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Am J Physiol Heart Circ Physiol 293: H343-H353, 2007. First published March 9, 2007; doi:10.1152/ajpheart.01371.2006
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Oxygen delivery by blood determines the maximal VO2 and work rate during whole body exercise in humans: in silico studies

Piotr Liguzinski and Bernard Korzeniewski

Faculty of Biochemistry, Biophysics, and Biotechnology, Jagiellonian University, Kraków, Poland

Submitted 15 December 2006 ; accepted in final form 7 March 2007


   ABSTRACT
TOP
 ABSTRACT
THEORETICAL PROCEDURES
 THEORETICAL RESULTS
 DISCUSSION
 APPENDIX
 REFERENCES
 
It has been proposed by Saltin (J Exp Biol 115: 345–354, 1985) that oxygen delivery by blood is limiting for maximal work and oxygen consumption in humans during whole body exercise but not during single-muscle exercise. To test this prediction quantitatively, we developed a static (steady-state) computer model of oxygen transport to and within human skeletal muscle during single-muscle (quadriceps) exercise and whole body (cycling) exercise. The main system fluxes, namely cardiac output and oxygen consumption by muscle, are described as a function of the "primary" parameter: work rate. The model is broadly validated by comparison of computer simulations with various experimental data. In silico studies show that, when all other parameters and system properties are kept constant, an increase in the working muscle mass from 2.5 kg (single quadriceps) to 15 kg (two legs) causes, at some critical work intensity, a drop in oxygen concentration in muscle cells to (very near) zero, and therefore oxygen supply by blood limits maximal oxygen consumption and oxidative ATP production. Therefore, the maximal oxygen consumption per muscle mass is significantly higher during single-muscle exercise than during whole body exercise. The effect is brought about by a distribution of a limited amount of oxygen transported by blood in a greater working muscle mass during whole body exercise.

oxygen transport; oxygen consumption; muscle exercise

MECHANICAL WORK IN SKELETAL muscle is driven by ATP hydrolysis. Oxidative phosphorylation in mitochondria is the main process responsible for ATP production in oxidative human skeletal muscles under most steady-state or quasi-steady-state conditions. Oxygen is a substrate for oxidative phosphorylation (in particular for cytochrome c oxidase), and under hypoxic or anoxic conditions its concentration may significantly affect the ATP-producing system. Therefore, quantitative description of the system of oxygen transport from lungs to muscle mitochondria is very important for our understanding of muscle functioning during work and of development of fatigue.

Plenty of experimental data have been collected concerning the oxygen transport to and within human skeletal muscle during single-muscle exercise and whole body exercise (e.g., cycling). They comprise oxygen concentration in arteries, veins, capillaries, and myocytes at rest and at different exercise intensities; dependence of the blood flow through working muscle; oxygen extraction from blood; difference between arterial and venous oxygen concentration and cardiac output on work intensity and oxygen consumption; relationship between maximal pulmonary oxygen consumption and maximal cardiac output; dependence of the diffusion constant for oxygen transport from capillaries to mitochondria on oxygen consumption; and so forth (2–4, 9, 10, 12, 13, 15, 16, 25, 26, 30, 32, 33, 38, 40, 44, 45, 51, 52). These extensive studies allowed a much better understanding of the multistage process of oxygen transport from lungs to cytochrome c oxidase in mitochondria.

It has been proposed that the oxygen delivery by blood is not limiting for the rate of oxygen consumption by muscle (VO2m) during single-muscle exercise (2, 33, 34, 38, 39). On the other hand, it was proposed that during whole body exercise, the oxygen transport is limiting for the maximal oxygen uptake in working muscles (VO2m,max) and in lungs (VO2p,max) (34, 38, 39), and therefore the maximal oxygen consumption per muscle mass is significantly greater during single-muscle exercise (350–600 ml·kg–1·min–1 or 15–24 mM/min) (2, 33, 29, 35) than during whole body exercise (110–320 ml·kg–1·min–1 or 5–14 mM/min) (3, 10, 15, 37, 44, 47). The oxygen delivery by blood is a derivative of several factors, including cardiac output (dependent on heart activity and capillary resistance), hemoglobin (Hb) content in blood, and saturation of arterial blood with oxygen (dependent on pulmonary ventilation, at least under hypoxic conditions).

The main aim of the present in silico study was to test in a quantitative way whether an increase in the working muscle mass from 2.5 kg (single quadriceps) to 15 kg (two legs), with other parameters and system properties (including the dependence of cardiac output on work rate, Hb content in blood, and arterial blood saturation with oxygen) kept constant, will cause oxygen supply by blood to become limiting for maximal oxygen consumption and work rate. To achieve this goal, a static (steady-state) computer model of oxygen transport to and within working human skeletal muscle was developed. It quantitatively predicts that oxygen-delivery limitations cause the maximal oxygen consumption per muscle mass during single-muscle (quadriceps) exercise to be significantly greater than during whole body (cycling) exercise. This happens because during whole body exercise, the limited amount of oxygen transported by blood must be distributed in a greater working muscle mass than during single-muscle exercise. The predicted maximal pulmonary oxygen consumption and maximal work intensity are well within the range encountered in experimental studies.

