## Abstract

An anatomically realistic model for oxygen transport in cardiac tissue is introduced for analyzing data measured from isolated perfused guinea pig hearts. The model is constructed to match the microvascular anatomy of cardiac tissue based on available morphometric data. Transport in the three-dimensional system (divided into distinct microvascular, interstitial, and parenchymal spaces) is simulated. The model is used to interpret experimental data on mean cardiac tissue myoglobin saturation and to reveal differences in tissue oxygenation between buffer-perfused and red blood cell-perfused isolated hearts. Interpretation of measured mean myoglobin saturation is strongly dependent on the oxygen content of the perfusate (e.g., red blood cell-containing vs. cell-free perfusate). Model calculations match experimental values of mean tissue myoglobin saturation, measured mean myoglobin, and venous oxygen tension and can be used to predict distributions of intracellular oxygen tension. Calculations reveal that ∼20% of the tissue is hypoxic with an oxygen tension of <0.5 mmHg when the buffer is equilibrated with 95% oxygen to give an arterial oxygen tension of over 600 mmHg. The addition of red blood cells to give a hematocrit of only 5% prevents tissue hypoxia. It is incorrect to assume that the usual buffer-perfused Langendorff heart preparation is adequately oxygenated for flows in the range of ≤10 ml · min^{–1} · ml tissue^{–1}.

- capillary network
- Langendorff isolated heart
- hypoxia

schenkman et al. (37–39) have developed an optical spectrographic method for measuring in situ myoglobin (Mb) saturation and an isolated perfused guinea pig heart preparation (40, 41) to study intracellular oxygenation and its relationship with cardiac function. These studies raised a central question: What does the tissue-averaged measurement of Mb saturation reveal about oxygen tension in the tissue? The average Mb saturation provides an incomplete picture because of the great heterogeneity in the tissue. Because the oxymyoglobin saturation curve is a nonlinear function of oxygen tension, inverting the function to find the value of oxygen tension associated with the tissue-averaged Mb saturation does not produce an unbiased estimate of the average tissue oxygen tension (24). Here, the intracellular oxygen distribution is estimated by fitting predictions made using a three-dimensional model of oxygen transport in the myocardium to experimental measurements reported previously (40) and in a companion paper (41).

Oxygen transport in tissue is recognized to depend on a tissue's microcirculatory structure and associated hemodynamics (9, 18, 19, 31, 35, 36). Accurate mathematical models of transport-advection, diffusion, permeation, and reaction of highly diffusive species such as oxygen depend on computationally expensive and memory-intensive computer programs built on detailed representations of the three-dimensional microscopic structures of vascular tissues (8). Such computer models have yielded more reasonable descriptions of the expected oxygen distribution in skeletal muscle (18, 19, 35) and in cerebral cortex tissue (36) than are available from simplified Krogh cylinder-based geometric models (11, 28). For example, predicted probability distributions of intracellular Po_{2} in the rat cerebral cortex differ markedly when calculations based on an axially symmetric Krogh cylinder are compared with those based on a three-dimensional microcirculatory structure obtained from electron microscopy (36).

The major drawback of these complex three-dimensional models is the associated heavy computational requirement that makes quantitative analysis of experimental data difficult. Applications such as optimization of model solutions for data fitting and parameter estimation, along with exploratory investigations of model behavior, are not practical when single model calculations require intensive calculation on high-speed computers (8, 18).

The code used here for simulation of generalized three-dimensional physiological transport is implemented on a standard desktop personal computer using the MATLAB (MathWorks; Natick, MA) computing environment. It uses fast and efficient numerical methods (8) custom tailored to physiological transport problems in complex three-dimensional geometries, such as those depicted in Fig. 1. The method is based on a discrete representation of the tissue on a cubic lattice. However, in contrast to standard finite-element and finite-difference schemes, each element in the discretized system can represent an inhomogeneous combination of tissue types in this method. Thus this inhomogeneous volume method need not interpolate features of the anatomy onto the computational grid. This feature provides accurate solutions using relatively coarse grids with a lattice spacing of 4 μm or larger. This grid spacing is more than twice as large as that used in similar studies (18, 19), resulting in less than 1/10 the number of required grid points on a three-dimensional lattice. Because computational times increase with the number of grid points, the inhomogenous volume method is efficient and fast compared with other methods.

