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Am J Physiol Heart Circ Physiol 290: H2277-H2285, 2006. First published January 6, 2006; doi:10.1152/ajpheart.00547.2005
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Effect of decreased O2 supply on skeletal muscle oxygenation and O2 consumption during sepsis: role of heterogeneous capillary spacing and blood flow

Daniel Goldman,1 Ryon M. Bateman,2 and Christopher G. Ellis3

1Departments of Mathematical Sciences and Biomedical Engineering, New Jersey Institute of Technology, Newark, New Jersey; 2iCAPTURE Centre, University of British Columbia, Vancouver, British Columbia; and 3Department of Medical Biophysics, University of Western Ontario, London, Ontario, Canada

Submitted 24 May 2005 ; accepted in final form 23 December 2005


   ABSTRACT
TOP
 ABSTRACT
MATHEMATICAL MODEL
 RESULTS
 DISCUSSION
 GRANTS
 REFERENCES
 
One of the main aspects of the initial phase of the septic inflammatory response to a bacterial infection is abnormal microvascular perfusion, including decreased functional capillary density (FCD) and increased blood flow heterogeneity. On the other hand, one of the most important phenomena observed in the later stages of sepsis is an increased dependence of tissue O2 utilization on the convective O2 supply. This "pathological supply dependency" is associated with organ failure and poor clinical outcomes. Here, a detailed theoretical model of capillary-to-tissue O2 transport during sepsis is used to examine the origins of abnormal supply dependency. With use of three-dimensional arrays of capillaries with heterogeneous spacing and blood flow, steady-state O2 transport is simulated numerically during reductions in the O2 supply. Increased supply dependency is shown to occur in sepsis for hypoxic (decreased hemoglobin O2 saturation) and stagnant (decreased blood flow) hypoxia. For stagnant hypoxia, a reduction in FCD with decreasing blood flow is necessary to obtain the observed increase in supply dependency. Our results imply that supply dependency observed under normal conditions does not have its origin at the level of individual capillaries. In sepsis, however, diffusion limitation and shunting of O2 by individual capillaries occur to a degree that is dependent on the heterogeneity of septic injury and the arrangement of capillary networks. Thus heterogeneous stoppage of individual capillaries is a likely factor in pathological supply dependency.

inflammation; computational model; microcirculation; supply dependency; functional shunting

SEPSIS, IN WHICH A LOCAL BACTERIAL infection produces a systemic inflammatory response, is a clinical disease that often leads to multiple organ failure and death (28). Despite some recent advances in the treatment of sepsis (6, 35), it remains one of the leading causes of mortality, and many of its basic mechanisms are not fully understood (5, 13, 20, 26). One of the most well-known features of sepsis (in humans and animal models) is its effect on the microcirculation, including increased capillary stopped flow and decreased functional capillary density (FCD) (4, 14, 25), increased blood flow heterogeneity, and loss of flow regulation resulting in blood flow maldistribution (14, 33, 41). Severe sepsis often results in a decreased ability of tissues to extract O2 from the blood (12, 37, 39). Although the exact mechanism is not known, it has been suggested that decreased O2 extraction in sepsis is related to microcirculatory disturbances, e.g., increased heterogeneity of capillary spacing and blood flow and the resulting increased heterogeneity of local O2 supply (21, 22, 43, 44).

On the basis of experimental measurements of FCD, capillary hemodynamics, and oxyhemoglobin saturation (So2) in rat skeletal muscle (14), we previously constructed a computational model to study the effect of changes in capillary O2 supply parameters on tissue oxygenation and O2 consumption under normal and sepsis conditions (16). Because stoppages of individual capillaries, rather than entire networks, are observed in this muscle during sepsis (14), the model uses a relatively large, three-dimensional (3D) array of parallel capillaries to simulate O2 transport for normal (or control) and severe sepsis conditions. The key feature of the model is that it allows study of steady-state O2 transport under the conditions of heterogeneous capillary spacing and blood flow observed during sepsis (14), including possibly important effects such as diffusive interactions between capillaries, decreased O2 consumption due to localized decreases in tissue oxygenation, and diffusion limitation of O2 transport (31, 32). For measured baseline O2 supply conditions, previous simulation results showed that the intrinsic (or maximum) O2 consumption rate increases in sepsis, tissue oxygenation decreases and becomes more heterogeneous, and these effects increase with the degree and heterogeneity of capillary flow disturbances. Simulations also showed that capillaries with relatively fast flow, which increase in number in sepsis, act only partly as shunts and play a significant role in O2 delivery. Although levels of tissue O2 partial pressure (PO2) low enough to decrease O2 consumption ("dysoxia") were not found, results indicated that, in sepsis, tissue would be more susceptible to dysoxia if the overall O2 supply were decreased. This suggested an additional area of investigation, because cardiac output and respiratory function can decrease in severe sepsis (3).

One of the main characteristics of advanced or late-stage sepsis is a decreased ability of tissues to extract O2 from the blood (7). In healthy individuals, O2 extraction [or total O2 consumption for a given tissue volume (VO2)] is essentially independent of the O2 supply rate (QO2) down to a critical QO2 (QO2,crit) below which VO2 begins to fall: above QO2,crit, there is an "aerobic plateau," and, below QO2,crit, VO2 is "supply dependent." Because the tissue is unable to extract 100% of the O2 supply, the measured slope of VO2 vs. QO2 during O2 supply dependency (msd) is ~0.75. Septic patients and nonseptic individuals have a similar VO2-QO2 relation, but septic patients have a much higher QO2,crit for the same baseline VO2. They also extract a smaller proportion of the O2 supplied, resulting in a lower msd (~0.6) and a larger positive slope for the aerobic plateau (mplateau). The higher QO2,crit and decreased ability to extract O2 in sepsis have led to the concept of "pathological supply dependency," which has been found in humans (11, 43), animal models (21, 30), and various tissues, including muscle, in sepsis (9).

