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1Program in Applied Mathematics, University of Arizona, Tucson 85721; and 2Department of Physiology, University of Arizona, Tucson, Arizona 85724
Submitted 26 January 2003 ; accepted in final form 25 April 2003
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ABSTRACT |
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 TOP ABSTRACT METHODSRESULTSDISCUSSIONDISCLOSURESREFERENCES |
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capillaries; conducted response; flow regulation; motor units; oxygen diffusion; theoretical model
50-fold, with a 25-fold increase in blood flow
between states of rest and maximal exercise in the knee extensor muscles
(4). As the activity level of a
skeletal muscle is increased, additional muscle fibers are recruited in groups
(motor units), which are activated in a predetermined sequence until the
desired force of contraction is achieved
(9,
18). Each motor unit consists
of a single motor neuron that innervates 20–2,000 fibers distributed
throughout the muscle (5,
6). A mammalian muscle
typically contains 50–300 motor units. Oxygen consumption in inactive
motor units remains near the resting level. Therefore, oxygen demand in a
partially activated muscle is highly nonuniform, with fibers belonging to
active motor units interspersed with quiescent fibers.
Oxygen is delivered to skeletal muscle fibers by convective transport in
blood flowing in capillaries that run approximately parallel to the fibers and
by diffusion from the capillaries to surrounding muscle fibers. The
availability of oxygen to a given muscle fiber is therefore dependent on the
number of nearby capillaries that are perfused and the rate of blood flow in
those capillaries. Krogh (24)
proposed that blood flow is controlled at the level of individual capillaries,
such that the number of flowing capillaries varies in response to local oxygen
demand. According to this concept, flow in each capillary is regulated by
active contraction of the capillary or of a precapillary sphincter located at
the entrance to the capillary. With regard to skeletal muscle, however, most
evidence suggests that capillary perfusion is regulated primarily by the
arterioles. The group of capillaries fed by a single terminal arteriole is
then the smallest functional unit for control of blood flow
(21). Such a microvascular
unit (MVU) typically contains 20–25 capillaries and supplies a region of
muscle tissue 100 x 200 µm in cross section and with length 500
µm along the fibers (8,
25), as shown in
Fig. 1. The region perfused by
one MVU necessarily encompasses portions of several adjacent muscle
fibers.
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A spatial mismatch is evident between the functional units of muscle activation and of flow regulation. Each motor unit consists of many dispersed muscle fibers, each of which is supplied by a number of different MVUs along its length (12). Conversely, each MVU supplies several adjacent muscle fibers, which typically belong to different motor units (33). Given this spatial mismatch, how is capillary blood flow regulated to meet local oxygen requirements? More specifically, how many capillaries must be perfused to ensure adequate oxygen supply to active muscle fibers at a given level of muscle activation?
Fuglevand and Segal (13) addressed these questions using a theoretical modeling approach. They considered a region in the cross section of a skeletal muscle that contained a number of motor units and MVUs. Individual fibers dispersed throughout the muscle were grouped into physiologically representative motor units. Each MVU was assumed to supply a region 1 x 1 mm or 0.5 x 0.5 mm in cross section (3). Perfusion of a MVU was assumed to occur whenever any muscle fiber lying within its region was activated. With these assumptions, the model predicted a highly nonlinear relationship between the fraction of active fibers and the fraction of perfused MVUs, with nearly all MVUs perfused when a small fraction of the muscle fibers was activated. If, instead, the functional units for blood flow control coincided with those of muscle activation (3), a linear relationship would be expected. These results therefore demonstrated that perfusion of skeletal muscle depends critically on the precise spatial relationship between MVUs and motor units.
This model (13) has evident limitations. First, the assumed size of the MVUs in the muscle cross section (1 x 1 mm or 0.5 x 0.5 mm) was larger than the size suggested by observations (0.1 x 0.2 mm) (25). Second, all capillaries in a MVU were assumed to flow whenever any fiber in the region was activated. Both assumptions could lead to overestimation of the number of capillaries that must be perfused when a given fraction of fibers has been recruited. In reality, MVU perfusion probably depends on levels of oxygen, potassium, and other metabolites released during muscle activity, and activation of a single fiber does not necessarily stimulate perfusion of all adjacent capillaries. Furthermore, the diffusive spread of oxygen through muscle tissue allows the possibility that an active muscle fiber may receive adequate oxygen supply from capillaries belonging to nonadjacent MVUs.
