Published ahead of print on December 4, 2003, doi:10.1164/rccm.200305-702OC
© 2004 American Thoracic Society A Mathematical Model of Tissue–Blood Carbon Dioxide Exchange during HypoxiaPulmonary and Critical Care Medicine Division, George Washington University, Washington, DC Correspondence and requests for reprints should be addressed to Guillermo Gutierrez, M.D., Ph.D., Director, Pulmonary and Critical Care Medicine Division, George Washington University, 2150 Pennsylvania Avenue, N.W., Suite 5-404, Washington, DC 20037. E-mail: ggutierrez{at}mfa.gwu.edu
A two-compartment mass transport model of tissue CO2 exchange is developed to examine the relative contributions of blood flow and cellular hypoxia (dysoxia) to increases in tissue and venous blood CO2 concentration. The model assumes perfectly mixed homogeneous conditions, steady-state equilibrium, and CO2 production occurring exclusively at the tissues. The behavior of the model is compared with published data derived from an isolated dog hindlimb preparation subjected to either reductions in blood flow (ischemic hypoxia) or decreases in arterial PO2 (hypoxic hypoxia). The results of the model corroborate the experimental finding of greater venous and tissue CO2 concentrations with ischemic hypoxia than with hypoxic hypoxia. The model also predicts increases in tissue CO2 concentration under conditions of adequate O2 supply if CO2 transfer from tissue to blood becomes impaired. Consequently, from a theoretical perspective, it appears that increases in the tissue or venous blood CO2 concentration are neither sensitive nor specific markers of tissue dysoxia. The results of the model support the notion that changes in tissue and venous blood CO2 concentration during dysoxia reflect primarily alterations in vascular perfusion and not scarcity in cellular energy supply.
Key Words: CO2 production • oxygen delivery • oxygen consumption • tissue oxygenation • venous PCO2
Dysoxia, defined as a deficit in aerobic ATP production in relationship to cellular energy requirements (1), occurs when cellular energy flux is limited either by decreases in O2 supply or by impaired mitochondrial aerobic capacity (2). The tissue CO2 concentration ([CO2]t) increases during dysoxia as hydrogen ions generated by anaerobic sources of energy are buffered by bicarbonate (3). Consequently, measures of CO2 accumulation in tissues and in venous blood have been proposed as clinical markers of dysoxia (4). Techniques to monitor increases in peripheral CO2 include gastric mucosal PCO2, sublingual capnography, and the arteriovenous PCO2 difference (av The physiologic significance of increases in [CO2]t is controversial because CO2 can be generated by aerobic and anaerobic biochemical processes. During tissue dysoxia, a rise in [CO2]t may result from lactic acid buffering by bicarbonate ("anaerobic CO2") or from flow stagnation as CO2 produced during pyruvate oxidation ("aerobic CO2") accumulates in poorly perfused tissues (10).
Vallet and colleagues (11) explored this question by measuring venous PCO2 in isolated dog hindlimb preparations subjected to comparable decreases in O2 delivery (
Model Development CO2 transport from tissue to blood involves several complex, time-dependent, mass-transport processes. CO2 diffuses out of cells into the interstitial fluid where it is found dissolved, bound to bicarbonate as carbonic acid, and bound to proteins as carbamate. CO2 in blood is also distributed among these moieties both in plasma and inside the red blood cells (15). Convective transport by the circulation carries CO2 to the lungs for excretion into the atmosphere.
The mass transport model shown in Figure 1
is proposed as a first approximation to these processes. The model consists of two compartments: a tissue compartment and a vascular compartment. The following assumptions are made in the formulation of the model: (1) Blood is a homogeneous mixture of erythrocytes and plasma, (2) perfectly mixed compartments, (3) constant physical properties, (4) CO2 production (
The rate of change of [CO2] in the tissue compartment is a function of  CO2 and of the rate of CO2 transfer between the tissue and vascular compartments. This relationship may be described as follows:
For the vascular compartment, the rate of change of [CO2]v depends on blood flow per unit volume of tissue (
 O2 and O2 changed according to the nonlinear paradigm described by Cain (16) for resting animal preparations. A detailed development of the equations governing the input functions of the model is shown in the Appendix.