Part of the motivation for the present study was to examine how changes in FCD and heterogeneity of capillary blood flow in sepsis are related to increased supply dependency. A previous model study using statistical theory showed that increased heterogeneity in microvascular perfusion, leading to a mismatch between local O2 supply and O2 demand, can result in an increased QO2,crit and a decreased msd (44). However, this work did not directly address the source of the relevant increases in heterogeneity and, in particular, whether these increases are primarily due to individual capillary stoppages or loss of flow regulation at the arteriolar level.

In the present study, we use our computational model of spatially heterogeneous capillary O2 transport in skeletal muscle to calculate how decreasing QO2 from baseline values affects tissue oxygenation and VO2 under control and severe sepsis conditions. We present results for tissue O2 distributions, minimum tissue PO2, tissue fraction at risk for dysoxia, and VO2, including two different methods for decreasing QO2: stagnant hypoxia (reduced flow velocity) and hypoxic hypoxia (reduced So2 at the capillary entrance). We discuss the differences between control and sepsis and make comparisons with a simpler, single-capillary (modified Krogh) model. Finally, we interpret our results in terms of the importance of capillary stopped flow in producing the observed increases in supply dependency during sepsis.


   MATHEMATICAL MODEL
TOP
 ABSTRACT
 MATHEMATICAL MODEL
RESULTS
 DISCUSSION
 GRANTS
 REFERENCES
 
Capillary geometry and blood flow. We previously described a three-dimensional spatially heterogeneous capillary-array model for the rat extensor digitorum longus (EDL) skeletal muscle that incorporates observed changes in FCD and capillary blood flow velocity distributions during sepsis (16). On the basis of experimental data for a 24-h sepsis model (14) and a tissue region that is 400 µm long and ~17,500 µm2 in cross-sectional area, 16 normal, 8 fast, and 8 stopped vessels are used to represent the control case and 7 normal, 7 fast, and 13 stopped vessels are used to represent the (severe) sepsis case. A muscle fiber construction that produces a realistic degree of spatial heterogeneity (18) was used to randomly place parallel capillaries in the tissue domain. Figure 1A shows how the 24 concurrently flowing capillaries for the control case are distributed in the cross section where the fast- and normal-flow capillaries are selected randomly. For the sepsis case (Fig. 1B), fast-, normal-, and stopped-flow capillaries are selected from the same set used for the control case, but fast-flow vessels are constrained to one quadrant to simulate observed increases in flow heterogeneity. This "clustering" of fast-flow capillaries is designed to simulate the most heterogeneous septic injury that is observed at the capillary level and one that would be most likely to result in increased O2 supply dependency.


Figure 1
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  		Fig. 1. Distribution of capillaries in tissue cross-section in control case (A) and during sepsis (B).

 
The tube hematocrit (HT) is fixed at 0.25, the average experimental value (14), and the discharge hematocrit (HD) is then obtained from an empirically derived formula (34). For control and sepsis cases, baseline red blood cell (RBC) velocities (vRBC) are initially fixed at 130 µm/s for normal-flow (14) and 500 µm/s for fast-flow (16) capillaries. Because there is evidence that total flow decreases in the EDL during sepsis (25), global adjustments are made in vRBC to obtain a baseline sepsis case with 64% of the RBC flow in the baseline control case.

Capillary-to-tissue O2 transport. The O2 transport model (15–19) describes transport of O2 in blood and tissue, including O2 bound to hemoglobin (Hb) and dissolved in the blood, as well as the effect of tissue myoglobin (Mb). In the capillaries, O2 transport is given by a time-dependent equation for So2

Formula 1(1)
where vb is mean blood velocity, R is vessel radius, {xi} is the local axial coordinate, and {theta} is the local circumforantial coordinate. The two terms in parentheses represent the volume- and flow-weighted O2-carrying capacities of blood: Pb is blood PO2, {alpha}b and Formula 1b are volume- and flow-weighted blood O2 solubilities, and CHb is O2-binding capacity of the Hb solution inside RBCs. Pb(S) = P50[S/(1 – S)]1/n (where S is O2 saturation and P50 is O2 half-saturation pressure) is the blood PO2 obtained by inverting the Hill equation for the Hb-O2 equilibrium binding curve. The blood-tissue O2 flux (j) is given by

Formula 2(2)
where Pw is tissue PO2 at the vessel surface and k is a mass transfer coefficient that depends on hematocrit (18).

For all simulations, S is fixed in all inlet segments by the parameter Sa. Although, in reality, Sa is not the same for all capillaries, this is a simplifying assumption that allows us to focus on heterogeneities in capillary spacing and blood flow and could be justified for capillaries supplied by the same arteriole. For baseline cases (control and sepsis), the blood velocities and hematocrits needed in Eq. 1 are assigned on the basis of experimental data as described above. For the cases with reduced O2 supply, vRBC and Sa are each decreased separately from baseline, with all other O2 transport parameters (e.g., hematocrit) fixed at baseline values. This procedure does not explore the full range of possible O2 transport parameters; however, it does cover the relevant range for sepsis for the cases corresponding to separate decreases in cardiac output and respiration.

