Theoretical Analysis of Cerebral Venous Blood Hemoglobin Oxygen Saturation as an Index of Cerebral Oxygenation during Hypothermic Cardiopulmonary Bypass: A Counterproposal to the "Luxury Perfusion" Hypothesis

In this model, we assume blood and brain are in thermal equilibrium. Our earlier simulation studies have shown that this assumption will limit this analysis to times other than the initial periods of cooling and rewarming. [5]Thus, temperature T refers to both blood and brain temperature (degree Celsius).

Oxygen solubility in blood (alpha^bl) (ml O²*symbol* 100 ml sup -1 *symbol* mmHg sup -1) depends on hemoglobin concentration and temperature. [6]Let H = hemoglobin concentration (g *symbol* 100 ml sup -1). Then, Equation 1. We use H = 8 g *symbol* 100 ml sup -1 in the calculation of alpha^blduring CPB.

(Equation 1) gives alpha^bl= 2.9 *symbol* 10 sup -3 ml O sub 2 *symbol* 100 ml sup -1 *symbol* mmHg sup -1 for H = 15 g *symbol* 100 ml sup -1 and T = 37 degrees Celsius. This value is virtually identical to oxygen solubility in brain tissue (3.0 *symbol* 10 sup -3 ml O²*symbol* 100 ml sup -1 *symbol* mmHg sup -1). [7]Thus, we use Equation 1with H = 15 g *symbol* 100 ml sup -1 to predict oxygen solubility in brain as a function of temperature. [6].

The hemoglobin oxygen dissociation curve is described by the Hill equation [6]: Equation 2S^O2(unitless) = fractional hemoglobin oxygen saturation and P^O2(mmHg) = oxygen partial pressure. The unitless Hill coefficient (2.8) gives the hemoglobin oxygen dissociation curve its sigmoid shape. P⁵⁰(mmHg) = the oxygen partial pressure at which hemoglobin oxygen saturation equals 0.50. Over the clinically relevant range of temperatures, diffusion primarily limits oxygen transfer from hemoglobin to brain. [8]Therefore, the mathematical model does not include chemical reaction kinetics. Equation 2is inaccurate for saturations less than 0.30. To ensure this inaccuracy does not affect our results, we did not perform simulations for which hemoglobin oxygen saturation is less than 0.30. We also use the inverse of Equation 2in the calculations, Equation 3.

Currently, two different hypothermic acid-base strategies are used during CPB, alpha-stat and pH-stat. With the pH-stat technique, arterial blood carbon dioxide partial pressure (Pa^CO2) is maintained at approximately 40 mmHg when corrected to temperature in vivo. With alpha-stat technique, Pa^CO2is maintained at approximately 40 mmHg when measured at 37 degrees Celsius. As temperature decreases, differences in Pa^CO2between techniques increases. Hemoglobin P sub 50 decreases with decreasing temperature, but increases with increasing Pa^CO2. Therefore, two equations are needed to describe the effect of temperature on P⁵⁰, depending on whether alpha-stat or pH-stat blood gas management is used [6]: Equation 4where P^age50equals the P⁵⁰at T = 37 degrees Celsius that is age appropriate.

Changes in Equation 2as a function of temperature can be predicted from Equation 4, because the Hill coefficient does not change with temperature. [8,9]To include fetal hemoglobin in our model, [9,10]Equation 5.

(Equation 2, Equation 3, Equation 4, Equation 5), were not developed using data from CPB. However, it is reasonable to assume they apply during CPB, because CPB per se does not affect P⁵⁰. [11]Indeed, P⁵⁰values predicted by Equation 4agree with available data from studies in humans. Coetzee and Swanepoel found P⁵⁰to equal 11.6 mmHg in adults undergoing CPB at 25.4 degrees Celsius (alpha-stat management). [11]This value to close to the P⁵⁰= 12.0 mmHg predicted by Equation 4.