Generally, the purpose of the present article is not to compare the impact of different factors (cardiac output, pulmonary ventilation, Hb content, capacity of oxidative phosphorylation, etc.) on the maximal pulmonary oxygen consumption and maximal work intensity but to estimate quantitatively the effect of an increased working muscle mass on the system when all parameters and system properties (including the phenomenological dependence of cardiac output on work rate) are kept constant.


   THEORETICAL PROCEDURES
TOP
 ABSTRACT
 THEORETICAL PROCEDURES
THEORETICAL RESULTS
 DISCUSSION
 APPENDIX
 REFERENCES
 
Model. The computer steady-state model of oxygen delivery to, transport within, and consumption by exercising human skeletal muscle(s) developed in the present study explicitly includes cardiac output and muscle blood flow, pulmonary and muscle oxygen consumption, oxygen binding to Hb and myoglobin (Mb), oxygen diffusion within muscle, and blood pH. The two independent variables in the model are venous oxygen concentration and oxygen concentration in myocyte cytosol. The "standard" version of the model refers to relatively well-trained (endurance training) male subjects. The parameters and variables used in the model are shown in Table 1 and Table 2, respectively (see APPENDIX).


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  		Table 1. Parameter values used in the model

 

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  		Table 2. Variables used in the model

 
Within the model, oxygen-consuming tissues are divided into two groups: working muscle(s) and other tissues. It is assumed that the weight of all respiring tissues is 50 kg and that the weight of working muscle(s) is either 2.5 kg during single-muscle exercise (estimated values: 2.3–2.7 kg; Refs. 2, 29, 32, 33, and 38) or 15 kg during whole body exercise (estimated values: 13–19.6 kg; Refs. 23, 44, 46, and 48) at a total body mass of ~75 kg.

The arterial oxygen concentration is assumed to be constant (2, 33) (at least under normoxic conditions; the present version of the model applies for neither hypoxic nor hyperoxic conditions), and the venous oxygen concentration in working muscle, but not in other tissues, is considered. Therefore, it is assumed that lungs have a high enough capacity to maintain a constant arterial oxygen concentration regardless of the value of cardiac output and venous oxygen concentration (8). However, one should be aware that this assumption may be invalid in hypoxia or hyperoxia.

The most basic property of the present model is that the two key fluxes, cardiac output (Q) and muscle oxygen consumption per muscle mass (VO2m,s), are expressed as functions of the mechanical total work rate (either general or per muscle mass). The phenomenological dependencies of these variables on work rate were extracted from different experimental data (compare Fig. 1). With some additional assumptions, these dependencies allowed us to generate all other relevant properties of the system.


Figure 1
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  		Fig. 1. Simulated (lines) and experimental (points) dependence of muscle blood flow (Qm) (A), cardiac output (Q) (B), muscle oxygen consumption (VO2m) (C), and pulmonary oxygen consumption (VO2p) (D) on total work rate (W). Assumed single-muscle (quadriceps) mass is 2.5 kg. Experimental data come from the following sources: gray circles, Refs. 2 and 38; closed circles, Ref. 33; open circles (2-leg cycling exercise in sitting and supine positions), Ref. 45; gray diamonds, Ref. 16; and open diamonds, Ref. 4.

 
Work intensity. In the present study, both general (whole body) work rate (W, in Watts) and work rate per muscle mass (Ws, in Watts per kilogram) were considered. The total work rate is used. During the single-muscle exercise protocol, the total work rate was a sum of the loaded work (measured by an ergometer) and unloaded work (unloaded kicking) (2, 33, 38). The unloaded work was estimated from the extrapolation of the linear part of the dependence of muscle blood flow (Qm) and muscle oxygen consumption per muscle mass on work rate (W or Ws) to the resting muscle blood flow and resting muscle oxygen consumption, respectively. The estimated value is 6.4 W/kg (16 W for 2.5 kg of quadriceps) during single-muscle exercise. On the same basis, it was roughly estimated that unloaded work during cycling is relatively very small (45). In all figures presenting simulations and experimental points concerning single-muscle exercise, the total (loaded + unloaded) work rates are shown. Within the model, the maximal total work rate during single-muscle exercise equals 46 W/kg of muscle (115 W = 99 W of loaded work + 16 W of unloaded work per 2.5 kg of quadriceps) (33).

It is assumed within the model that the entire working muscle group is activated to the same extent at a given work intensity. Therefore, the model does not explicitly take into account the differentiated activation of particular muscle fibers or muscles. Of course, this is only a rough approximation of the reality; however, it seems sufficient for the needs of the present, semiquantitative model.