With this simulation method, multiple solutions are obtained in a fairly reasonable amount of time (overnight) on desktop personal computers, providing fits to the experimental data on oxygen transport in isolated perfused guinea pig hearts reported by Schenkman et al. (40, 41). The mean Mb saturation (S̄_{Mb}) and venous Po_{2} predicted by the model of oxygen transport in the myocardium is matched to the average time courses measured for several experiments (40, 41) by finding an optimal value for a single free parameter: the rate of oxygen consumption within the cardiomyocytes. To our knowledge, this study represents the first application of this advanced class of three-dimensional oxygen transport model to parameter estimation and optimization-based fitting of experimental data.

## METHODS

*Model anatomy.* The assumed idealized microscale geometry of the myocardium is illustrated in Fig. 1. In the plane perpendicular to the major axis of the muscle fibers, cylindrical fibers are arranged on a hexagonal lattice (Fig. 1*A*). The center-to-center distance between nearest neighboring fibers is , and each muscle fiber diameter is assumed to be 25 μm. Capillary vessels, aligned parallel to the major axis of the fiber, are placed equidistant to three neighboring fibers. For the regular geometry depicted in Fig. 1, the spacing between adjacent axial capillaries is 16 μm. The continuous space occupied by neither capillary nor muscle fiber is treated as interstitium, which makes up ∼20% of the tissue volume (20). This idealized geometric arrangement results in a capillary density of 3,000 capillaries/mm^{2}, similar to measurements made in the rat [3,400–3,600 capillaries/mm^{2} (7, 34)], rabbit [3,300–3,400 capillaries/mm^{2} (44)], dog [3,100–3,800 capillaries/mm^{2} (6)], and guinea pig [2,000–4,000 capillaries/mm^{2} (17, 21)] left ventricle. Capillaries are assumed to have a constant diameter of 4 μm.

An interconnected network is constructed by placing 105 cross-connecting anastomoses between neighboring axial vessels in the 18-capillary network. The anastomoses have a length of 16 μm and branch at 90° angles from axial vessels, connecting nearest neighbor axial vessels. The positions of the anastomoses are determined randomly, resulting in a mean distance between anastomoses of 62 μm, with a SD of 42 μm. These morphometric statistics are similar to those reported by Kassab et al. (26) for the pig left ventricle. The resulting three-dimensional network is illustrated in Fig. 1*B*. In the image shown in Fig. 1*B*, the four central muscle fibers are depicted as solid gray cylinders.

As shown in Fig. 1, a region of dimensions 72 × 84 × 1,000 μm is used for the simulation of oxygen transport in the tissue. At the center axial position (*z* = 500 μm), a feeding arteriole is connected to one of the axial capillaries. Draining venules are placed at *z* = 166 μm and *z* = 833 μm. The axial vessels are topologically connected at the *z* = 0 μm and *z* = 1,000 μm positions, resulting in a network that is periodic in the *z* direction, with flow permitted across the *z* = 0 plane. This arrangement of one feeding arteriole and two draining venules approximates the observation that at the smallest vessel order, the ratio of the number of venules to arterioles is ∼1.75 (26).

The network construction results in fractional volumes of 0.042, 0.220, and 0.738 ml/ml tissue for the microvascular space, interstitial space, and parenchymal cell (myocyte) space, respectively.

The results presented below are based on a single realization of the network structure, with random cross-connecting anastomoses arranged as depicted in Fig. 1. By generating multiple realizations of the network geometry, it was found that the predicted mean tissue Mb saturation and venous outflow oxygen tension varied <2% between realizations. Thus simulation based on this single realization is judged sufficient for the analysis presented here.