Net oxygen transport from blood to tissue is described using Fick's law of diffusion for a parallel plane-layer geometric configuration [6,12]Equation 6where CMR^O2is given in ml O²*symbol* 100 ml sup -1 *symbol* mmHg sup -1; alpha^br= oxygen solubility in brain (ml O²*symbol* 100 ml sup -1 *symbol* mmHg sup -1); Pv^Osub 2 (mmHg) = cerebral venous blood oxygen partial pressure; Pin^Osub 2 (mmHg) = interstitial oxygen partial pressure; and D (cm²*symbol* min sup -1) = the net oxygen diffusion coefficient in the blood and tissue. Equation 6assumes Pv^O2equals the cerebral end capillary oxygen partial pressure. The 11,400 cm²*symbol* 100 g sup -1 term equals the sum of arteriole, capillary, and venule surface areas available for oxygen exchange. [6,13]The 27 *symbol* 10 sup -4 cm term specifies the sum of capillary and tissue thicknesses. [6,12]The 13 (unitless) is a correction for the increase in oxygen diffusivity that Schacterle and colleagues showed was necessary for Fick's law to attain the level of oxygen transport measured experimentally. [14]We recognize this correction factor, determined at normothermia, could decrease with decreasing temperature. The implications of this possibility are considered in the Discussion section, below. The equation for D is [15]Equation 7.

At all temperatures, CMR^O2can be determined by mass balance for oxygen [6]: Equation 8where CBF = regional cerebral blood flow (ml *symbol* g sup -1 *symbol* min sup -1); Sa^O2(unitless) = arterial blood hemoglobin oxygen saturation; Pa^O2(mmHg) = arterial blood oxygen partial pressure; and Sv^O2is unitless. We use Pa^O2= 300 mmHg in all simulations.

At normothermia, cerebral oxygen consumption varies with the interstitial oxygen partial pressure (Pin^O2) [14]: Equation 9where 2.0 mmHg specifies the interstitial oxygen partial pressure at which CMR^O2is 50% of the maximal (i.e., normal) value (CMR^O2max). Thus, as the interstitial oxygen partial pressure increases, CMR^O2asymptotically approaches its maximal (normal) value. Recently, Back and colleagues, using microelectrodes in normothermic rats, found cerebral interstitial oxygen partial pressure to equal 28 mmHg. [16]Substituting this value into Equation 9, at an interstitial oxygen partial pressure of 28 mmHg, CMR^O2is predicted to be 93% of its maximal (normal) value. We consider 93% to be a value clinically indistinguishable from normal.

An important issue is how the relation described in Equation 9changes with temperature. It is well known CMR^O2, and therefore CMR^O2max, decreases with decreasing temperature. For all simulations, CMR^O2max is that appropriate for temperature. Specifically, the relation between CMR^O2and temperature in adults and infants during CPB can be described by [1,3,6]Equation 10. In both cases, CMR^O2was determined under steady-state, "high-flow" conditions. Hence, we assume these CMR^O2measurements represent maximal normal values.

In contrast, the 2.0 mmHg term in Equation 9does not change with temperature. It represents the oxygen partial pressure that needs to be maintained at the cerebral mitochondria. As analyzed by Willford and colleagues, [6]the intracellular oxygen gradient needed to maintain mitochondrial oxygenation probably does not depend on temperature. In brief, neither the interstitial-intracellular surface area for diffusion nor the distance over which oxygen diffusion occurs are likely to depend on temperature. Furthermore, a decreased oxygen diffusion rate proportionately compensates for an increased solubility for oxygen at lower temperatures.

The algebraic equations (1-10) were solved simultaneously using the Systat (Evanston, IL) statistics program implementation of the Simplex direct-search method. [17]Solutions for CMR^O2were constrained to be less than or equal to CMR^O2max with a penalty function. [17]In particular, a nearly infinite value was added to the cost function whenever the constraint was not satisfied. Iterations were continued until CMR^O2and Sv^O2were calculated to within 0.01%. Cerebral blood flow was varied to achieve different CMR^O2.