Cardiac output and muscle blood flow. An empirical phenomenological dependence of cardiac output (liters per minute) on work rate (Watts) was extracted from experimental data (Fig. 1B)

Formula 1(1)
where Qo is blood flow through other tissues (including nonworking muscles) equal to 4.72 and 3.32 l/min for single-muscle and whole body exercise, respectively, whereas Qm,rest,s is the resting blood flow through the working muscle group per muscle mass (l·min–1·kg–1). AQ (130 l/min) and BQ (1,615.5 W) are fitted phenomenological constants. Of course, the muscle blood flow through working muscle mass (liters per minute) equals (compare Fig. 1A)

Formula 2(2)
The total resting cardiac output is assumed to be 5.0 l/min (3, 9, 37, 45). It must be emphasized that the above phenomenological dependence of cardiac output on work rate implicitly involves such factors as capillary resistance.

Oxygen consumption. It is assumed that oxygen consumption per muscle mass in working muscle equals (see Fig. 1C)

Formula 3(3)
where VO2m,s,rest is the resting muscle oxygen consumption, VO2m,s,lin is the linear component of the oxygen consumption in working muscle equal to

Formula 4(4)
(where kVO2lin is an appropriate proportionality coefficient), and VO2m,s,over is the nonlinear component of the oxygen consumption in working muscle equal to

Formula 5(5)
where kVO2over is an appropriate proportionality coefficient, whereas Aover is the work rate per muscle mass above which the nonlinear component of oxygen consumption appears. The nonlinear component is added only when Ws > Aover.

The whole body (pulmonary) oxygen consumption (VO2p) is expressed as follows (see Fig. 1D)

Formula 6(6)
where VO2a,s = 8% of (VO2m,s – VO2m,s,rest) is work-dependent oxygen consumption by "auxiliary" tissues that are activated during exercise (heart, respiratory muscles, posture-stabilizing muscles), whereas VO2p,0 is the "basic" (resting) oxygen consumption by other tissues (including auxiliary tissues and nonworking muscles) that equals 0.290 and 0.238 l/min for single-muscle and whole body exercise, respectively. The total (whole body, pulmonary) resting oxygen consumption equals 0.3 l/min (3, 38, 45).

Oxygen diffusion in muscle. The rate of oxygen diffusion from capillaries to mitochondria is described as follows (32)

Formula 7(7)
where kD is the muscle oxygen diffusion constant that equals

Formula 8(8)
where kDr is the proportionality constant between the muscle oxygen consumption and the muscle oxygen diffusion constant. The above description of the oxygen "diffusion system" in muscle is a "black box" description. The real system is composed of several stages (e.g., oxygen unloading from Hb, diffusion from red cells to and through capillary walls, diffusion from capillary walls to mitochondria) and depends on several factors. When the work rate and oxygen consumption increase, some of these factors (e.g., shortening of the blood transition time through capillaries and thus a decrease in oxygen unloading; see Ref. 28) decrease the phenomenological diffusion constant kD, whereas other factors (e.g., opening of capillaries and increase in their diameter, increase in Mb mobility) increase this constant. What really matters within the model is the resultant effect of all these factors. It was found (32) that the phenomenological diffusion constant increases linearly with oxygen consumption (Eq. 8). Therefore, one can suppose that the effect of the factors that increase the phenomenological diffusion constant predominates over the effect of the factors that decrease it.

Oxygen concentration. Oxygen concentration is expressed in either millimoles or milliliters per liter. The relation between these units is as follows

Formula 9(9)

Oxygen binding by Hb and blood pH. The total oxygen concentration in blood (Ct) is described as the sum of oxygen associated with Hb and oxygen dissolved in plasma

Formula 10(10)
where [Hb] is the Hb concentration, HbO2 is the fraction of oxygenated Hb, {alpha} is the oxygen solubility in blood plasma, and PO2 is oxygen partial pressure in plasma.

The previously published (1) standard oxygen-Hb dissociation curve was used for determining the amount of oxygen associated with Hb and dissolved in plasma

Formula 11(11)
where K1 = 51.87074, K2 = 129.8325, K3 = 6.828368, K4 = –223.7881, K5 = 27.95300, K6 = –258.5009, K7 = 21.84175, and K8 = –119.2322. Within the present version of our model, the Hb saturation with oxygen in artery is constant and equals 97.0%.

Plasma pH dependence of dissociation curve has been included as PO2 correction (43) to better approximate oxygen partial pressure in capillaries

Formula 12(12)
where {Delta}pH is the difference between resting blood pH (= 7.4) and actual blood pH.

The arterial pH (pHa) is kept constant (7.4), and the muscle venous pH is described as linearly decreasing with increasing muscle oxygen consumption

Formula 13(13)
where kpH (= 0.2) is the maximal difference between muscle arterial and venous pH. The capillary blood pH has been approximated as the average of the arterial and venous pH.