*Flow simulation.* To calculate the flows throughout the network, it is assumed that vessel resistance is proportional to length divided by the fourth power of diameter. In doing this, any potential nonlinearities associated with constitutive properties of the perfusate [such as the effective viscosity of blood (32, 33)] are ignored. However, in this study, oxygen transport in a buffer-perfused system and in a system perfused with a red blood cell suspension with 5% hematocrit (fractional volume of red blood cells) is simulated. The approximation of constant effective viscosity is expected to be valid at these low hematocrits.

The distribution of flow throughout the network is computed by setting the pressure at the two output nodes to zero. Because the system is linear, the absolute value of the pressure drop between the input and output nodes is arbitrary. To simulate different measured flows, the flows in the network are scaled to match a given experimental flow, measured in milliliters per minute per gram of tissue. (Tissue density of 1.05 g/ml is assumed.) At a tissue flow of F = 10 ml/min (per ml tissue), the mean flows carried in axial vessels and cross-connecting vessels are 6.0 × 10^{4} and 5.6 × 10^{4} μm^{3}/s per vessel, respectively.

*Transport equations.* Oxygen transport in cardiac tissue is simulated using the capillary arrangement depicted in Fig. 1. It is assumed that the arterial perfusate enters the arterial inlet, flows throughout the network, and exits at the venous outlets. Perfusate velocities are calculated by assuming a linear pressure-flow relationship throughout as described above. For the intravascular region, the transport equations are expressed in terms of oxygen content (C_{T}), the total molar concentration [hemoglobin (Hb) bound plus unbound] of oxygen in blood vessels, and the oxygen partial pressure in the blood vessels (P_{b}). The molar concentrations of unbound oxygen in the interstitium and vascular space are assumed to be linearly related to partial pressures measured (in mmHg): C_{i} = α_{i}P_{i} and C_{b} = α_{b}P_{b}, where α_{i} and α_{b} are the oxygen solubilities in the interstitial fluid and buffer perfusate, respectively, and P_{i} is the oxygen partial pressure in the interstitial space.

Transport proceeds in the vessels according to the advection equation (1) where *t* is time, is the velocity of the perfusate, *p*_{w} is the permeability of the vessel wall, and *R* is the radius of the vessel. Passive permeation between the intra- and extravascular spaces occurs through the second term on the right-hand side, where the integral is over the circumference of the vessel. The velocity in a given vessel is set to the cross-sectional average, calculated as the flow in the vessel divided by the cross-sectional area. Intravascular concentration gradients are explicitly considered in only the mean flow directions (8). The effects of radial intravascular concentration gradients are modeled using a finite radial resistance to oxygen transport (16, 22, 45) rather than explicitly calculating radial gradients within vessels.

The total oxygen concentration in the vessel is expressed as (2) where H is the volume fraction of red blood cells in the perfusate, α_{c} is oxygen solubility in the red blood cells, and C_{Hb} is the monomeric concentration of oxygen-binding sites in the red blood cells. The oxygen saturation of Hb (S_{Hb}) is described by a Hill equation (3) where P_{50,Hb} is the half-saturation partial pressure of Hb and *n*_{H} is the Hill exponent.

Oxygen transport in the interstitial space is governed by the standard diffusion equation (4) with no binding or consumption terms, where *D*_{i} is interstitium O_{2} diffusivity.

Within muscle fibers, oxygen is consumed and binds with Mb. With the assumption that the binding and unbinding with Mb are rapid enough to maintain equilibrium saturation, the governing equation in the muscle fiber space can be expressed as (31) (5) where C_{f} and *D*_{f} are the free oxygen concentration and molecular diffusion coefficients, C_{Mb} and *D*_{Mb} are the bulk (total bound plus unbound) Mb concentration and diffusion coefficients, and *G* is the rate of O_{2} consumption in muscle fibers. The oxygen saturation of Mb is given by the first-order Hill equation (6) where α_{f} is the oxygen solubility in the muscle fiber space and P_{50,Mb} is the oxygen-Mb half-saturation value.

Oxygen is assumed to be consumed according to Michaelis-Menten kinetics (7) where *G*_{M} is the maximal rate of oxygen consumption and *K*_{M} is the effective Michaelis-Menten constant. The above equations assume that sites of oxygen binding and consumption are homogeneously distributed within cardiomyocytes.