For reductions in vRBC, two cases are studied: constant and decreasing FCD. The latter case represents the derecruitment observed experimentally as the RBC supply to a capillary bed is reduced. On the basis of experiments by Lindbom and Arfors (27), FCD is decreased by specifying uniform distributions of normal (90–170 µm/s) and fast (300–700 µm/s) velocities and using a threshold velocity (40 µm/s for control and 80 µm/s for sepsis) below which RBC flow is assumed to stop. The numbers of flowing capillaries for several reductions in vRBC are shown in Table 1. In general, decreasing FCD with vRBC will decrease the efficiency of O2 transport and, therefore, decrease O2 extraction, with higher threshold velocities producing greater decreases in O2 extraction.


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  		Table 1. Decreases in FCD with vRBC

 
O2 transport in the tissue is given by a time-dependent equation for PO2

Formula 3(3)
where {alpha} and D are the tissue O2 solubility and diffusivity and CMb, DMb, and SMb are the O2 binding capacity, diffusivity, and O2 saturation of Mb. Michaelis-Menten consumption kinetics are used to describe the dependence of O2 consumption on PO2 [M(P) = M0P/(P + Pc), where M is tissue O2 consumption rate], and the equilibrium Mb saturation is given by SMb(P) = P/(P + P50,Mb), where P50,Mb is Mb half-maximal pressure. At the vessel-tissue interface, a flux boundary condition on P is applied using Eq. 2, and no-flux boundary conditions are applied on the outer tissue surfaces in the z direction.

On the outer tissue surfaces in the x and y directions, periodic (16, 19) or no-flux (18) boundary conditions are specified to represent two extremes of the effect of neighboring capillary networks on the O2 distribution within the computational domain, i.e., the region being modeled. With periodic boundary conditions, the neighboring tissue is effectively tiled with identical copies of the computational domain. In the case of sepsis, the groups of seven fast-flow capillaries are evenly spaced in the surrounding tissue (Fig. 2A). With no-flux boundary conditions, the neighboring tissue is tiled with multiple versions of the computational domain reflected at the domain boundaries. Regions containing the group of seven fast-flow capillaries are thus clustered in groups of four (Fig. 2B). The difference in effective capillary spacing between the periodic and zero-flux boundary conditions is due to the corner location of the group of fast-flow capillaries, i.e., the heterogeneity of capillary spacing in the network being considered. Once the boundary conditions are specified, steady-state solutions to the O2 transport Eqs. 1–3 are calculated numerically using a finite-difference method (15).


Figure 2
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  		Fig. 2. Boundary conditions in x and y directions during sepsis. Periodic (A) and no-flux (B) conditions give different effective arrangements of capillary networks in and around the computational domain. {blacksquare}, Group of clustered fast-flow capillaries. For periodic conditions, surrounding capillary networks are identical, and maximum distance between any tissue point and fast-flow capillaries is ~115 µm. For no-flux conditions, surrounding networks are a reflection of the network considered, and maximum distance between any tissue point and fast-flow capillaries is increased by ~50%.

 
Model parameters. Most of the parameters used in our O2 transport calculations (Table 2) are the same as those used previously (16). As reported previously (14), M0 is estimated by comparing measured capillary O2 extraction ratios (14) with numerical calculations of the extraction ratios for several values of M0. For control, this yields the M0 used previously: 1.5 x 10–4 ml O2·ml–1·s–1. For sepsis, M0 was previously estimated to be 5.26 x 10–4 ml O2·ml–1·s–1 under the assumption that baseline O2 supply did not decrease (16). However, in the present work, we have assumed a 36% decrease in baseline O2 supply in sepsis (25). Therefore, for consistency with measured O2 extraction ratios (14), it was necessary to use a reduced value: M0 = 3.98 x 10–4 ml O2·ml–1·s–1.


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  		Table 2. Biophysical parameters for O2 transport

 

   RESULTS
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 ABSTRACT
 MATHEMATICAL MODEL
 RESULTS
DISCUSSION
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 REFERENCES
 
For each case studied, O2 delivery parameters and the PO2 boundary conditions in the x and y directions are fixed, and numerical calculations are performed to obtain the steady-state distributions of O2 in the capillaries and muscle tissue. Periodic and no-flux boundary conditions are considered to simulate the extremes of the effect of neighboring capillary networks on the O2 distributions in the tissue surrounding the capillary network being considered. Tissue O2 distributions at baseline O2 supply (Formula 3O2) for control and sepsis cases with periodic boundary conditions are shown in Fig. 3. At baseline, the sepsis case already has lower minimum tissue PO2 (Pmin, 17.9 vs. 40.0 mmHg for control), higher spatial heterogeneity of tissue PO2 {standard deviation/mean [CV(PO2)] of 0.22 vs. 0.04 for control}, and higher total Formula 3O2 (4.04 x 10–9 ml O2/s vs. 1.54 x 10–9 ml O2/s for control). No-flux boundary conditions have little effect on the control case but decrease Pmin and increase spatial heterogeneity for the sepsis case (Fig. 4). The difference between results for periodic and no-flux boundary conditions depends on the heterogeneity of the capillary arrangement inside the computational domain. This is one reason why the sepsis case, with fast-flow capillaries clustered in one corner, is more sensitive to the imposed boundary conditions.