Oxygen binding by Mb. The total oxygen concentration in myocytes (Cz,t) has been described as the sum of oxygen associated with Mb and oxygen dissolved in cytoplasm

Formula 14(14)
where [Mb] is the total Mb concentration, MbO2 is the fraction of oxygenated Mb, {alpha}T is oxygen solubility in cytoplasm, and PO2z is oxygen partial pressure in muscle cytoplasm.

The oxygen-binding curve for Mb was used for determining the fraction of oxygenated Mb in myocyte

Formula 15(15)
where P50 is Mb half-saturating oxygen partial pressure.

Differential equations. The model contains two differential equations describing changes over time of total venous oxygen concentration (Cv,t) and mean total myocyte oxygen concentration

Formula 16(16)

Formula 17(17)
where VDIFF,s is oxygen diffusion from capillaries to mitochondria per muscle mass. It was assumed that oxygen consumption per muscle mass is identical along the muscle. Therefore, the mean total capillary oxygen concentration (Cc,t) was assumed to be the average of the total muscle arterial (Ca,t) and total muscle venous oxygen concentration

Formula 18(18)
Therefore, Cc,t is a unique function of Cv,t (at constant Ca,t) and therefore the term (Ca,t – Cv,t) in Eq. 16 can be replaced with the term 2(Cc,t – Cv,t) if one prefers to emphasize that blood gets to veins from capillaries and not from arteries. Certainly, the assumption that oxygen concentration is equal in all capillaries (and cells) within a given muscle is only a rough approximation of the reality.

One could ask why differential equations instead of analytical formulas are used in a static (steady-state) model. We have decided in favor of this for three main reasons: 1) the description of our model cannot be purely analytical, because iterative procedures are necessary to calculate the free and Hb-bound oxygen concentration/partial pressure from the total oxygen concentration in blood at a given pH; 2) in our opinion, the mass balance appearing explicitly in differential equations is very useful for the intuitive understanding of the system; and 3) we intend to transform the present static model into a dynamic model that takes into account the changes of system variables over time (see DISCUSSION).

Computer simulations. The set of differential equations was integrated numerically by using the fourth-order Rosenbrock method. In both sets of simulations concerning either single-muscle exercise or whole body exercise, in subsequent simulations the total work intensity per muscle mass was increased by 0.1 W/kg and the simulation lasted until steady-state variable values were reached. It was assumed implicitly that all muscle fibers/muscles in the working muscle group are activated in parallel to the same extent when work rate increases. For single-muscle exercise, the experimentally studied range of work intensity per muscle mass (0–48 W/kg, 0–120 W for 2.5 kg of quadriceps) was analyzed. On the other hand, in the simulations concerning whole body exercise, the work intensity was increased until the oxygen concentration dropped to zero (for the standard version of the model, this critical work intensity was equal to 22.5 W/kg, which corresponded to the general work intensity of 337.5 W performed by 15 kg of working muscles).


   THEORETICAL RESULTS
TOP
 ABSTRACT
 THEORETICAL PROCEDURES
 THEORETICAL RESULTS
DISCUSSION
 APPENDIX
 REFERENCES
 
The comparison of the experimental data concerning the dependence of the oxygen consumption and blood flow on work rate during single-muscle exercise and whole body exercise shows that these dependences are unique under both conditions (exercise types). This can be seen in Fig. 1. Therefore, the "kinetics" of both oxygen consumptions and blood flow can be approximated by the same mathematical formulas (Eqs. 1–6). This conclusion justifies the studying of the effect of the working muscle mass on the system behavior when all other system properties and parameter values are kept constant.

The simulated dependence of the oxygen extraction from blood (in %) on work rate during single-muscle exercise and the simulated dependence of the difference between muscle arterial and venous oxygen concentration on the pulmonary oxygen consumption during whole body exercise agree well with experimental dependencies. The comparison between simulated and experimental dependencies is given in Figs. 2 and 3. The axes have different units in both figures, because such units were used in the original experiments sited. However, the fact that one unique model is able to reproduce well both sets of experimental data testifies that the same set of parameters and system properties applies to both single-muscle exercise and whole body exercise.


Figure 2
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  		Fig. 2. Simulated and experimental dependence of muscle oxygen extraction (%) on W during single-muscle exercise. Assumed muscle mass is 2.5 kg. Line represents simulations. Experimental data are from the following sources: gray circles, Refs. 2 and 38; closed circles, Ref. 33.

 

Figure 3
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  		Fig. 3. Simulated and experimental dependence of arteriovenous oxygen (a-v) difference on VO2p. Line represents simulations. Experimental data are from the following sources: gray circles, Ref. 10; open circles, Ref. 9 (untrained subjects); closed circles, Ref. 9 (trained subjects); gray diamonds, Ref. 37; open diamonds, Ref. 3; closed diamonds, Ref. 15.

 
The model also predicts correctly the relationship between cardiac output and pulmonary oxygen consumption encountered in experimental studies. This is demonstrated in Fig. 4. The good agreement between theoretical predictions and experimental data confirms the correctness of the cardiac output/work rate and pulmonary oxygen consumption/work rate relationships (Eqs. 1–6) presented in Fig. 1, B and D.