On the basis of these considerations, the objective of the present study was to develop a more realistic theoretical model to examine the relationship between muscle fiber activation and capillary perfusion in skeletal muscle. The model includes a two-dimensional simulation of diffusive oxygen transport from capillaries to fibers in a muscle cross section to predict levels of tissue oxygenation. The mechanisms by which activation of muscle fibers leads to capillary perfusion are not well established. In the model, the effects of four hypothetical mechanisms were explored. In the first two mechanisms, capillaries adjacent to active fibers are activated, independent of oxygen levels, as in the model of Fuglevand and Segal (13). To show explicitly the effects of capillary grouping into MVUs, the following two cases were considered: in case A, any individual capillary adjacent to an active fiber is perfused ("adjacent capillary mechanism"); in case B, any MVU containing one or more capillaries adjacent to an active fiber is perfused ("adjacent MVU mechanism"). In the other mechanisms considered, perfusion of MVUs was assumed to depend on tissue oxygen levels. Two alternative hypotheses regarding this response were considered: in case C, flow is initiated in a MVU whenever the partial PO2 adjacent to the arteriole feeding that MVU drops below a threshold value ("arteriolar response mechanism"). This hypothesis assumes direct arteriolar responsiveness to local PO2 levels (10, 20, 36). In case D, flow is initiated whenever the average level of PO2 in the tissue adjacent to capillaries of the MVU drops below a threshold value ("conducted response mechanism"). This hypothesis assumes the existence of mechanisms for transferring information about tissue oxygenation levels at the capillary level upstream to the feeding arteriole (19, 38), possibly via conducted electrical responses (32, 34). For each mechanism considered, the model was used to predict the number of perfused capillaries and oxygen levels in tissue as a function of the fraction of active muscle fibers.
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METHODS |
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TOPABSTRACTMETHODSRESULTSDISCUSSIONDISCLOSURESREFERENCES |
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Geometry of motor units and MVUs. The grouping of capillaries into MVUs and muscle fibers into motor units could not be determined from the photomicrographs. Appropriately sized MVUs were therefore generated by dividing the muscle cross section into 18 domains 100 x 200 µm in size, with every capillary inside a given domain grouped into the same MVU (Fig. 3). The MVUs contained 11–22 capillaries depending on the capillary density of the tissue. The muscle fibers lying within the tissue region were randomly assigned to 10 equal groups, each group being dispersed through the region. To show the effects of increasing levels of muscle activation, these groups were sequentially activated and the corresponding changes in oxygen levels and MVU activation simulated, as described below. This assumption does not correspond directly to the sequential activation of realistic motor units, where units with few fibers are typically activated earlier and units with many fibers later (27). However, introducing heterogeneous motor unit sizes in the present model would not alter the results for any given fraction of active fibers, as long as fibers belonging to each motor unit are randomly dispersed through the tissue region.
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Oxygen transport and consumption. Under any given set of conditions, each capillary was designated as perfused or unperfused. In perfused capillaries, the PO2 was set at a fixed level of 45 mmHg, representative of capillary PO2 levels in active skeletal muscle. Because oxygen transport in a single muscle cross section is considered, it is not possible to include explicitly the decline of PO2 along capillaries within the context of this model. The diffusion of oxygen through tissue was considered to be unaffected by the presence of unperfused capillaries. Similarly, each muscle fiber was designated as active or inactive. The basal oxygen demand of inactive fibers was assumed to be Minactive = 1 cm3O2 (100 cm3 · min)–1 (11). The oxygen demand of active muscle fibers was assumed to be Mactive = 16 cm3O2 (100 cm3 · min)–1, close to the maximal rate of working of trained rat hindlimb muscle (30). The relationship between the oxygen consumption and tissue PO2 was described by Michaelis-Menten kinetics with half-maximal consumption at a tissue PO2 of P0 = 1 mmHg (7, 29). To assess the overall level of tissue oxygenation, the "oxygen deficit" was defined as the difference between the total oxygen demand of the region and the total consumption computed according to the Michaelis-Menten relationship divided by the total demand. Consumption falls significantly below demand only in regions of low PO2, so the oxygen deficit is a functional index of the amount of hypoxic tissue.
Computation of tissue PO2. A two-dimensional
time-dependent diffusion equation was used to compute the distribution of
tissue PO2 within the simulated muscle cross section
 | (1) |
denotes a partial
derivative. The quantity D
was assumed to be a constant, where
D is the diffusion coefficient of oxygen in tissue (1.5 x
10–5
cm2·s–1)
(1), and is the
solubility (4 x 10–5
cm3O2·cm–3·mmHg–1).
Within each muscle fiber, the oxygen demand M0 was set to
Minactive or Mactive as appropriate.