Figure 3
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  		Fig. 3. Tissue PO2 probability distribution functions for stagnant hypoxia with constant functional capillary density (FCD) and periodic boundary conditions, where V is red blood cell velocity (vRBC). A: control, with baseline sepsis case shown for comparison. B: sepsis.

 

Figure 4
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  		Fig. 4. Tissue PO2 probability distribution functions during sepsis for stagnant hypoxia with constant FCD and no-flux boundary conditions.

 
Figures 3 and 4 also show the effect of decreasing Formula 3O2 via vRBC, where FCD is held constant. For control, PO2 distributions become broader as Formula 3O2 decreases, but Pmin does not reach zero; and Formula 3O2 remains nearly constant, until Formula 3O2 becomes very small (~10% of baseline). For sepsis (Fig. 3B), PO2 distributions also become broader as Formula 3O2 decreases, but Pmin decreases more quickly than for control and becomes almost zero when Formula 3O2 is only 50% of baseline (75% for no-flux boundary conditions). This indicates that, for sepsis, Formula 3O2 begins to decrease sharply at a much higher relative Formula 3O2 than for control (50–75% of baseline compared with ~10%).

Calculated Formula 3O2-Formula 3O2 curves for control and sepsis cases with periodic boundary conditions are compared in Fig. 5 for: 1) stagnant hypoxia with constant FCD, 2) stagnant hypoxia with decreasing FCD, and 3) hypoxic hypoxia. For all three types of hypoxia, Formula 3O2 is initially flat for control and then drops sharply as Formula 3O2 approaches zero; for sepsis, Formula 3O2 initially decreases faster and then begins to drop sharply at a lower relative Formula 3O2. Calculated Formula 3O2-Formula 3O2 curves for control and sepsis with no-flux boundary conditions are shown in Fig. 6. No-flux boundary conditions increase mplateau, cause supply dependency to begin at a higher O2 supply, and decrease msd.


Figure 5
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  		Fig. 5. O2 extraction-O2 supply curves for stagnant and hypoxic hypoxia for periodic boundary conditions in the control case (A) and during sepsis (B).

 

Figure 6
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  		Fig. 6. O2 extraction-O2 supply curves for stagnant and hypoxic hypoxia for no-flux boundary conditions in the control case (A) and during sepsis (B).

 
Figures 5 and 6 show that for sepsis, with periodic and no-flux boundary conditions, the curves for hypoxic and stagnant hypoxia with decreasing FCD are nearly identical, whereas the curve for stagnant hypoxia with constant FCD is slightly higher. Because experimentally the type of hypoxia imposed has not been observed to affect the shape of Formula 3O2-Formula 3O2 curves (38), the result here indicates that decreasing, rather than constant, FCD is the appropriate assumption for stagnant hypoxia. Therefore, for sepsis with stagnant hypoxia, only decreasing FCD is considered below. For control, on the other hand, constant (rather than decreasing) FCD gives a better match between the Formula 3O2-Formula 3O2 curves for stagnant and hypoxic hypoxia. However, because the curve for hypoxic hypoxia lies between the two curves for the stagnant hypoxia cases, the conclusion is that the threshold velocity chosen for the control case (40 µm/s) was too large. Rather than iterate to find a self-consistent threshold velocity, we consider stagnant hypoxia with constant FCD to be a reasonable approximation for the control case.

For sepsis, calculations show a greater effect of hypoxic than of stagnant hypoxia on Pmin: 4.3 vs. 7.2 mmHg for periodic conditions with a 35% decrease in Formula 3O2. Hypoxic hypoxia also produces a higher fraction of tissue with PO2 <1 mmHg: 0.18 vs. 0.15 for periodic conditions with a 60% decrease in Formula 3O2 and 0.20 vs. 0.17 for no-flux conditions with a 50% decrease in Formula 3O2. The criterion PO2 <1 mmHg is used to identify tissue at risk for dysoxia, because in our Michaelis-Menten description of O2 consumption kinetics, a critical PO2 of 0.5 mmHg is assumed. Calculations of CV(PO2) indicate slightly less heterogeneity of tissue PO2 caused by hypoxic than by stagnant hypoxia: 0.606 vs. 0.614 for periodic conditions and 0.816 vs. 0.900 for no-flux conditions, both with a 50% decrease in Formula 3O2.

In addition to the above-mentioned results, the following parameters of the Formula 3O2-Formula 3O2 curves are shown in Table 3 for control and sepsis cases: mplateau, msd, and the critical O2 extraction ratio (O2ERcrit = Formula 3O2,crit/Formula 3O2,crit, where Formula 3O2,crit is critical Formula 3O2). These parameters, which were derived by dual-line regressions on the calculated data points (44) (Fig. 7), are mainly affected by the physiological conditions (control vs. sepsis) and, in sepsis, by the boundary conditions (periodic vs. no-flux) and do not vary greatly between hypoxic and stagnant hypoxia.


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  		Table 3. Parameters of Formula 3O2-Formula 3O2 curve for stagnant and hypoxic hypoxia

 

Figure 7
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  		Fig. 7. Dual-line regression of O2 extraction-O2 supply curve for sepsis during stagnant hypoxia with decreasing FCD calculated using the 3-dimensional model and no-flux boundary conditions. ERcrit, critical O2 extraction ratio; ysd, regression line for supply-dependent region; ypl, regression line for aerobic plateau.