Figure 4
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  		Fig. 4. Simulated and experimental relationship between Q and VO2p. Experimental data are from the following sources: gray circles, Ref. 45 (2-leg exercise in sitting and supine positions); open circles, Ref. 9 (untrained subjects); closed circles, Ref. 9 (trained subjects); gray diamonds, Ref. 37; open diamonds, Ref. 3; closed diamonds, Ref. 15.

 
The comparison of the computer simulations made for single-muscle exercise and whole body exercise shows clearly that, whereas for the working muscle mass of 2.5 kg the oxygen delivery by blood does not limit the maximal oxygen consumption and work rate (for the parameter values accepted within the model), for the working muscle mass equal to 15 kg oxygen concentration falls to zero at some critical work intensity and the oxygen consumption (and oxidative ATP supply) cannot increase any further. This key theoretical result is shown in Fig. 5. This figure presents the simulated dependence of the free (plasma or cytosolic) and total (free and Hb-bound or Mb-bound) oxygen concentration/partial pressure in muscle artery, vein, capillaries, and myocytes on work rate during single-muscle exercise (Fig. 5, A–C) and whole body exercise (Fig. 5, D–F). The simulated (lines) total (plasma plus Hb-bound) oxygen concentration at different work intensities in muscle artery and vein during single-muscle exercise agrees well with that encountered in experimental studies (points) (Fig. 5A). No direct measurement of the capillary oxygen concentration in muscle has been performed, and therefore only the simulated dependence of the capillary oxygen concentration on work rate is shown in Fig. 5A.


Figure 5
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  		Fig. 5. Simulated and experimental dependence of arterial, venous, capillary, and myocyte oxygen concentration on W during single-muscle exercise (A–C) and whole body exercise (D–F). A and D: total [plasma and hemoglobin (Hb)-bound] blood oxygen concentration in muscle artery (Ca,t), capillaries (Cc,t; mean), and vein (Cv,t). Lines represent simulations. Experimental data are from the following sources: gray circles, Refs. 2 and 38; closed circles, Ref. 33; open circles, Ref. 32 (work rate estimated from oxygen consumption). B and E. oxygen partial pressure in muscle artery (PO2a), capillaries (PO2c; mean), vein (PO2v), and myocyte cytosol (PO2z; mean). Experimental data are from the following sources: rest (0 W), Ref. 30; work, Ref. 32. C and F: mean myoglobin (Mb)-bound and cytosolic free oxygen concentration in myocytes. Experimental data are from the following sources: rest (0 W), Ref. 30; work, Ref. 32.

 
The simulated dependence of the (mean) oxygen partial pressure in blood plasma in muscle artery, vein, and capillary and in myocyte cytosol during single-muscle exercise shows that vein, capillary, and plasma oxygen pressure decreases significantly with work rate at low work intensities and that this decrease is much smaller at higher work intensities. This can be seen in Fig. 5B. The mean cytosolic oxygen partial pressure first decreases with work rate and then stabilizes at a quasi-plateau (~4 Torr), in accordance with experimental data (31, 32). It is worth emphasizing that the simulated oxygen concentration does not drop (very close) to zero and therefore does not become limiting for oxygen consumption rate in the range of work rate (up to ~50 W/kg, 120 W/2.5 kg of quadriceps) studied in experiments concerning the single-muscle exercise.

Computer simulations predict that the Mb-bound and cytosolic oxygen concentration in myocytes does not fall to zero even at the highest work intensities that are reached during single-muscle exercise. This is demonstrated in Fig. 5C. One can see that the predicted Mb-bound oxygen concentration agrees fairly well with experimental data obtained by Richardson et al. (33). Additionally, even at the highest work intensity, Mb is still about half-saturated with oxygen. Therefore, there is still some significant reservoir of oxygen for a further increase in maximal oxygen consumption per muscle mass and work rate per muscle mass (and work rate).

The situation is significantly different during whole body exercise. The maximal general work rate (Wmax) is higher here (~340 W) than in the case of single-muscle exercise, although the maximal work rate per muscle mass (Ws,max) is smaller (~23 W/kg). The general pattern of the dependence of the (mean) oxygen concentration/partial pressure in muscle artery, vein, capillaries, and cells is rather similar to that simulated for single-muscle exercise. This is shown in Fig. 5, D–F, which is analogous to Fig. 5, A–C but presents computer simulations concerning the dependence of the oxygen concentration/partial pressure on work intensity for whole body exercise. However, the most striking difference between single-muscle exercise and whole body exercise is the fact that in the latter case at some critical work rate the mean cytosolic oxygen concentration drops (very close) to zero (Fig. 5, E and F). Therefore, of course, no further increase in rate of oxygen consumption per muscle mass and in sustained work rate (driven by oxidative ATP production) is possible. The simulated properties of the system presented in Fig. 5, D–F have not been measured experimentally during whole body exercise. Therefore, these simulations offer a completely new knowledge, especially in relation to the termination of exercise when oxygen concentration drops to zero.