The boundary conditions at the vessel wall-interstitial fluid interface and at the interstitial fluid-muscle fiber interface characterize the exchange between these regions. At the vessel wall interface, the following condition is applied (8) where **n̂**_{w} is the outward-pointing unit normal vector on the vessel wall surface and *p*_{w} is the permeability of the vessel wall. *Equation 8* corresponds to passive permeation across the vessel wall. A similar condition governs transport across the muscle fiber boundaries (9) where **n̂**_{f} is the outward-pointing unit normal vector on the muscle fiber surface and *p*_{f} is the fiber boundary permeability.

A glossary of variables used in the transport model is listed in Table 1.

*Parameter values.* The biophysical parameters for the oxygen transport model are listed in Table 2. Apart from *G*_{M}, the maximal Michaelis-Menten rate of O_{2} consumption in the myocytes, all parameters are obtained from literature and are not treated as adjustable. Estimation of *G*_{M} based on experimental data is discussed below.

The solubility parameters α_{b}, α_{c}, α_{i}, and α_{f} define the oxygen solubility in the distinct regions of the idealized tissue geometry. The value for solubility in saline at 37°C (1) is used for the oxygen solubility of the buffer. For interstitial fluid, the value for solubility obtained for plasma (12) is used. All other solubility values are obtained directly from the citations indicated in Table 2.

Because sheep's blood is the source of red blood cells for the red blood cell perfusate used in the companion study (41), the Hb half-saturation value reported for sheep blood, P_{50,Hb} = 43.5 mmHg (23), is used to fit data from the red blood cell experiments.

Estimates of oxygen diffusivity in muscle tissue vary from as low as 9.0 × 10^{–7} cm^{2}/s (4) to as high as 2.41 × 10^{–5} cm^{2}/s (10). Because the solubility value of α_{f} = 1.74 × 10^{–6} M/mmHg measured by Mahler et al. (29) is used for oxygen solubility in the muscle cell, Mahler et al.'s estimate for oxygen diffusivity in the cell, *D*_{f} = 1.45 × 10^{–5} cm^{2}/s, is used as well. This value is reasonably close to the molecular diffusion coefficient of oxygen in water, 2.4 × 10^{–5} cm^{2}/s. For the interstitial space, the diffusivity value for an aqueous solution is used. Estimates of Mb diffusivity in muscle are equally variable; here, the recent estimate of Jürgens et al. (24, 25), 2.2 × 10^{–7} cm^{2}/s, is adopted.

The oxygen-Mb half-saturation point is set to P_{50,Mb} = 2.39 mmHg, as measured by Schenkman et al. (37).

For microvessel wall permeability, the value reported Shah and Mehra (43), *p*_{w} = 250 μm/s for buffer-perfused capillaries, is used. This permeability value corresponds to a mass transfer coefficient of α_{b} *p*_{w} = 3.4 × 10^{–8} mol·cm^{–2}·s^{–1}·mmHg^{–1}. The permeability-surface area product associated with this value of microvessel permeability is *PS* ≈ 500 ml/min (per ml tissue). For the flows studied here, the ratio *PS*/F, where F is the flow, is in the range of 50 and higher. Thus the transport characteristics are flow limited (5) and are not sensitive to the assumed permeability value. In the absence of quantitative estimates of the muscle fiber wall permeability, the muscle fiber membrane permeability is set equal to the vessel wall permeability, *p*_{f} = *p*_{w}, which is likely to be an underestimate because the membrane is not expected to be a significant barrier to oxygen transport. However, because the fibers have greater total surface area than microvessels, *PS* for the fiber membrane is ∼2 × 10^{4} ml·min^{–1}·ml tissue^{–1}, a value that is very large compared with the diffusive transport resistance. Therefore, the results presented below are insensitive to the assumed value of *p*_{f}.

The Michaelis-Menten constant for oxygen consumption is set to the value used by Goldman and Popel (19), *K*_{M} = (0.5 mmHg)α_{f} = 0.87 μM.