 

   DISCUSSION
TOP
 ABSTRACT
 MATHEMATICAL MODEL
 RESULTS
 DISCUSSION
GRANTS
 REFERENCES
 
Our experiment-based simulations of O2 transport in the EDL show the importance of increased O2 consumption, decreased overall O2 supply, and increased O2 supply heterogeneity during sepsis. At baseline, i.e., under average measured conditions, Formula 3O2 is 36% lower and M0 is 165% higher in sepsis than in control, leading to a greater susceptibility to hypoxia. This susceptibility is increased by the loss of FCD and greater blood flow heterogeneity in sepsis. By incorporating these factors into our computational model, we have shown that increased capillary-level heterogeneity during sepsis can lead to an increase in supply dependency similar to that observed during sepsis (9, 21, 30, 43).

Because local O2 consumption depends on PO2, the supply dependency found here in sepsis is a direct result of insufficient O2 delivery to certain tissue regions. The role of heterogeneity in sepsis can be seen in the fact that when Formula 3O2 was decreased by 50% (stagnant hypoxia), Pmin decreased by 83% but mean tissue PO2 only decreased by 31%. For the control case, a 50% decrease in Formula 3O2 resulted in an 11% decrease in Pmin and a 6% decrease in mean PO2. Another measure of O2 delivery heterogeneity, CV(PO2), increased 136% in sepsis when Formula 3O2 was reduced by 50%, in contrast to a 75% increase in the control case. The differences between results for periodic and no-flux boundary conditions also show the importance of heterogeneity in sepsis. In particular, when the fast-flow capillaries are clustered to represent a heterogeneous septic injury, tissue oxygenation is impaired, and this effect is increased when nearby capillary networks are not able to deliver O2 to regions containing only stopped- and normal-flow capillaries. For stagnant hypoxia with no-flux boundary conditions, Fig. 8 shows the change in the spatial distribution of PO2 at the venous end of the tissue domain during the development of supply dependency in sepsis.


Figure 8
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  		Fig. 8. Spatial distribution of tissue PO2 at venular end of tissue domain at onset of supply dependency in sepsis. O2 supplies corresponding to 100% (A), 65% (B) and 40% (C) of baseline are shown for stagnant hypoxia with decreasing FCD and no-flux boundary conditions. As O2 supply decreases, clustered fast-flow capillaries act as O2 shunts, and tissue supplied by normal and stopped capillaries becomes dysoxic and stops consuming O2.

 
Our simulation results for sepsis also show that tissue PO2 is somewhat affected by the way in which Formula 3O2 is decreased. In particular, given the same overall reduction in Formula 3O2, muscle PO2 is affected more by changes in Sa than by changes in vRBC. The implication of these results is that Sa is the most important variable to maintain near baseline to maximize tissue O2 delivery during sepsis.

Comparison with a modified Krogh model. The above-described results differ somewhat from those that would be obtained using a Krogh-type, single-capillary model of O2 transport (24), which does not take into account the spatial heterogeneity of capillary spacing and blood flow. To show this, we use a modified Krogh model (29) that includes PO2-dependent consumption, intravascular resistance, and the appropriate values of M0, average tissue cylinder radius, and average blood flow per capillary for our control and sepsis cases. Mb, which has been shown to have a minimal effect under steady-state conditions (29), is not included. For the control case, PO2 distributions obtained with the modified Krogh model are similar to those obtained with the 3D capillary-array model, implying effectively little supply heterogeneity. However, for sepsis (Fig. 9), the Krogh model gives narrower distributions and higher Pmin than the 3D model, especially when no-flux boundary conditions are used, implying a greater role for supply heterogeneity. The Krogh model predicts increased supply dependency for sepsis; however, it does not capture the pathological supply dependency found in the 3D model with no-flux boundary conditions (Fig. 10) and gives a slope of ~1 for the supply-dependent portion of the Formula 3O2-Formula 3O2 curves. The Krogh model also predicts smaller mplateau values. For stagnant hypoxia, the Krogh model gives mplateau = 0.00065 and 0.0056 for control and sepsis, respectively, whereas the 3D model gives mplateau = 0.0054 and 0.061 for periodic boundary conditions and mplateau = 0.0058 and 0.126 for no-flux conditions. Comparison of these results with the measurements of Humer et al. (21), which gave mplateau = 0.002 for control and mplateau = 0.039 for endotoxemia, suggests that the results of the 3D model are more consistent with experiment.


Figure 9
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  		Fig. 9. Tissue PO2 probability distributions during sepsis for baseline O2 supply and 50% stagnant hypoxia with decreasing FCD calculated using modified Krogh (K) and full [three-dimensional (3D)] models.

 

Figure 10
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  		Fig. 10. O2 extraction-O2 supply curves for hypoxic hypoxia calculated using modified Krogh and full 3-D models.