In addition to the above-presented main theoretical result obtained in the present study, it has been theoretically tested how changes in the intensity of blood flow affect the maximal pulmonary oxygen consumption and maximal cardiac output. The intensity of blood flow was manipulated through changes in the cardiac output/work rate dependence; in subsequent simulations the intensity of cardiac output at all work rates was either increased or decreased by some relative factor (e.g., 10%). The simulated relationship between the maximal pulmonary oxygen consumption and maximal cardiac output was compared with the experimental one. One can see (Fig. 6) that the model correctly predicts this dependence, suggesting a close connection between both variables. However, it is not true that maximal cardiac output determines maximal pulmonary oxygen consumption. Rather, both variables are determined by the overall system properties, especially by the subtle balance between oxygen delivery by blood and oxygen consumption. It should be stressed that similar results were obtained when oxygen delivery was changed by changing Hb content in blood or arterial blood saturation with oxygen, whereas the cardiac output/work rate relationship is kept constant (not shown).


Figure 6
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  		Fig. 6. Simulated and experimental relationship between maximal pulmonary oxygen consumption (VO2p,max) and maximal cardiac output (Qmax). Experimental data are from the following sources: gray circle, Ref. 10; open circles, Ref. 3; closed circles, Ref. 15; open diamonds, Ref. 13; closed diamonds, Ref. 25.

 

   DISCUSSION
TOP
 ABSTRACT
 THEORETICAL PROCEDURES
 THEORETICAL RESULTS
 DISCUSSION
APPENDIX
 REFERENCES
 
In the present article, a rather simple steady-state model of oxygen transport to and within working human skeletal muscles is developed. The model is broadly validated through comparison of computer simulations with experimental data. It was demonstrated that a single unique quantitative description reproduces well the system behavior during single-muscle exercise and whole body exercise, and therefore the same model can be applied to describe both types of exercise. The main aim of the present study was to test in a theoretical way what will happen if the working muscle mass is increased from 2.5 kg (single-muscle exercise) to 15 kg (whole body exercise) when all other parameter values and system properties (including the phenomenological dependence of cardiac output on work rate) are kept constant.

The most important finding of the theoretical studies carried out using the model is that oxygen supply by blood is limiting for the maximal work rate and muscle oxygen consumption during whole body exercise but not during single-muscle exercise, as proposed previously by Saltin (38). The maximal muscle oxygen consumption per muscle mass is much greater during single-muscle exercise (350–600 ml·kg–1·min–1) (2, 29, 33, 35) than during whole body exercise (110–320 ml·kg–1·min–1) (estimated from Refs. 3, 10, 15, 37, 44, and 47). Computer simulations show that when the working muscle mass is increased from 2.5 kg (single-muscle exercise) to 15 kg (cycling), the oxygen concentration falls (very close) to zero at some critical work rate, and therefore the work rate and oxygen consumption cannot be further increased. As a consequence, the maximal work rate per muscle mass is significantly higher during single-muscle exercise (at least 50 W/kg) than during whole body exercise (~22.5 W/kg). Similarly, the maximal oxygen consumption per muscle mass is significantly higher during single-muscle exercise (at least 600 ml·kg–1·min–1) than during whole body exercise (~270 ml·kg–1·min–1). This happens for all other parameters kept constant. The maximal general work rate and pulmonary oxygen consumption during whole body exercise (~340 W and 4.3 l/min, respectively) are well within the physiological range in humans. This fact suggests that the model developed in the present study is able to reproduce at least semiquantitatively the properties of the modeled system.

The only effect caused by an increased working muscle mass is that the limited amount of oxygen transported by blood must be distributed in a greater muscle mass. In other words, less oxygen can be delivered per working muscle mass. Therefore, in principle, the general qualitative conclusion that during whole body exercise the oxygen supply by blood to working muscles is more restricted than during single-muscle exercise is straightforward and in fact trivial.

However, this does not mean that the predictions of the model are trivial as well. The model allows us to estimate the quantitative effect of the increase in the working muscle mass on the system properties. For instance, it shows that the oxygen concentration falls (very close) to zero at some particular value of work rate and oxygen consumption. This value agrees very well with the value encountered in experimental studies. This conclusion is by no means trivial and does not result from some assumptions made in advance or simply from the mass conservation property (and therefore does not involve circular reasoning). The value of the maximal work rate and oxygen consumption is a derivative of the subtle balance between the dependences of cardiac output (at constant blood Hb content and arterial oxygen concentration) and oxygen consumption on work rate (and the dependence of the phenomenological diffusion constant on oxygen consumption). This value cannot be extracted from experimental data by the human brain alone; therefore, quantitative models well-based on experimental data are necessary. The model also shows that the maximal oxygen consumption and work rate are very sensitive to the form/shape of the dependence of oxygen consumption and cardiac output on work rate; i.e., slightly different forms/shapes can cause the oxygen concentration to fall to zero at very different values of the work rate (not shown). This property has little to do with the mass conservation property. Therefore, the model provides a new quantitative insight into the system functioning.