The equations of the model were programmed using Microsoft Excel. Model-predicted changes in the av
Model Input Parameters The input parameters used in the model simulations are shown in Table 1 . Most of the values in that table conform to the experimental conditions reported by Vallet and colleagues (11). These include arterial PCO2 of 34 mm Hg, arterial O2 saturation of 0.98 (corresponding to the reported arterial PO2 of 100 Torr), and limb respiratory quotient of 0.6. Because the hemoglobin concentration and arterial pH were not reported by Vallet and colleagues (11), values for dog blood of hemoglobin concentration of 150 g · L-1 and pH of 7.35 were assumed. Initial O2 and O2 values of 10 ml · kg-1 · min-1 and 4.8 ml · kg-1 · min-1, respectively, along with a critical O2 extraction ratio (ERO2) of 0.65 were obtained from Vallet and colleagues (11). These values resulted in a (O2)critical of 6.9 ml · kg-1 · min-1, a number equal to that reported by Vallet et al (11) for HH.
Model Input Functions Input functions for the model consist of decreases in  or arterial PO2 and the associated changes in O2, O2, and CO2. Figure 2
shows O2 as a function of for IH and as a function of arterial PO2 for HH. Decreases in and arterial PO2 replicate the conditions experienced by the experimental preparation of Vallet and colleagues (11). Decreases in during IH are linearly related to O2, whereas changes in PO2 during HH conform to the shape of the oxyhemoglobin dissociation curve. As shown in Figure 3
, the parameters of Table 1 result in input functions that closely approximate the experimental O2–O2 and O2–ERO2 curves. The discrete points in these graphs correspond to the experimental data of Vallet and colleagues (11) for the IH and HH experiments.
The model defines O2 as the sum of CO2 produced by aerobic and by anaerobic energy processes. Aerobic CO2 depends solely on O2 and the tissue's respiratory quotient. Anaerobic CO2 is a function of the parameter FCO2, a user-defined variable that relates anaerobic CO2 to the maximal rate of anaerobic ATP production (see Appendix). Alterations in FCO2 produce different rates of anaerobic CO2 when O2 is less than (O2)crit.
The effect of FCO2 on
Comparison of the Model Predictions to Experimental Data Figure 6 shows predicted changes in av  PCO2 for identical decreases in O2 for the IH and HH conditions. The experimental data of Vallet and colleagues (11) are shown as discrete points. Although both groups are subject to exactly equal degrees of tissue dysoxia, the model predicts a substantial increase in avPCO2 during IH, whereas this parameter remains constant during HH.
Model Sensitivity to FCO2 The sensitivity of the predicted [CO2]v and [CO2]t to FCO2 was tested by running multiple simulations with the initial conditions of Table 1 and a range of FCO2 values. Figure 7 shows changes in venous blood and tissue [CO2] for IH and HH as FCO2 varies from 0.01 to 1.00. The latter corresponds to maximal anaerobic CO2 production, the condition in which each mole of anaerobic ATP generates a mole of CO2. At any given FCO2, decreases in O2 result in different [CO2] for IH and HH. In the case of IH, the model predicts progressively larger increases in [CO2]v and [CO2]t with increasing FCO2. Conversely, in the case of HH, [CO2]v and [CO2]t increase only when FCO2 is more than 0.30, although to a much lesser degree than during IH.
Model Sensitivity to Kv The tissue CO2 diffusion coefficient Kv is another user-defined variable. It determines the rate of CO2 transfer from the tissues to the vascular space. According to Equation 5B (Appendix), the parameter Kv has no effect on venous [CO2], and whatever value is selected for Kv will not alter predicted changes in av PCO2. A value of 0.05 was chosen in these simulations as a reasonable estimate for Kv. This choice was based on the sensitivity analysis shown in Figure 8
for Kv values ranging from 0.10 to 0.001, with other input parameters being those shown in Table 1.
Conversely, Kv is an important determinant of tissue [CO2]. As shown in Figure 8, lower Kv values (corresponding to decreases in tissue CO2 diffusivity) result in increased initial [CO2]t for the IH and HH conditions, even though at that point the tissues are not dysoxic. In other words, according to the model, it may be possible to detect increases in tissue PCO2 resulting solely from decreases in CO2 diffusivity, irrespective of the degree of tissue dysoxia. As O2 decreases, [CO2]t diverges for the two conditions tested, with HH (dashed line) being consistently lower than corresponding IH values (solid line).