The least reliable parts of the mathematical model are the oxygen solubility in brain, oxygen diffusion (Equation 6and Equation 7), and cerebral oxygen transport (Equation 8). We therefore repeated our analysis using a simplified model that did not include these terms. We assume that oxygen diffusion and transport are sufficient for Pv^O2= Pin^O2. Then, solving Equation 9for Pin^O2and substituting the result into Equation 2, Equation 11. This simplified model predicts the relation between Sv^O2and the temperature-appropriate CMR^O2without including oxygen diffusion, transport, or solubility in brain.

We simulated relations between temperature-appropriate normal CMR^O2versus Sv^O2for adults and infants undergoing CPB at 37 degrees Celsius, 27 degrees Celsius, or 17 degrees Celsius, with alpha-stat or pH-stat acid-base management. In each case, to create a decreasing Sv^O2, cerebral blood flow was varied (decreased). Figure 1shows the complete CMR^O2versus Sv^O2relation during three common CPB scenarios: 37 degrees Celsius adult, 27 degrees Celsius (alpha-stat) adult, and 17 degrees Celsius (alpha-stat) infant. Table 1shows predicted Sv^O2values below which CMR sub O²will decrease to less than 90% of temperature-appropriate normal CMR^O2, under all simulated conditions. The results are nearly identical using the simplified mathematical model (Table 1), which omitted terms for oxygen diffusion, transport, and solubility in brain. Collectively, these simulations show that as temperature and P⁵⁰decrease, Sv^O2must be maintained at progressively greater values to maintain temperature-appropriate CMR^O2. Limitation of oxygen transfer from hemoglobin to brain becomes a progressively more important determinant of CMR^O2as temperature decreases.

During normothermic CPB, decreasing cerebral blood flow results in decreasing Sv^O2. Nevertheless, Figure 1shows CMR^O2remains nearly normal until Sv^O2approaches very low values. During normothermic CPB, Sv^O2could decrease to less than 30% before CMR^O2would be significantly reduced (i.e., to < 90% of normal CMR^O2) (see Comments).

In stark contrast, the CMR^O2versus Sv^O2relation is markedly different during CPB at 17 degrees Celsius (alpha-stat). Decreasing cerebral blood flow results in decreasing Sv^O2. However, in contrast to normothermia, CMR^O2significantly decreases. At 17 degrees Celsius, CMR^O2varies with Sv^O2in an almost linear fashion once Sv^O2is less than 95%. Thus, during profoundly hypothermic CPB, impaired transfer of oxygen from hemoglobin limits oxygen availability to the brain. At 17 degrees Celsius, to maintain cerebral aerobic metabolism, cerebral blood flow must be maintained at values that preserve high Sv^O2.

Sv^O2(measured by jugular venous catheters or near-infrared spectroscopy) is commonly used as an index of the adequacy of cerebral oxygenation during CPB. It is widely believed that Sv^O2values greater than the normal (normothermic) value of 60% [3,18]indicate that, during hypothermic CPB, CMR^O2is being preserved with a relative excess of cerebral blood flow. [1,3,4]Our results suggest that this interpretation may be wrong. The CMR^O2versus Sv^O2relation changes with temperature. To maintain CMR^O2at temperature-appropriate values, the interstitial oxygen partial pressure must remain essentially constant at all temperatures. Consequently, the cerebral venous oxygen partial pressure must also remain essentially constant. However, hemoglobin Phosphorus⁵⁰decreases with temperature. As a result, to maintain constant cerebral venous oxygen partial pressure, Sv^O2must be maintained at progressively greater values as temperature decreases. Thus, appropriate clinical interpretation of Sv^O2during hypothermic CPB depends dramatically on the temperature of the patient.