To minimize the effect of boundary artifacts, periodic boundary conditions
were imposed at the edges of the domain, i.e., levels of
PO2 at each edge of the domain were assumed to match
those at the corresponding point on the opposite edge. A finite-difference method was used to solve Eq. 1. A 300 x 300 grid was used, corresponding to a spacing of 2 µm between grid points, with a 0.5-ms time step. PO2 values at all grid points lying within perfused capillaries were set to 45 mmHg. The governing equations were discretized using central differences and solved using an explicit scheme. The computations were continued until the absolute difference in tissue PO2 between successive time steps was <2 x 10–4 mmHg at each grid point. The tissue PO2 was assumed to be at a steady state when this condition was met. In the initial state, all fibers are assumed to be inactive. It was assumed that a single, randomly chosen MVU was perfused to meet the quiescent oxygen demand in this case. The number of active fibers was then increased as already described, and the perfused capillaries were determined according to the criteria described below. The steady-state solution was found for each combination of active fibers and perfused capillaries.
Activation sequence of motor units and MVUs. For each of the three
configurations considered (Fig.
2), increasing activation of the muscle was simulated by
sequential recruitment of spatially dispersed groups of muscle fibers.
Starting with the case in which all fibers are inactive, simulations were
carried out for cases where the active fraction of the total muscle cross
section area increased in steps of 0.1. With each successive increase in
the number of active fibers, the new steady-state distribution of
PO2 in the tissue cross section was calculated according
to the procedure described above with additional MVUs or individual
capillaries perfused as necessary to maintain tissue oxygenation using the
chosen mechanism of blood flow regulation. In each case, the variation of the
fraction of perfused capillaries with the fraction of active fiber area was
determined and presented in graphical form.
The four mechanisms of capillary perfusion considered were implemented in the model as follows (Fig. 4).
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Case A: adjacent capillary mechanism. Any capillary lying adjacent to an active muscle fiber was assumed to be perfused, independent of PO2 levels. This mechanism makes no reference to MVUs.
Case B: adjacent MVU mechanism. Any MVU containing a capillary lying adjacent to an active fiber was assumed to be perfused, independent of PO2 levels.
Case C: arteriolar response mechanism. For each MVU, the feeding arteriole was represented by a straight segment 200 µm in length, lying 75 µm outside the region supplied by the MVU (Fig. 3). The minimum tissue PO2 along the segment was estimated, and the arteriole was assumed to dilate and perfuse the MVU if the minimum PO2 fell below a specified critical level.
Case D: conducted response mechanism. For each MVU, the tissue PO2 at the location of all capillaries in the MVU was computed. The corresponding arteriole was assumed to dilate as a result of conducted responses, perfusing the capillaries of the MVU, if the average PO2 at the capillaries fell below a specified level. A range of critical PO2 values was considered for the arteriolar response mechanism and the conducted response mechanism.
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RESULTS |
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TOPABSTRACTMETHODSRESULTSDISCUSSIONDISCLOSURESREFERENCES |
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A comparison of Fig. 5, left and right, shows the predicted effect of increasing the fraction of active muscle fibers from 0.2 to 0.42. In this instance, two additional MVUs were perfused according to the criteria of the conducted response mechanism, and the fraction of perfused capillaries increased from 0.42 to 0.54. This increase in oxygen supply was not enough to completely balance the increase in demand, and the oxygen deficit increased from 0.06 to 0.09.
Effects of capillary grouping into MVUs. Figure 6 shows the fraction of perfused capillaries as a function of active fiber fraction for the adjacent capillary and adjacent MVU mechanisms. For a given active fiber fraction, perfusion control by MVUs leads to more perfused capillaries than individual capillary perfusion because whenever any capillary is perfused, all other capillaries in the same MVU are necessarily also perfused. The oxygen deficit is low in all cases and does not differ appreciably between the two assumed mechanisms. These results show that the grouping of capillaries into functional units triggers the perfusion of additional capillaries not required to maintain adequate tissue oxygenation.
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Effects of mechanisms controlling capillary perfusion. The predicted fractions of perfused capillaries as a function of active fiber fraction are shown in Fig. 7 for the arteriolar response mechanism and the conducted response mechanism. Relative to the adjacent MVU mechanism (Fig. 6), both the arteriolar response and the conducted response mechanisms result in substantially lower fractions of perfused capillaries for a given level of muscle activation, together with higher levels of oxygen deficit. In the case of the conducted response mechanism, the oxygen deficit varied with muscle fiber activation but was not >0.11. In contrast, the oxygen deficit was not well controlled when the arteriolar response mechanism was assumed, but generally increased with increasing activation, despite the fact that the fraction of perfused capillaries was comparable in both cases.
