple {LA−, H+} through MCT-1. The enlarged plasma-to-RBC [LA−] gradient also favors the physical diffusion of lactate through the membrane. Since the increase of the plasma-to-RBC [LA−] gradient favors both the inward physical diffusion of lactate through the membrane and the inward transport of {LA−, H+} through MCT-1, the influx of LA− associated with the plasma-to-RBC [LA−] gradient in the mathematical model should be regarded as the combination of both.

The understanding of the efflux of LA− may involve the negative electric charge inside RBCs (~−10 mV at the inside of the red cell membrane forcing diffusible anions out), which is caused by various anions, particularly the non-diffusible anions of hemoglobin Hb−. This negative charge forces diffusible anions out and resists the inward diffusion of LA− even for the resting state (Donnan equilibrium). However, the couple (LA−, H+), transported through MCT-1, as a whole is electrically neutral. Therefore, there is no net electric force acting on the couple. Furthermore, it was shown that an equilibrium for other permeating anions (An−) and for H+ is reached within a second after a sudden rise in plasma concentration like at exhaustion, whereas such an equilibrium for LA– can only be attained after a few minutes due to the slow transport [11].

Another possibility for the forces acting on the distribution could be a preferred direction of MCT-1 transportation. Indeed, the different K m values (Michaelis-Menten constant) for efflux and influx in RBCs may suggest asymmetric behaviors of MCT-1 [18]. This preference should also be taken into account in the balance between the gradients of [LA−] and [H+] for the resting state.

The above ideas for the possible mechanism of the efflux of LA− and for the reason why the efflux gets stronger as [LA]RBC increases are still speculative. We do not claim that we have fully understood the physical and chemical mechanism for the efflux of LA−. We leave this interesting issue open to further discussions. The detailed physical and chemical mechanism of the efflux of LA− remains to be figured out in future research.

On the potential applications of the model

The present model gives a simple and convenient way to calculate C 1 and C 2, and gives a detailed explanation and quantitative description for the entire dynamic processes of lactate transport to RBCs during and after exercises. According to Eq. 1, C 1 is the increase of [LA]RBC per unit time associated with a unit difference [LA]plasma − [LA]RBC, while C 2 is the decrease of [LA]RBC per unit time associated with a unit [LA]RBC. The values of C 1 and C 2 could be determined by the best fit to observations as we did in “Methods” and “Results of the model and the comparison with observations.” However, it is not a simple matter to determine all the properties of the blood that would influence the transport of lactate to red blood cells and the values of C 1 and C 2. In any case, both C 1 and C 2 reflect the properties of the blood.

The basic idea of the model was to help to understand the dynamic processes of lactate transport (kinetics) to RBC and to give a quantitative description of these processes. Of course one limitation of the model is the fact that the model is a semi-empirical model because the influx rate coefficient C 1 and the efflux rate coefficient C 2 are determined by the best fit to observations. Constants (C 1 and C 2) serve for fitting and vary between different interventions/subjects. Nevertheless, C 1 and C 2 are constants independent of sampling time points for a given subject and a given intervention. Therefore, the model might be utilizable for further studies investigating lactate kinetics in different body/tissue compartments during exercise. It might help to further understand the dynamic processes of lactate distribution.

But further investigations are needed to clarify if the model is utilizable in these areas. Of course, for the potential applications, it would be necessary to standardize the protocol of the experimental study, e.g., the warm-up condition, the duration, the intensity and the procedure of the exercise as well as the time points of blood sampling, etc. It would also be necessary to standardize the range for fitting in the mathematical model.

Conclusions

A mathematical model, as a quantitative description for the dynamic process of lactate transport to red blood cells during and after exercise, is constructed. In addition to the influx of LA associated with the plasma-to-RBC [LA] gradient, an efflux of LA with a rate proportional to the lactate concentration in red blood cells is assumed. This simple model is justified by the good agreements between the model-predicted and the observed time courses of the individual [LA]RBC, as well as between the model-predicted and the observed ratio [LA]RBC·[LA] −1plasma at the peak of [LA]RBC for all 33 cases (11 subjects and 3 warm-up conditions). The two constants involved in this model, i.e., the influx rate coefficient C 1 and the efflux rate coefficient C 2, reflect the corresponding blood properties favoring the influx and the efflux of lactate crossing the membrane of RBCs, respectively. These properties cannot be easily observed directly but can be calculated by the present model. Further investigations are needed to clarify if the model is utilizable for studies investigating lactate kinetics in different tissue compartments during exercise and if it helps to understand the dynamic processes of lactate distribution.