During Normothermic ("Warm") Bypass. Our model predicts, that during normothermia, decreases in cerebral blood flow can be compensated for by increased oxygen transfer from hemoglobin to brain. This increased extraction can maintain CMR^O2until Sv^O2decreases to very low levels (< 30%). This prediction has been verified in a non-CPB baboon model of focal cerebral ischemia. [19]A 46% reduction of cerebral blood flow was associated with a decrease in Sv^O2from 60% to approximately 30%. However, CMR^O2did not change. Recently, Mutch and colleagues used graded CPB pump flow reductions in normothermic dogs to reduce cerebral blood flow (from 74 to 57 ml *symbol* 100 g sup -1 min sup -1). [20]Sv^O2decreased from 0.66 to 0.56, without a change in CMR^O2. [20]Thus, during normothermic CPB, increased oxygen off-loading from hemoglobin can compensate for significant decreases in cerebral blood flow. Within bounds, at normothermia, increased extraction can maintain normal cerebral oxygen metabolism.

During Moderately Hypothermic Bypass. During moderately hypothermic CPB, Sv^O2equals roughly 75% with alpha-stat management. [3]This greater than normal Sv^O2has been interpreted as indicating an excess of cerebral blood flow compared with metabolism (i.e., "luxury perfusion"). [3]On the other hand, the literature consistently shows that cerebral autoregulation is fairly well maintained under alpha-stat conditions at 27 degrees Celsius. [20,21]Teleologically, it seems strange the brain would autoregulate and maintain cerebral blood flow at an "excessive" level. Our model predicts the relatively high Sv^O2does not indicate the presence of "luxury perfusion." At 27 degrees Celsius, maintenance of an Sv^O2around 75% is, in fact, necessary to maintain CMR^O2at nearly (i.e., > 90%) its temperature-appropriate norm. Greater Sv^O2simply reflect a decreased Phosphorus⁵⁰and impaired oxygen transfer from hemoglobin to brain.

Nevertheless, within the customary range of systemic pressures and flows maintained during moderately hypothermic CPB, impairment of oxygen transfer is probably not clinically significant. Fedderson and colleagues decreased cerebral blood flow during hypothermic CPB (28 degrees Celsius) by decreasing mean arterial pressure. [22]Sv^Osub 2 was reduced from 71% to 55%. The model predicts that, at an Sv^O2of 55%, CMR^O2should be approximately 95% of the value of that with an Sv^O2of 71% (Figure 1). As predicted, Fedderson and colleagues found CMR^O2decreased from 1.0 to 0.9 ml O²*symbol* 100 g sup -1 *symbol* min sup -1. [22]Thus, at 27 degrees Celsius, it appears oxygen off-loading from hemoglobin is sufficient to maintain CMR^O2at temperature-appropriate values. This applies even when cerebral blood flow and Sv^O2are significantly decreased. Consequently, the 3 mmHg increase in hemoglobin Phosphorus⁵⁰resulting from pH-stat management is unlikely to meaningfully improve oxygen off-loading or increase CMR^O2under standard, moderately hypothermic CPB. Consistent with this latter hypothesis, both rabbit [23]and human [21]studies have found CMR sub O²to be equivalent under alpha-stat and pH-stat conditions at 27 degrees Celsius.

During Profoundly Hypothermic Bypass. During full-flow CPB at profound hypothermia (15-20 degrees Celsius), Sv^O2is typically 85% or greater. [1,2]This has been interpreted as indicating a state of "luxury perfusion." [1]However, our model predicts Sv^O2must be in this range to maintain CMR^O2at temperature-appropriate norms (Figure 1and Table 1). Profound hypothermia induces a decrease in hemoglobin Phosphorus⁵⁰, which causes marked impairment of oxygen off-loading from hemoglobin. Thus, only a relatively small amount of oxygen is transported from arterial blood to brain. Therefore, to maintain CMR^O2, Sv^O2must be relatively high, and cerebral blood flow must be large compared with CMR sub O². This has been observed in both human infants [1]and baboons. [24]Cerebral blood flow to CMR^O2ratios were three to four times greater during profound hypothermia than during normothermia.