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The results shown in Fig. 7 assume a minimum critical PO2 value of 5 mmHg for the arteriolar response mechanism and an average critical PO2 value of 7 mmHg for the conducted response mechanism. These levels were chosen to give comparable levels of capillary perfusion and oxygen deficit at low levels of activation. Further simulations were also carried out in which the assumed critical PO2 values were varied to examine the dependence of the results on the chosen values. For the arteriolar response mechanism, increasing the critical PO2 value did not result in better control of tissue oxygenation at high levels of muscle activation. Not all capillaries were perfused when all muscle fibers were activated, even when the critical PO2 value was increased to 11 mmHg. These results suggest that control of perfusion by arteriolar response to oxygen levels in adjacent tissue is relatively ineffective in matching oxygen demand and supply. The ability of the conducted response mechanism to control oxygen deficit over the full range of activation levels was maintained when the critical PO2 value was varied between 1 and 13 mmHg.
Effects of fiber and capillary geometry. Similar results were obtained for all three configurations considered (Figs. 6 and 7). The results for the idealized hexagonal structure did not differ in a systematic way from the results for configurations derived from photomicrographs. Thus the heterogeneity in fiber size and capillary spacing observed in vivo does not appear to have a large impact on the fraction of perfused capillaries needed to supply oxygen under given conditions.
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DISCUSSION |
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TOPABSTRACTMETHODSRESULTSDISCUSSIONDISCLOSURESREFERENCES |
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20 µm
(26). As expected, the grouping of capillaries into MVUs increases the number of capillaries that must be perfused at a given level of activation, relative to the case in which perfusion of each capillary is assumed to be individually controlled (Fig. 6). The results for the adjacent MVU mechanism are qualitatively similar to those obtained by Fuglevand and Segal (13), but the initial rate of increase in capillary perfusion as a function of muscle activation was less steep in the present study. The lower slope reflects the smaller domain size of each MVU assumed here, i.e., 100 x 200 µm, compared with 1 x 1 mm in the Fuglevand and Segal study, and the inclusion of effects of oxygen diffusion when determining which MVUs are perfused in response to increasing fiber activation.
The results shown in Figs. 6
and 7 demonstrate how inclusion
of PO2 as an explicit control signal influences
perfusion within the muscle cross section. Compared with the simulations in
which adjacent active fibers induced perfusion of MVUs, the simulations in
which oxygen levels were sensed led to a lower fraction of perfused
capillaries, particularly at low levels of muscle activation, while keeping
the oxygen deficit relatively low (Fig.
7). Moreover, because oxygen can diffuse to active fibers that are
near but not adjacent to the perfused capillary, this partially compensates
for the geometric mismatch between the functional units. Low oxygen levels
occur only when active fibers are a distance of 50 µm or more from the
nearest perfused capillary. The diffusion of oxygen from regions of higher
concentration to regions of lower concentration helps to minimize the effects
of the irregular shapes of the muscle fibers and randomly distributed
capillary locations in the observed muscle cross sections, giving results that
were similar to those for an idealized hexagonal arrangement with
approximately the same capillary density and fiber size.
Results obtained using this model depend on the assumptions made with regard to the mechanisms controlling perfusion of capillaries or MVUs. Several studies (2, 14, 39) have shown that stimulation of muscle fibers causes increased blood flow in adjacent capillaries. The mechanisms of this response are not known but likely involve a combination of mechanisms involving potassium, oxygen, adenosine, nitric oxide, other released metabolites, and/or neural feedback (16). Responses to oxygen levels probably play a crucial role. Arterioles are known to respond by constriction or dilation to changes in tissue oxygen levels (10, 20, 37). Furthermore, oxygen availability is the most important factor limiting the rate of muscle working in prolonged exercise, so arteriolar response to oxygen would provide the most direct feedback mechanism to match flow with metabolic needs. Control of MVUs was therefore assumed to depend on oxygen levels in this model, although other mechanisms, such as responses to changing extracellular potassium levels, could also be explored using a similar approach.
The model was used to examine the consequences of two different assumptions regarding the sites of oxygen sensing in the tissue. In the arteriolar response mechanism, arterioles were assumed to dilate when the minimum PO2 level along an arteriole falls below a threshold level of 5 mmHg. This parameter is difficult to estimate from available data on arteriolar responses to oxygen levels. The level of 5 mmHg was chosen to keep the oxygen deficit within a low range at low and moderate levels of activation. As already mentioned, oxygen deficit could not be controlled at higher levels of muscle activation, even if this threshold was increased. Similar results were obtained if the average rather than the minimum PO2 along the arteriole was used as a criterion, but a higher threshold level was needed.