## RESULTS

*Interpretation of data from buffer-perfused hearts.* To interpret the data over the range of arterial Po_{2} values studied by Schenkman et al. (40), the average time courses of flow, arterial and venous Po_{2}, and Mb saturation reported in Ref. 40 are considered. As described in the previous paper (40), the arterial pressure was held constant as the arterial Po_{2} was steadily ramped down over the course of 20 min while the hearts were paced at 240 beats/min. The experimental data curves plotted in Fig. 2, *left*, represent the means obtained from averaging the data from the seven experiments. With the use of the oxygen transport model, steady-state oxygen distributions were computed by integrating *Eqs. 1* and *5* until the time derivatives went to zero. The flow and arterial Po_{2} (curve labeled P_{A}) were set to the average values at equally spaced time points along the experimental curves; the adjustable parameter *G*_{M} was varied so that the model predictions of steady-state venous Po_{2} (curve labeled P_{V}) and mean Mb saturation S̄_{Mb} match the experimental values as closely as possible. Specifically, an estimate of *G*_{M} = 1.5 × 10^{–4} M/s was obtained by minimizing the normalized sum of square errors (10) at the baseline (*time 0* in Fig. 2, *B* and *C*). The estimated value was fixed and used to simulate the remainder of the time course. Thus the rate of oxygen consumption at all points in the tissue was calculated using Michaelis-Menten kinetics (*Eq. 7*) with a *K*_{M} of 0.5 mmHg and a fixed *G*_{M} value of 1.5 × 10^{–4} M/s over the 20-min time course.

For the seven buffer-perfused hearts, as reported by Schenkman (40), the measured mean baseline (*time 0*) flow, arterial and venous Po_{2}, and S̄_{Mb} are 7.67 ml/min per milliliter of tissue, 643 mmHg, 105 mmHg, and 0.72, respectively. When the flow and arterial Po_{2} are set to the measured mean values, and with the use of a consumption value of *G*m 1.5 × 10^{–4}M/s the model predicts values of venous Po_{2} and S̄_{Mb}, of 103 mmHg and 0.78. The corresponding probability distributions of intracellular Po_{2} and S_{Mb} are plotted in Fig. 3, *left*. As a measure of the fraction of the tissue that is normoxic, the volume fraction of the fibers for which C_{f} > *K*_{M} is computed. For this predicted oxygen distribution, the normoxic fraction is FN, meaning that the oxygen concentration is less than *K*_{M} (0.5 mmHg) for ∼20% of the tissue. Almost one-fifth of the tissue is hypoxic, even when venous Po_{2} is higher than 100 mmHg.

As illustrated in Fig. 2, the model reasonably reproduces the mean experimental data. However, P_{V} is consistently underestimated, whereas S̄_{Mb} is overestimated. A better fit to the S̄_{Mb} values may be obtained by increasing *G*_{M}; yet increasing *G*_{M} results in a greater underestimate of the P_{V} data. Equivalently, a closer match to the P_{V} data results in consistently higher predictions of S̄_{Mb}. Likely explanations for the observed differences between the model predictions and experimental measures of S̄_{Mb} are outlined in thediscussion.

*Interpretation of data from red blood cell-perfused hearts.* Plotted in Fig. 2, *right*, are the mean data obtained from the red blood cell-perfused hearts reported in the companion paper (41). As in the case of buffer-perfused hearts, arterial Po_{2} was ramped down over a period of 20 min, and flow, arterial and venous Po_{2}, and mean Mb saturation were measured. To analyze the experimental data, flow and arterial Po_{2} in the model were set to the average steady-state values at equally spaced time points along the experimental curves. Reasonable fits were obtained using *G*_{M} = 1.5 × 10^{–4} M/s, the same value, to within two significant digits, determined for the buffer-perfused hearts. In this case, the hematocrit was set to the experimentally determined value of 0.05. The resulting model simulation fits are plotted in the Fig. 2, *right* (open circles). Again, the P_{V} data are underestimated, S̄_{Mb} data are slightly overestimated, by the model simulations.