Suppose cerebral blood flow were, in fact, truly "luxurious" at profound hypothermia. Then, one would expect large reductions in cerebral blood flow to not affect CMR^O2. This does not seem to be the case. Kern and colleagues, studying infants at 22 degrees Celsius, decreased CPB flow to decrease cerebral blood flow. [4]Sv^O2decreased from 70% to 63%. Our model would predict a small ([nearly equal] 4%) decrease in CMR^O2based solely on this change in Sv^Osub 2 (Figure 1). In contrast, Kern and colleagues found CMR^O2decreased by 45%. This suggests that in vivo, there may be an even greater limitation of oxygen transfer from hemoglobin to brain than predicted from this model.

Canine data at 20 degrees Celsius also suggest that actual Sv sub O²values required to preserve CMR^O2are probably greater than predicted. Miyamoto and colleagues [25]decreased CPB flow to decrease cerebral blood flow. Sv^O2decreased from 83% to 37%. Our model would predict a 12% decrease in CMR^O2based solely on this change in Sv^O2. In contrast, these investigators found CMR sub O²decreased by 35%. Therefore, Sv^O2is probably an even less accurate index of the cerebral blood flow and CMR^O2relation during hypothermic CPB than our model predicts. We discuss possible reasons for the discrepancy in the Limitations of the Model section, below. Nevertheless, the point is clear: although cerebral blood flow and Sv^O2are relatively high during profound hypothermia, there is probably a significant limitation in oxygen transfer. High Sv^O2values cannot be considered as indicating "luxury perfusion." Sv sub O²is probably not a reliable index of CMR^O2during hypothermic CPB.

One might anticipate pH-stat management, by increasing Phosphorus⁵⁰, might allow a greater CMR^O2for a given Sv sub O²during profound hypothermia. However, at 17 degrees Celsius, the effect of pH-stat management to increase fetal hemoglobin Phosphorus sub 50 ([nearly equal] 2 mmHg) is negligible compared with the overwhelming effect of hypothermia to decrease it (Table 1). Thus, pH-stat management seems unlikely to better preserve CMR^O2.

Our simulations show that, during profoundly hypothermic CPB, impaired oxygen transfer from hemoglobin to brain can greatly affect cerebral oxygen metabolism (Figure 1and Table 1). However, this impairment is likely to be clinically important only during low flow CPB (i.e., when cerebral blood flow is very low). [4]An issue of current controversy is the lower limit of systemic perfusion necessary to provide adequate cerebral perfusion during profoundly hypothermic CPB. Monitoring Sv^O2, by jugular bulb saturation or near-infrared spectroscopy, may be a way to assess whether cerebral oxygen requirements are being met. However, what our simulations show are that a "normal" Sv^O2(i.e., [nearly equal] 60%) does not indicate an adequate (i.e., normal) CMR^O2during profound hypothermia. The Sv^O2must be maintained at high levels (i.e., > 90%) to fully support cerebral oxygen consumption at 17 degrees Celsius.

Sv^O2desaturation (to < 50%) occurs commonly during rewarming after both moderately [18,26]and profoundly [1]hypothermic CPB. This desaturation has been interpreted as showing a pathologic mismatch between cerebral blood flow and CMR^O2. [26]Simultaneous measurements of CMR^O2and cerebral blood flow show Sv^O2desaturations can, at times, reflect inadequate cerebral oxygen delivery. [26]However, the increase in Phosphorus⁵⁰with rewarming might also result in Sv^O2reduction, simply because of increased ease of oxygen transfer from hemoglobin to brain (Table 1).

The extreme simplicity of the mathematical model is both its strength and weakness. Our model only considers the temperature and age dependence of cerebral metabolism and hemoglobin Phosphorus⁵⁰(Table 1). The advantage of this simplicity is the simulation results can be easily interpreted. The drawback is many important physiologic processes (e.g., shunt and capillary recruitment) are necessarily excluded. Therefore, it is important to appreciate predicted Sv^O2at given levels of CMR sub O²(Figure 1and Table 1) should be considered as rough estimates only. It appears that our model underestimates the limitation of oxygen transfer from hemoglobin to brain during profound hypothermia. Actual Sv^O2values required to preserve CMR^O2may be higher than those given in Table 1for profoundly hypothermic CPB. However, determination of these estimates was not the goal of the study. Instead, our aim was to determine the effect of hypothermia on the relation between CMR^O2and Sv^O2. By limiting our model, we could examine the pure effect of temperature on cerebral metabolism and hemoglobin oxygen dissociation.