 
Capillary-level disturbances and pathological supply dependency. The results of our three-dimensional model for baseline control and sepsis conditions have been shown (16) to be in basic agreement with other measurements in skeletal muscle, such as those of Astiz et al. (2), Anning et al. (1), Sair et al. (36), and Vallet et al. (42). Using a modified Krogh model similar to that described above, Schumacker and Samsel (38) showed that, under normal conditions, capillary spacing (dK, diameter of a Krogh tissue cylinder) was a key parameter in determining the nature of supply dependency. In particular, for dK less than ~80 µm the supply-dependent slope of the Formula 3O2-Formula 3O2 curve was ~1 and supply dependency was not sensitive to the type of hypoxia imposed, as is also the case experimentally (10, 38). To match experimental findings for O2ERcrit (typically 0.6–0.75), it was necessary to postulate a functional arteriovenous shunt of ~30% (38).

Utilizing a statistical model in which distributions of local O2 demand-to-supply ratios were used to construct Formula 3O2-Formula 3O2 curves, Walley (44) investigated supply dependency. Humer et al. (21) used data on gut capillary transit time distributions in healthy and endotoxemic pigs to obtain O2ERcrit values close to those measured experimentally. Although its functional origin (i.e., whether it occurs at the capillary or arteriolar level) was not discussed, Walley's assumed O2 demand-supply distributions imply a degree of arteriovenous shunting of O2 (23). Usually, shunting of O2 indicates a direct flow from an arteriole to a venule that bypasses the capillaries; however, within the capillary bed, functional shunting can also occur if individual capillaries carry O2 through to the venous side, rather than deliver O2 to sites where it is needed in surrounding tissue (e.g., due to diffusion limitation).

In the present study, the average distance to the nearest normal- or fast-flow capillary (~dK/2) is ~14 µm for control and 22 µm for sepsis, with maximum distances of 39 and 67 µm for control and sepsis, respectively. Classic Krogh analysis can be used to show that diffusion limitation begins for dK/2 approximately equal to 125 and 81 µm for control and sepsis, respectively. Thus, although for the control case, shunting is highly unlikely at the capillary level, for sepsis, capillary shunting of O2 can occur if there is a slight increase in dK/2, as is the case for no-flux boundary conditions. Thus, for periodic boundary conditions, despite the imposed heterogeneity in capillary arrangement and the loss of FCD because of sepsis, we found msd close to 1 (i.e., ~100% of the O2 supplied is consumed), in agreement with the results of Schumacker and Samsel (38). The fast-flow capillaries provide some functional shunting of O2; however, this was not enough to affect msd or O2ERcrit, which we estimate to be ~1 for control and sepsis. Therefore, because msd and O2ERcrit did not decrease during sepsis in the present model, classic pathological supply dependency did not occur for periodic boundary conditions.

In contrast to the preceding results, for no-flux boundary conditions, which increase the maximum capillary-tissue distance (Fig. 2), the fast-flow capillaries provided enough functional shunting in sepsis to decrease msd and O2ERcrit well below 1. Therefore, for no-flux boundary conditions, pathological supply dependency did occur in sepsis. The present study assumed a 36% decrease in baseline blood flow in sepsis, which resulted in a lower M0 and, therefore, less potential for capillary shunting of O2 due to diffusion limitation. If baseline blood flow remains the same in sepsis, there would likely be more supply dependency than was found here. Two other characteristics of increased supply dependency in sepsis were found for periodic and no-flux boundary conditions: increased mplateau and decreased Formula 3O2,crit. This suggests that these two quantities are directly related to increased O2 transport heterogeneity at the individual capillary level, whereas msd and O2ERcrit depend on individual capillary stoppages and the arrangement and relative O2 supply of surrounding microvascular units.

The present model does not consider the influence of CO2 or pH, which could be expected to alter somewhat O2 transport in sepsis. In addition, the role of nitric oxide (NO) is not explicitly considered. NO is known to be upregulated in sepsis and to inhibit mitochondrial respiration, although its production also consumes O2. At a minimum, NO, similar to CO2 and pH, might be expected to alter the spatial distribution of O2 in tissue during sepsis. It has been proposed that overproduction of NO during sepsis could eventually lead to an inhibition of mitochondrial respiration and a decrease in O2 extraction (8, 40). Additional heterogeneities in capillary hematocrits and inlet saturations, not considered in the present model, might serve to further decrease O2 extraction and increase supply dependency in sepsis.

Using a detailed, experiment-based model of O2 transport in muscle during control and sepsis conditions, we have explored the effect on tissue oxygenation and O2 extraction of reducing the overall O2 supply. We found more sensitivity to O2 supply in sepsis than in control because of increased O2 consumption and decreased FCD. We also found that, in sepsis, tissue oxygenation is affected more by hypoxic than by stagnant hypoxia.

We have also used the model to explore the role of heterogeneities in capillary spacing and blood flow in producing supply dependency in muscle. Our results imply that the supply dependency observed under normal experimental conditions, with <100% O2 extraction, does not have its origin at the level of individual capillaries, because the absence of diffusion limitation (even when no-flux boundary conditions are imposed) prevents the necessary functional O2 shunting. Although our results show little functional shunting during sepsis for periodic boundary conditions, they show that diffusion limitation and shunting of O2 by individual capillaries can occur for no-flux boundary conditions. The most physiologically appropriate boundary conditions will depend on the local arrangement of microvascular units and their relative O2 supplies. However, the overall effect of nearby microvascular units should lie between those seen here for periodic and no-flux conditions. Therefore, it is expected that, in many cases, the pathological supply dependency seen for no-flux conditions will be relevant. In summary, we have shown that pathological supply dependency can occur within individual capillary networks and that loss of individual capillaries in sepsis likely plays a role. Full determination of the relative importance of individual capillary stoppages and impairment of local blood flow regulation will require further modeling based on new experimental data.