For the red blood cell case, the mean experimental baseline conditions for flow, arterial and venous Po_{2}, and S̄_{Mb} are 13.7 ml · min^{–1} · ml tissue^{–1}, 632 mmHg, 324 mmHg, and 0.94, respectively. The predicted values of venous Po_{2} and S̄_{Mb} are 283 mmHg and 0.99, and the predicted normoxic fraction is FN = 1.00, fully oxygenated. The predicted intracellular Po_{2} and S_{Mb} distributions for the baseline red blood cell case are plotted in Fig. 3, *right*.

*Predicted intracellular oxygen distributions.* In Fig. 4, the predicted intracellular distributions of Po_{2} and S_{Mb} for the model simulations are compared with the buffer-perfused and red blood cell-perfused data. In Fig. 4*A*, *left*, the probability distributions of intracellular Po_{2} are plotted for various levels of arterial oxygen content; 12 probability distribution functions are plotted, corresponding to 12 data points along the time course, as indicated in Fig. 2. The mean intracellular Po_{2} for each distribution is indicated by a solid bar. As the arterial oxygen content decreases, the distributions shift to lower values of Po_{2} and the means decrease. In Fig. 4*A*, *right*, the intracellular Po_{2} distributions for the red blood cell simulations are plotted.

The corresponding distributions of predicted intracellular Mb saturation are plotted in Fig. 4*B*. For the buffer-perfused simulations (Fig. 4*B*, *left*), the S_{Mb} distributions are bimodal over the entire range of arterial oxygen content. The mean value decreases as the arterial content decreases, and a larger fraction of the tissue becomes hypoxic. For the red blood cell S_{Mb} simulations (Fig. 4*B*, *right*), the distributions are unimodal, and tissue is entirely normoxic until the arterial oxygen content drops below ∼0.6 mM, corresponding to a P_{A} = 45 mmHg at H = 0.05.

Because the predicted Mb distributions are bimodal, with peaks at S_{Mb} = 0 and S_{Mb} = 1, for the buffer-perfused hearts, the normoxic fraction FN (volume fraction for which C_{f} > *K*_{M}) is linearly related to the mean saturation S̄_{Mb}. This behavior is illustrated in Fig. 5*A*, where the predicted FN is plotted versus S̄_{Mb}. For the red blood cell-perfused case, the predicted FN is higher than the buffer-perfused case at a given value of S̄_{Mb}. In Fig. 5*B*, the maximal rate of left ventricular pressure development, averaged over the seven buffer experiments and nine red blood cell experiments, is plotted versus measured S̄_{Mb} for the buffer-perfused hearts (40) and red blood cell-perfused hearts (41). The model prediction of normoxic fraction as a function of S̄_{Mb} (Fig. 5*A*) is quite similar to the trend in measured mechanical performance (Fig. 5*B* and Fig. 2*A* of Ref. 41).

## DISCUSSION

The major finding of the present study is that there is a substantial proportion (∼20%) of the cardiac tissue that is hypoxic (Po_{2} < 0.5 mmHg) when guinea pig hearts are perfused in the usual Langendorff manner with arterial buffer equilibrated with 95% oxygen (arterial Po_{2} > 600 mmHg). The addition of red blood cells to give a hematocrit of only 5% prevents the localized hypoxia.

Detailed three-dimensional modeling of oxygen transport in cardiac tissue elucidates important differences between the expected oxygen content in hearts perfused with red blood cell-containing and red blood cell-free perfusates. As observed experimentally (40) when cell-free crystalloid buffer is used as the perfusate, there is a high venous Po_{2} but relatively low mean tissue Mb saturation. Model analysis reveals that at the baseline conditions reported by Schenkman (40), substantial fractions of the tissue (∼20%) are hypoxic (Po_{2} < 0.5 mmHg) while the arterial Po_{2} is over 600 mmHg and the mean baseline venous oxygen tension is 105 mmHg. Yet, although one-fifth of the tissue is hypoxic, the mean oxygen tension within myocytes is predicted to be 223 mmHg for the average baseline steady-state conditions observed (40). The high degree of heterogeneity predicted by the model accounts for the observations of low Mb saturation along with high venous Po_{2}. As illustrated in Fig. 3, the intracellular Po_{2} can range between 0 and 600 mmHg when arterial Po_{2} is set to P_{A} = 680 mmHg.