The greatest simplification and potential limitation in our model concerns net oxygen diffusion from blood to brain. Both experimental results and theoretical analyses of cerebral oxygen transport are (almost exclusively) limited to the case of the normothermic adult. Therefore, we could not make this part of our model more realistic. [6]Nevertheless, we can make some predictions about how model error might affect our results. In the complex model, we assumed cerebral venous oxygen partial pressure equals the cerebral end capillary oxygen partial pressure. In the simplified model, we assumed the cerebral venous oxygen partial pressure also equals the cerebral interstitial oxygen partial pressure. Results from the detailed and simplified models closely matched (Table 1). Therefore, the detailed model essentially assumed that oxygen diffusion and transport are sufficient for cerebral venous oxygen partial pressure to equal the interstitial oxygen partial pressure. The quantitative model discrepancy at profound hypothermia suggests that under these conditions cerebral venous oxygen partial pressure probably exceeds interstitial oxygen partial pressure.

The discrepancy may relate to the diffusivity of oxygen in vivo. For example, as described in Materials and Methods, above, researchers have found it necessary to increase the oxygen diffusion term (13-fold) to match the level of oxygen transport measured experimentally at normothermia. [14]Suppose this correction term for oxygen diffusivity in vivo were to decrease with decreasing temperature. Then, for a given decrease in Sv^O2, CMR^O2would be less than those predicted in Figure 1and Table 1. However, there are several other explanations. For example, examining Equation 1and Equation 6, the model relation for the temperature dependence of oxygen solubility in brain could overestimate the solubility during profound hypothermia. Alternatively, nonequilibrium chemical reactions in erythrocytes [27]may limit oxygen delivery during profound hypothermia. Nevertheless, both models indicate that oxygen transfer from blood to brain is impaired during profound hypothermia. That clinical and experimental studies show that oxygen transfer may be worse than predicted only substantiates the model conclusions that luxury perfusion may not exist during hypothermic CPB.

Our model shows that hemoglobin Phosphorus⁵⁰strongly influences the relation between cerebral blood flow and metabolism. Consequently, experiments to test predictions of this model should consider differences between animals and humans in baseline Phosphorus⁵⁰and in the effect of temperature on Phosphorus⁵⁰. For example, the pig is commonly used in studies of cerebral physiology under profoundly hypothermic conditions. However, both normothermic Phosphorus⁵⁰, and the temperature dependence of Phosphorus⁵⁰, differ dramatically from that of humans. [28]Porcine Phosphorus⁵⁰decreases much less with temperature than does human Phosphorus⁵⁰. For example, predicted Phosphorus⁵⁰for adult humans at 17 degrees Celsius under pH-stat conditions equals 9.3 mmHg. At comparable conditions in pigs, Phosphorus⁵⁰equals 17.1 mmHg. [28]This latter value exceeds human Phosphorus⁵⁰at 27 degrees Celsius (Table 1), where, as we have discussed, impairment of oxygen off-loading from hemoglobin is probably not significant. Therefore, the relation between CMR^O2and cerebral blood flow under hypothermic conditions probably differs significantly between pigs and humans. Species, such as the pig, with values for hemoglobin Phosphorus sub 50 and/or Phosphorus⁵⁰temperature dependencies that differ from that of humans are not appropriate to test the predictions of this model.

We used computer simulation to predict the relation between changes in Sv^O2and CMR^O2. The relation is temperature dependent. Sv^O2per se may not be an accurate measure of CMR^O2during hypothermic CPB. Investigators of cerebral physiology during hypothermic CPB should consider these limitations when interpreting Sv^O2data.