However, as we discussed before (see INTRODUCTION), we do not claim that we are the first to propose that oxygen delivery by blood is limiting for work rate and oxygen consumption during whole body exercise in humans; this opinion is shared by many (but not all) physiologists. However, the model is able 1) to support this hypothesis quantitatively; 2) to explain explicitly how and why the behavior of the system changes when the working muscle mass increases; 3) to predict the concentration/partial pressure of oxygen in arteries, veins, capillaries, and myocytes during whole body exercise (Fig. 5, D–F) (and thus to offer a completely new knowledge); and 4) to explain why the cellular oxygen concentration falls to zero at maximal work rate only during whole body exercise, although during both single-muscle and whole body exercise the capillary oxygen concentration at maximal work rate is similar (see Fig. 5). (This happens because the diffusion constant kD depends on the oxygen consumption per muscle mass that is much higher at the maximal work rate during single-muscle exercise than during whole body exercise). Last but not least, a correct formal quantitative description of the system guarantees that we understand the system well. Generally, all good models are deeply based on experimental data, and therefore the predictions made by using such models result indirectly from these data. The main role of the models is to help to extract quantitative system properties form experimental measurements.

It was postulated that in quadrupeds the oxygen supply by blood is not limiting for maximal pulmonary oxygen consumption (41). This proposal was based on the observation that the whole animal maximal sustained oxygen uptake is quite similar, when recalculated for the mitochondria volume, to the maximal oxygen uptake in isolated mitochondria (however, see discussion in Ref. 19). However, this conclusion was based on the assumption that the maximal capacity of oxidative phosphorylation in working muscles corresponds to the maximal capacity in isolated mitochondria. On the other hand, it was proposed that, due to the so-called parallel activation (or each-step-activation) mechanism of the regulation of oxidative phosphorylation (17–19), the capacity of oxidative phosphorylation in intact muscle may be several times greater than in isolated mitochondria (18, 22). If this mechanism is correct, the conclusion that in quadrupeds oxygen supply to muscles is not limiting for oxygen consumption may be false.

The present study strongly suggests that the oxygen delivery by blood is limiting for the maximal oxygen consumption and work rate during whole body exercise. The oxygen delivery constitutes a derivative of several factors, including cardiac output (dependent among others on capillary resistance), Hb concentration in blood, alveolar ventilation, and arterial blood saturation with oxygen. All these factors could, at least in principle, contribute to determining the oxygen delivery rate.

There is an ongoing discussion whether cardiac output is the primary factor determining the maximal pulmonary oxygen uptake during exercise in humans (39) or whether there are other factors that contribute significantly to determining uptake (50). However, the developed model and the present theoretical study are only designated to analyze the role of the working muscle mass, and all other parameters and system properties (including the dependence of cardiac output on work rate, Hb content in blood, and arterial blood saturation with oxygen) are kept constant. The model does not take into account explicitly the capillary resistance for blood flow; it enters implicitly into the phenomenological dependence between blood flow and work intensity. The present version of the model assumes that oxygen supply is not limiting for oxygen consumption and work rate during single-muscle exercise under standard conditions. However, this assumption may be false under some other conditions, e.g., in hypoxia.

One should clearly distinguish between a potential control exerted by some factors over the maximal oxygen uptake and work rate and the actual effect of some factor(s) exerted in a certain particular situation. The potential control determines what would happen if, for instance, cardiac output, Hb content in blood, or arterial blood saturation with oxygen changes by some value. The actual effect concerns the change in the behavior of the system if, for instance, the working muscle mass is increased from 2.5 kg to 15 kg and all other parameter values and system properties are kept constant. In the present model we only deal with the actual effect under some particular conditions, and therefore we do not enter the above-discussed polemics whether cardiac output is the primary determinant of maximal pulmonary oxygen consumption or not.

The present theoretical studies suggest that the maximal cardiac output is not a fixed parameter but is a system variable (like VO2p,max and Wmax), the value of which results from the overall system properties, namely from a subtle balance between oxygen delivery by blood and oxygen consumption by working muscles (compare Fig. 6). This conclusion seems to be obvious for VO2p,max and Wmax: when oxygen concentration drops (very close) to zero at a certain critical work intensity, the oxygen consumption and sustained work intensity cannot further increase. However, if, under some particular conditions, cardiac output is a unique function of work rate (all other parameter values and system properties are kept constant), the maximal work intensity will determine the maximal cardiac output. As can be seen from the experimental points presented in Fig. 1 (and Fig. 4), the cardiac output/work rate (and cardiac output/pulmonary oxygen consumption) relationship is not saturable (the line is only slightly bent downward), and therefore there is no defined-in-advance maximal cardiac output.