In buffer-perfused hearts, a large fraction of the tissue can be hypoxic even when mean intracellular oxygen tension is >200 mmHg. Thus mean tissue oxygen tension is higher in hearts perfused with buffer at 95% O_{2} than in vivo blood-perfused hearts at an arterial Po_{2} of 90–100 mmHg. However, the fraction of the tissue that is normoxic and the mean Mb saturation is higher in the red blood cell-perfused hearts. Therefore, the low Mb saturation (72%) in guinea pig hearts perfused with buffer at high arterial Po_{2} (over 600 mmHg) reported by Schenkman et al. (40) is consistent with a high mean tissue oxygen tension (over 200 mmHg) and high venous Po_{2} (over 100 mmHg). However, the present results indicate that neither the mean nor the minimum tissue Po_{2} is equal to the venous Po_{2}.

Because of this behavior, care must be taken when interpreting either mean Mb saturation or mean tissue Po_{2} as a measure of tissue oxygenation level. The model helps to resolve observed differences in mean tissue Mb saturation and oxygen tension between buffer-perfused and red blood cell-perfused hearts because it accounts for the spatially heterogeneous distribution of intratissue oxygen and Mb. Krogh cylinder-based models, which account for distributed transport along the major axis of a capillary but do not consider the complex geometry of the microcirculatory network, fall short of reproducing the data plotted in Fig. 4.

Axially distributed Krogh cylinder models, with physiologically realistic values for capillary permeability, predict substantially higher values for mean tissue Mb saturation than those reported in Fig. 2*C* for both the buffer case and the red blood cell case. For example, the baseline measured venous Po_{2} of 105 mmHg is predicted from a simulation of buffer-perfused tissue transport using a three-region Krogh cylinder model with *G*_{M} = 1.25 × 10^{–4} M/s and input values of P_{A} = 643 mmHg and F = 7.67 ml · min^{–1} · ml tissue^{–1}. However, for these values, the Krogh model predicts that S̄_{Mb} = 0.99. By increasing *G*_{M} to 1.5 × 10^{–4} M/s, S̄_{Mb} is reduced to 0.82, with P_{V} = 35 mmHg. In analyzing the experimental results, the key difference between a Krogh model and the network model is that, in the Krogh model, hypoxia occurs only when P_{V} drops to values much lower that 100 mmHg. Because the network model effectively represents a heterogeneous distribution of pathway lengths and flows, regions in the tissue can be hypoxic while P_{V} remains relatively high.

The modeling results suggest that the observed mechanical performance in isolated guinea pig hearts is closely related to the predicted intracellular oxygen distributions. Figure 5*A* shows that the predicted normoxic fraction of the tissue is higher at any given value of S̄_{Mb} for the fits to the red blood cell-perfused case than for the fits to the buffer-perfused data. Similarly, the maximal rate of left ventricular pressure development is higher in the red blood cell-perfused hearts than in the buffer-perfused hearts for all observed values of S̄_{Mb} (Fig. 5*B*). The similarity between Fig. 5, *A* and *B*, suggests that *1*) the heterogeneous distribution of intracellular O_{2} influences physiological performance and *2*) neither the value of mean Mb saturation nor mean tissue Po_{2} reveals all of the information about intracellular O_{2}. At a given value of mean Mb saturation, the red blood cell-perfused hearts have a lower mean tissue Po_{2} than the buffer-perfused hearts, but a higher normoxic fraction, and show superior mechanical performance.