   GRANTS
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 ABSTRACT
 MATHEMATICAL MODEL
 RESULTS
 DISCUSSION
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REFERENCES
 
This work was supported by the Whitaker Foundation (D. Goldman), Canadian Institutes of Health Research Grant MOP-49416 (C. G. Ellis), and a Heart and Stroke Postdoctoral Fellowship (R. M. Bateman).


   FOOTNOTES
 

Address for reprint requests and other correspondence: D. Goldman, Dept. of Mathematical Sciences, New Jersey Institute of Technology, Univ. Heights, Newark, New Jersey 07102 (e-mail: dgoldman{at}oak.njit.edu)

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
 MATHEMATICAL MODEL
 RESULTS
 DISCUSSION
 GRANTS
 REFERENCES
 

  1. Anning PB, Sair M, Winlove CP, and Evans TW. Abnormal tissue oxygenation and cardiovascular changes in endotoxemia. Am J Respir Crit Care Med 159: 1710–1715, 1999.[Abstract/Free Full Text]
  2. Astiz M, Rackow EC, Weil MH, and Schumer W. Early impairment of oxidative metabolism and energy production in severe sepsis. Circ Shock 26: 311–320, 1988.[Web of Science][Medline]
  3. Balk RA. Optimum treatment of severe sepsis and septic shock: evidence in support of the recommendations. Dis Mon 50: 168–213, 2004.[CrossRef][Medline]
  4. Bateman RM, Jagger JE, Sharpe MD, Ellsworth ML, Mehta S, and Ellis CG. Erythrocyte deformability is a nitric oxide-mediated factor in decreased capillary density during sepsis. Am J Physiol Heart Circ Physiol 280: H2848–H2856, 2001.[Abstract/Free Full Text]
  5. Bateman RM, Sharpe MD, and Ellis CG. Bench-to-bedside review: microvascular dysfunction in sepsis—hemodynamics, oxygen transport, and nitric oxide. Crit Care 7: 359–373, 2003.[CrossRef][Web of Science][Medline]
  6. Bernard GR, Vincent JL, Laterre PF, LaRosa SP, Dhainaut JF, Lopez-Rodriguez A, Steingrub JS, Garber GE, Helterbrand JD, Ely EW, and Fisher CJ Jr. Efficacy and safety of recombinant human activated protein C for severe sepsis. N Engl J Med 344: 699–709, 2001.[Abstract/Free Full Text]
  7. Boekstegers P, Weidenhofer S, Kapsner T, and Werdan K. Skeletal muscle partial pressure of oxygen in patients with sepsis. Crit Care Med 22: 640–650, 1994.[Web of Science][Medline]
  8. Brealey D and Singer M. Mitochondrial dysfunction in sepsis. Curr Infect Dis Rep 5: 365–371, 2003.[Medline]
  9. Bredle DL, Samsel RW, Schumacker PT, and Cain SM. Critical O2 delivery to skeletal muscle at high and low PO2 in endotoxemic dogs. J Appl Physiol 66: 2553–2558, 1989.[Abstract/Free Full Text]
  10. Cain SM. Oxygen delivery and uptake in dogs during anemic and hypoxic hypoxia. J Appl Physiol 42: 228–234, 1977.[Abstract/Free Full Text]
  11. Cain SM and Curtis SE. Experimental models of pathologic oxygen supply dependency. Crit Care Med 19: 603–612, 1991.[Web of Science][Medline]
  12. Chittock DR and Russell JA. Oxygen delivery and consumption during sepsis. Clin Chest Med 17: 263–278, 1996.[CrossRef][Web of Science][Medline]
  13. Cohen ES and Hollenberg SM. Tissue oxygenation and sepsis. Crit Care Med 29: 1479–1480, 2001.[CrossRef][Web of Science][Medline]
  14. Ellis CG, Bateman RM, Sharpe MD, Sibbald WJ, and Gill R. Effect of a maldistribution of microvascular blood flow on capillary O2 extraction in sepsis. Am J Physiol Heart Circ Physiol 282: H156–H164, 2002.[Abstract/Free Full Text]
  15. Goldman D. Simulations of capillary network oxygen transport during transient ischemia in the presence and absence of tissue myoglobin. Adv Exp Med Biol 510: 355–359, 2003.[Web of Science][Medline]
  16. Goldman D, Bateman RM, and Ellis CG. Effect of sepsis on skeletal muscle oxygen consumption and tissue oxygenation: interpreting capillary oxygen transport data using a mathematical model. Am J Physiol Heart Circ Physiol 287: H2535–H2544, 2004.[Abstract/Free Full Text]
  17. Goldman D and Popel AS. Computational modeling of oxygen transport from complex capillary networks. Relation to the microcirculation physiome. Adv Exp Med Biol 471: 555–563, 1999.[Web of Science][Medline]
  18. Goldman D and Popel AS. A computational study of the effect of capillary network anastomoses and tortuosity on oxygen transport. J Theor Biol 206: 181–194, 2000.[CrossRef][Web of Science][Medline]
  19. Goldman D and Popel AS. A computational study of the effect of vasomotion on oxygen transport from capillary networks. J Theor Biol 209: 189–199, 2001.[CrossRef][Web of Science][Medline]