In the present model, the dependence of the respiration rate on cytosolic oxygen concentration is not included; it is simply assumed that muscle work is terminated when cytosolic oxygen concentration drops to zero. Such a simplification is justified because the half-saturation constant of oxidative phosphorylation for oxygen is very low, well below 1 µM (11).

Also, the assumption that the mean oxygen concentration in capillaries (being the average of the arterial and venous oxygen concentration) is equal in all capillaries obviously constitutes a significant simplification. It is possible to develop more realistic, but at the same time more complex, descriptions of the spatial distribution of oxygen concentration in capillaries in the whole muscle. However, such descriptions would have to be based on a greater number of assumptions and would be more difficult to interpret. For instance, the problem of what fraction of muscle cells must be in anoxia to terminate the exercise would arise. Another related assumption is that the oxygen concentration is the same in all muscle cells. However, it s likely that oxygen is not distributed homogeneously within the muscle, and therefore the use of a mean cytosolic oxygen concentration, unique for the whole muscle, is certainly only a rough approximation.

In the above discussion it was implicitly assumed that the modeled system is a perfect steady-state system. However, this assumption is not entirely accurate. At high work intensities cardiac output progressively increases over time, and therefore the measured values of cardiac output may be underestimated. This may be one of the reasons why the measured cardiac output/work rate dependence is slightly bent down at high work intensities (see Fig. 1B). Because, as discussed above, the maximal work rate and oxygen consumption are very sensitive to the form/shape of the cardiac output/work rate and oxygen consumption/work rate dependences, this would lead to an increase in maximal work rate and maximal pulmonary oxygen consumption. However, the pulmonary and muscle oxygen consumption also increase over time above the so-called lactate threshold (the so-called slow component of the oxygen consumption on-kinetics) (52). This phenomenon would result in a decrease in maximal work rate and maximal pulmonary oxygen consumption. Generally, these two effects counterbalance each other, although it is not clear whether they cancel each other entirely.

From many experimental data relevant to the present theoretical study that are available in the literature, only some were taken for the analysis. Our aim was to use data that are a good reference point for model testing, contain as complete sets of experimental points as possible, are easy to extract from the original studies, concern relatively well-trained male subjects, and are "typical." However, the choice of the data was certainly to some extent arbitrary, and one should be aware that much more data concerning the analyzed system are available in the literature (see for example Refs. 12, 26, 40, 44, 51, and 52).

The computer model developed in the present study is a static (steady-state) model, because it does not involve the behavior of the system during rest-to-work and work-to-rest transitions. In the future it is planned to join the present model with the model of oxidative phosphorylation in skeletal muscle developed previously by Korzeniewski and co-workers (20, 21) and to transform the present static model into a dynamic model, taking into account changes in system variables over time.

Some other models of the oxygen transport to and within working skeletal muscle have been developed: (5–8, 14, 24, 27, 36, 42, 49). However, these models are designated to different purposes and have different philosophies underlying them from those of our model. None of them has been used to produce theoretical studies similar to those performed in the present study.

Summing up, in the present in silico study a static model of oxygen transport to and within working human skeletal muscle is developed. Within the model, cardiac output and muscle oxygen consumption are described as phenomenological functions of work rate. The model is able to reflect properly the relevant properties of the system both during single-muscle exercise and whole body exercise. Computer simulations predict that an increase in the working muscle mass from 2.5 kg (single quadriceps) to 15 kg (two legs), with all other parameters and system properties kept constant, will cause oxygen concentration in muscle cells to drop (very near) to zero at some critical work intensity, and therefore oxygen supply by blood limits maximal oxygen consumption and oxidative ATP production. As a consequence, the maximal oxygen consumption per muscle mass is greater during single-muscle exercise than during whole body exercise, as proposed originally by Saltin (38). This is caused by the fact that the same limited oxygen amount delivered by blood must be distributed in a greater working muscle mass during whole body exercise.


   APPENDIX
TOP
 ABSTRACT
 THEORETICAL PROCEDURES
 THEORETICAL RESULTS
 DISCUSSION
 APPENDIX
REFERENCES
 
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   ACKNOWLEDGMENTS
 
We are grateful to Jerzy A. Zoladz for a stimulating discussion.


   FOOTNOTES
 

Address for reprint requests and other correspondence: B. Korzeniewski, Faculty of Biochemistry, Biophysics, and Biotechnology, Jagiellonian Univ., ul. Gronostajowa 7, 30-387 Kraków, Poland (e-mail: benio{at}mol.uj.edu.pl)

The costs of publication of this article were defrayed in part by the payment of page charges. The article must therefore be hereby marked "advertisement" in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.


   REFERENCES
TOP
 ABSTRACT
 THEORETICAL PROCEDURES
 THEORETICAL RESULTS
 DISCUSSION
 APPENDIX
 REFERENCES
 

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