Many clinical and experimental studies support the notion that increases in venous PCO2 are temporally related to the development of tissue dysoxia. Large increases in mixed venous blood PCO2 have been measured during cardiopulmonary resuscitation in humans, a phenomenon associated with moderate increases in blood lactic acid concentration (17). Increases in mixed venous blood PCO2 under these extreme conditions of global ischemia have been attributed to buffering of anaerobically generated lactic acid by endogenous bicarbonate. Experimental animals undergoing cardiopulmonary resuscitation also show substantial increases in blood lactate concentration, mixed venous blood PCO2, and the venoarterial PCO2 difference (18).
The hypothesis that a rise in tissue PCO2 could serve as a marker of tissue dysoxia was advanced by Grum and colleagues (19) who measured increases in tissue PCO2 in the isolated dog intestine made dysoxic. Other experimental and clinical studies have confirmed Grum and colleagues findings (5–8). Implicit in these studies is the assumption that increases in tissue and venous PCO2 correlate closely with worsening degrees of tissue dysoxia. Conversely, it is possible that such increases in PCO2 may be the result of blood flow stagnation and the accumulation in tissue of "aerobic" CO2. Schlichtig and Bowles (10) attempted to separate the individual contribution of aerobic and anaerobic CO2 production to portal venous PCO2 and intestinal mucosal PCO2 in a canine model of bowel ischemia. They noted that increases in venous PCO2 were produced initially by oxidative phosphorylation in portions of the intestine perfused at very low flow, but once intestinal mucosal The results of the CO2 exchange model presented here approximate closely the experimental data of Vallet and colleagues (11), lending support to the notion that increases in tissue and venous blood PCO2 are related mainly to decreases in blood flow, not to tissue dysoxia.
Critique of the Model The assumption of perfect mixing for the tissue and blood compartment ignores the heterogeneous nature of CO2 transport from mitochondria to red blood cell. CO2 concentration gradients are likely to exist within regions of tissue and blood, information that could be deduced by more complex morphometric models of CO2 transport. This type of detailed analysis, one that would provide the basis for evaluating individual components of this complex transport process, is beyond the scope of this model. The purpose of this analysis was to develop relatively simple relationships to elucidate the mechanisms resulting in the differences noted in the HH and IH experiments. As such, a detailed description of the various components of the CO2 transport process would have added little to the information derived from a much simpler model. By considering blood as a homogeneous compartment, the model ignores the partitioning effect of red blood cells and of carbonic anhydrase on reaction velocity and vascular CO2 concentration. This assumption is justified by the steady-state nature of the model, in which all chemical and diffusive processes are assumed to have reached equilibrium. Thus, the model is not applicable to rapidly varying conditions in which time-dependent changes in CO2 are paramount. The time-independent nature of the model also justifies the assumption of nonpulsatile, constant blood flow. The assumption that CO2 production occurs exclusively in the tissue compartment presumes the negligible movement of H+ from cytosol to blood. This not a realistic assumption, as during dysoxic conditions, H+ will migrate from tissue to blood bound to lactate, where it may dissociate and bind to bicarbonate to produce CO2. As a first-order approximation, however, this is not an unreasonable assumption given the difficulties in establishing the rate of CO2 formation from lactate-bound H+.
The two-compartment, lumped-parameter model analysis has been used by others to describe tissue O2 exchange (20, 21) and appears well suited as a first-degree approximation of steady-state tissue CO2 transport. A more detailed, time-dependent, multicompartment model would have been very difficult to formulate and its results suspect because many of the input parameters for a model of such complexity are poorly defined under conditions of extreme O2 supply deprivation. On the other hand, this model has a distinct advantage of having few parameters to describe tissue CO2 transport. Among these parameters are
Increases in [CO2]v Occur Mainly during IH
The Behavior of
Under conditions of O2 supply deprivation, however, It may be possible, however, to estimate the fraction of the O2 debt paid by anaerobic sources of energy by considering that in the intestines approximately half of all H+ resulting from the hydrolysis of anaerobically produced ATP binds to bicarbonate to produce CO2 (3). This percentage may be even lower in skeletal muscle, given that the creatine kinase reaction also participates in the anaerobic production of ATP while consuming H+. Assuming a 50% conversion of anaerobic ATP into CO2, an FCO2 of 0.03 corresponds to an anaerobic ATP production equal to 6% of the potential O2 debt.