  20. Hinshaw LB. Sepsis/septic shock: participation of the microcirculation. Crit Care Med 24: 1072–1078, 1996.[CrossRef][Web of Science][Medline]
  21. Humer MF, Phang PT, Friesen BP, Allard MF, Goddard CM, and Walley KR. Heterogeneity of gut capillary transit times and impaired gut oxygen extraction in endotoxemic pigs. J Appl Physiol 81: 895–904, 1996.[Abstract/Free Full Text]
  22. Ince C. Mechanism of impaired oxygen extraction in sepsis. Shunt and the PO2 gap. Minerva Anestesiol 65: 337–339, 1999.[Medline]
  23. Ince C and Sinaasappel M. Microcirculatory oxygenation and shunting in sepsis and shock. Crit Care Med 27: 1369–1377, 1999.[CrossRef][Web of Science][Medline]
  24. Krogh A. The number and distribution of capillaries in muscles with calculations of the oxygen pressure head necessary for supplying tissue. J Physiol 52: 409–515, 1919.[Free Full Text]
  25. Lam C, Tyml K, Martin C, and Sibbald W. Microvascular perfusion is impaired in a rat model of normotensive sepsis. J Clin Invest 94: 2077–2083, 1994.[Web of Science][Medline]
  26. Lehr HA, Bittinger F, and Kirkpatrick CJ. Microcirculatory dysfunction in sepsis: a pathogenetic basis for therapy? J Pathol 190: 373–386, 2000.[CrossRef][Web of Science][Medline]
  27. Lindbom L and Arfors KE. Mechanisms and site of control for variation in the number of perfused capillaries in skeletal muscle. Int J Microcirc Clin Exp 4: 19–30, 1985.[Web of Science][Medline]
  28. Martin GS, Mannino DM, Eaton S, and Moss M. The epidemiology of sepsis in the United States from 1979 through 2000. N Engl J Med 348: 1546–1554, 2003.[Abstract/Free Full Text]
  29. McGuire BJ and Secomb TW. A theoretical model for oxygen transport in skeletal muscle under conditions of high oxygen demand. J Appl Physiol 91: 2255–2265, 2001.[Abstract/Free Full Text]
  30. Nelson DP, Samsel RW, Wood LD, and Schumacker PT. Pathological supply dependence of systemic and intestinal O2 uptake during endotoxemia. J Appl Physiol 64: 2410–2419, 1988.[Abstract/Free Full Text]
  31. Piiper J and Haab P. Oxygen supply and uptake in tissue models with unequal distribution of blood flow and shunt. Respir Physiol 84: 261–271, 1991.[CrossRef][Web of Science][Medline]
  32. Piiper J and Scheid P. Diffusion limitation of O2 supply to tissue in homogeneous and heterogeneous models. Respir Physiol 85: 127–136, 1991.[CrossRef][Web of Science][Medline]
  33. Piper RD, Pitt-Hyde M, Li F, Sibbald WJ, and Potter RF. Microcirculatory changes in rat skeletal muscle in sepsis. Am J Respir Crit Care Med 154: 931–937, 1996.[Abstract]
  34. Pries AR, Secomb TW, Gaehtgens P, and Gross JF. Blood flow in microvascular networks. Experiments and simulation. Circ Res 67: 826–834, 1990.[Abstract/Free Full Text]
  35. Rivers E, Nguyen B, Havstad S, Ressler J, Muzzin A, Knoblich B, Peterson E, and Tomlanovich M. Early goal-directed therapy in the treatment of severe sepsis and septic shock. N Engl J Med 345: 1368–1377, 2001.[Abstract/Free Full Text]
  36. Sair M, Etherington PJ, Curzen NP, Winlove CP, and Evans TW. Tissue oxygenation and perfusion in endotoxemia. Am J Physiol Heart Circ Physiol 271: H1620–H1625, 1996.[Abstract/Free Full Text]
  37. Samsel RW and Schumacker PT. Oxygen delivery to tissues. Eur Respir J 4: 1258–1267, 1991.[Abstract]
  38. Schumacker PT and Samsel RW. Analysis of oxygen delivery and uptake relationships in the Krogh tissue model. J Appl Physiol 67: 1234–1244, 1989.[Abstract/Free Full Text]
  39. Schumacker PT and Samsel RW. Oxygen delivery and uptake by peripheral tissues: physiology and pathophysiology. Crit Care Clin 5: 255–269, 1989.[Web of Science][Medline]
  40. Singer M and Brealey D. Mitochondrial dysfunction in sepsis. Biochem Soc Symp 66: 149–166, 1999.[Medline]
  41. Tyml K, Yu J, and McCormack DG. Capillary and arteriolar responses to local vasodilators are impaired in a rat model of sepsis. J Appl Physiol 84: 837–844, 1998.[Abstract/Free Full Text]
  42. Vallet B, Lund N, Curtis SE, Kelly D, and Cain SM. Gut and muscle tissue PO2 in endotoxemic dogs during shock and resuscitation. J Appl Physiol 76: 793–800, 1994.[Abstract/Free Full Text]
  43. Vincent JL and De Backer D. Oxygen transport—the oxygen delivery controversy. Intensive Care Med 30: 1990–1996, 2004.[CrossRef][Web of Science][Medline]
  44. Walley KR. Heterogeneity of oxygen delivery impairs oxygen extraction by peripheral tissues: theory. J Appl Physiol 81: 885–894, 1996.[Abstract/Free Full Text]



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