Although the terminal arterioles are the primary site for the control of flow in MVUs, they are not necessarily the main sites for sensing of tissue oxygen levels (19). The conducted response mechanism is based on the alternative assumption that capillaries can sense oxygen levels. According to this assumption, low PO2 levels trigger a response that is transmitted to the arteriole, causing dilation. Evidence for such an information transfer mechanism has been presented by Tyml et al. (38), and it may involve conducted electrical responses (32, 34). In the model, perfusion of a feeding arteriole was assumed to occur when the average capillary PO2 fell to <7 mmHg. This parameter cannot be estimated directly from available experimental data, and the assumed value was chosen to keep the oxygen deficit low over the full range of muscle activation.
The results (Fig. 7) show that the mechanisms regulating MVU perfusion affect the ability of the system to deliver oxygen as needed despite the spatial mismatch between MVUs and motor units. In the arteriolar response mechanism, it is assumed that perfusion of a MVU was triggered by a decline in PO2 levels adjacent to the terminal feeding arteriole. Because this arteriole typically flows through a region perfused by capillaries from other MVUs (Fig. 3), it does not necessarily respond to a decline in PO2 in the region of the MVU that it feeds. As a result, tissue hypoxia is not well controlled and the oxygen deficit tends to increase at high levels of activation, without triggering the perfusion of additional MVUs. In the conducted response mechanism, it is assumed that perfusion of an arteriole is determined by PO2 levels adjacent to the capillaries belonging to the MVU fed by the arteriole. This mechanism leads to closer matching between perfusion of MVUs and fiber activation, with better control of the tissue oxygen deficit. The results favor the assumption that conducted responses from capillaries to arterioles play a role in regulating perfusion of skeletal muscle. However, the involvement of other mechanisms, including the arteriolar response mechanism, cannot be ruled out based on the present results.
An important simplification in the model is the assumption of a fixed PO2 level in all perfused capillaries. According to this assumption, each capillary has two possible states in the model, unperfused and perfused, and the rate of blood flow in the capillary does not otherwise enter into the model. In reality, the PO2 of each perfused capillary varies along its length, depending on its flow rate and the rate at which oxygen is extracted from it. The flow rate may vary with the level of muscle activation. Simulation of such a system is possible in principle (31) but would add greatly to the complexity of the model and to the number of arbitrary assumptions that would be necessary.
One consequence of this simplification is that the model does not allow quantitative prediction of the variation of blood flow rate with muscle activation. Experimental data (15, 35) show an approximately linear increase in blood flow as a function of contractile force or oxygen demand. Such a gradual increase would be difficult to explain if the number of perfused capillaries increased in a highly nonlinear way, as predicted by Fuglevand and Segal (13). The present model (with the conducted response mechanism) predicts variation in the number of perfused capillaries with muscle activation that is qualitatively similar to the reported experimental variation in blood flow with level of muscle activation, i.e., roughly linear with a positive intercept on the vertical axis. This is consistent with the experimental data if it is assumed that once a capillary is perfused, its flow rate remains approximately constant with further increases in muscle activation because overall flow rate is then proportional to the number of perfused capillaries.
A further simplification in the model is the assumption of a fixed rate of oxygen consumption in active muscle fibers. In reality, oxygen consumption rates of muscle fibers can vary depending on the level of activation (28), with a gradual increase from the basal rate to a maximal rate of consumption (23). Actual variations in fiber oxygen consumption rates may be wider than the 16-fold variation between resting and active rates assumed here. Such variations would increase the spatial heterogeneity in oxygen consumption rates but would not lead to qualitatively different behavior from that predicted here.
In conclusion, the model shows that the spatial mismatch between functional units of capillary perfusion and motor units leads to substantial heterogeneity of oxygen levels in partially activated skeletal muscle. The relationship between the level of muscle activation and the fraction of perfused capillaries needed to achieve adequate tissue oxygenation depends on the mechanism by which perfusion of MVUs is controlled. A control mechanism based on the sensing of oxygen levels by capillaries and transmission of this information to the feeding arterioles via conducted responses was found to lead to adequate tissue oxygenation over the full range of muscle activation, with the fraction of perfused capillaries increasing gradually with increasing activation.
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DISCLOSURES |
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ACKNOWLEDGMENTS |
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FOOTNOTES |
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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.
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, Adjacent capillary mechanism; 
, adjacent MVU
mechanism.