The Effect of Tissue Diffusion on [CO2]t As shown by Figure 8, decreases in Kv result in increased tissue CO2 concentration but have no effect on vascular CO2 content. Therefore, increases in [CO2]t produced by alterations in Kv will go unnoticed by measures of venous PCO2, although they may be detected by methods that measure tissue PCO2, such as gastric tonometry or sublingual capnometry. Another useful insight provided by Figure 8 is that tissues with low Kv may experience elevations in [CO2]t under fully aerobic conditions, the result of steep tissue–blood CO2 concentration gradients.
According to the foregoing analysis, clinical interpretation of the arteriovenous PCO2 gradient and tissue PCO2 concentration should be done with caution and always in the context of the clinical condition resulting in altered O2 supply. The model of tissue CO2 exchange described here corroborates the findings of Vallet and colleagues (11) and others (13, 14) in that venous and tissue CO2 increases during IH but not during HH. These results support the hypothesis that increases in tissue CO2 and in the arteriovenous PCO2 gradient reflect only microcirculatory stagnation, not tissue dysoxia. Thus, from the theoretical perspective provided by the model, increases in tissue and venous PCO2 are insensitive markers of tissue dysoxia and merely reflect vascular hypoperfusion. Moreover, an increase in tissue CO2 also appears to be a nonspecific marker of tissue dysoxia, as the potential exists for [CO2]t to rise under normoxic metabolic conditions, as the result of decreased tissue diffusivity of CO2. As such, this rise in [CO2]t represents a false-positive measure of tissue dysoxia, although it could well portend the initial stages of microvascular derangement, such as those seen in sepsis and in other pathologic conditions that may affect the diffusion of CO2 in tissue (22).
Mathematical Development of the Two-compartment CO2 Exchange Model Two compartment model equations. The rate of change of [CO2] in the tissue compartment shown in Figure 1 is a function of CO2 and of the mass transfer of CO2 between tissue and vascular compartments. This relationship may be described as follows:
For the vascular compartment, the rate of change of [CO2]v depends on blood flow per unit volume of tissue (
 t , v, and a denote the transformed variables. The initial [CO2] values for the tissue and vascular compartment, respectively, are [CO2]t(0) and [CO2]v(0). Solving these equations for [CO2]v(S) and [CO2]t(S) yields the following:
Multiplying the terms of Equations 3A and 3B by S and letting S
Input Functions
The model employs the critical O2 extraction ratio (ERO2crit), defined as ERO2crit =
The model computes values for
Implicit in Equation 6B is a progressive rise in ERO2 as
Modeling Decreases in
HH is simulated by maintaining blood flow constant at the initial value used for the IH simulations (
During the simulations, arterial PCO2 and pH are held constant at the initial values of 34 and 7.35 mm Hg, respectively (Table 1). These values correspond to a [CO2]a of 41.5 ml/dl. Venous pH is allowed to change according to the following expression:
Determination of
Computation of the Aerobic
O2 > O2crit
Computation of the Anaerobic
Under these conditions of O2 scarcity, the rate of cellular ATP production is the sum of mitochondrial and anaerobic ATP production. The latter is derived from glycolysis, the creatine kinase, and the adenylate kinase reactions:
It is nearly impossible to define a function to describe accurately the rate of CO2 formation from anaerobically derived H+. For the purposes of this model, an approximation to this function may be obtained by first calculating the maximal d(ATP)anaerobic/dt, defined as the rate of anaerobic ATP production needed to fulfill the O2 debt. An expression for maximal d(ATP)anaerobic /dt may be developed by noting that each molecule of O2 consumed during aerobic phosphorylation produces six ATP molecules (27). This relationship implies a switch by the tissues from free fatty acid to glucose consumption, a metabolic strategy resulting in a more efficient use of scarce O2 molecules (28).
Introducing the term FCO2, a user-defined variable that relates anaerobic
It should be noted that (
Determination of Venous Blood PO2 and PCO2
The value for PO2 corresponding to a given SO2 at pH = 7.40 and PCO2 = 40 mm Hg is calculated from the following relationship describing the oxyhemoglobin dissociation curve (29):
Blood PCO2 is calculated on the basis of the Henderson-Hasselbach equation (30):
Where the CO2 content of plasma, [CO2]plasma, is defined by Douglas (31) as
This article has an online supplement, which is accessible from this issue's table of contents online at www.atsjournals.org Conflict of Interest Statement: G.G. has no declared conflict of interest. Received in original form May 27, 2003; accepted in final form November 22, 2003
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