Because the mean flows under baseline conditions reported by Schenkman et al. (40, 41) are substantially higher for the red blood cell case than for the buffer case, the model was used to predict the behavior of buffer-perfused hearts at high flows. Steady-state oxygen distributions were predicted for F = 15 ml · min^{–1} · ml tissue^{–1} at P_{A}s of 640, 560, 480, 400, 320, 240, 160, and 80 mmHg. The resulting predicted mean Mb saturation and normoxic fraction (plotted as gray squares in Fig. 5*A*) reveal that at the high P_{A} values (560 and 640 mmHg), hypoxia may be prevented at this high flow. As observed for the predictions made for buffer-perfused hearts at lower flow, FN varies approximately linearly with mean Mb saturation in the tissue.

The model introduced here is based on an anatomically realistic description of the microcirculatory network and uses physiologically realistic values for all parameters, including permeability values. The simulation results reported here were obtained using only one free parameter, the Michaelis-Menten maximum rate of oxygen consumption *G*_{M}. Because the data from both red blood cell and buffer experiments were fit with the same value, *G*_{M} = 1.5 × 10^{–4} M/s, the agreement between the model predictions and experimental measures of S̄_{Mb} is promising. However, the model does not reproduce the observed data perfectly. Several short-comings may account for the consistent overestimate of mean Mb saturation by the model. Diffusive shunting of oxygen between arterioles and venules would reduce the effective Po_{2} difference between the feeding vessels and draining vessels at the microcirculatory level. This effect would tend to reduce the Mb saturation levels in the tissue for given levels of arterial and venous Po_{2}. However, direct diffusive shunting from arterial to venous blood is expected to be negligible at the high flows reported here and not likely to account for the observed 5–10% differences between measured and predicted S̄_{Mb} (27).

A greater contribution to the consistent differences between measured and predicted values is likely to be that left ventricular flow (and oxygen consumption) tends to be higher than the average for the whole heart. Thus the appropriate flow may be higher and the P_{V} lower for the region of tissue sampled with the optical probe than the whole organ measured values. In addition, the vascular network and the anatomic region used here (Fig. 1) is not large enough to entirely capture the heterogeneity of the whole organ system. It has been shown that at low flows, in skeletal muscle, over one-half of the oxygen delivered to the tissue may permeate out of arterioles (35). While not nearly so much permeation out of arterioles is expected at the high flows observed in the isolated heart, the oxygen tension at the arterial inlet to the capillary network is likely to be lower than that measured at the inlet from the aorta. Therefore, a model that includes a larger network (including vessels larger than the smallest capillaries) may be required to obtain an improved match to the experimental data.

Other sources of heterogeneity that are ignored in the present model may be key in making future improvements. For example, the microscopic anatomy of cardiac tissue is not as regular as the idealized geometry used in the present model (Fig. 1). A realistic model that accounts for irregularities in the shape of cardiomyocytes and the capillary network would result in a greater degree of heterogeneity in the intracellular oxygen distribution. Also, modeling nonlinearities associated with rheological properties of the perfusate may result in transport that is more heterogeneous than that of the present model. Another possibly significant factor influencing the heterogeneity of intracellular cardiac oxygenation, sites of oxygen consumption (mitochondria), are not homogeneously distributed throughout the intracellular space. The possible consequences of these additional sources of heterogeneity in the system will be explored in later work.

The recognition that there are substantial hypoxic areas in the usual buffer-perfused Langendorff heart preparation aids in the interpretation of data in many previous papers. The release of adenosine in such preparations is probably due to the hypoxic zones (14, 42). In addition, the observation that adenosine is produced by ecto 5′-nucleotidase acting on extracellular AMP (13) may be explained. Hypoxia may result in a leakage of AMP from cells that does not occur during normoxia.

In conclusion, an anatomically realistic simulation of data from buffer-perfused and red blood cell-perfused guinea pig hearts demonstrated a high degree of heterogeneity in buffer-perfused hearts. The simulations were performed using established parameter values from the literature, with only one free parameter: the Michealis-Menten maximum oxygen consumption *G*_{M}. A single value of the free parameter was chosen and used for all the simulations for both the buffer- and red blood cell-perfused conditions. In buffer-perfused hearts ∼20% of the tissue is hypoxic (Po_{2} < 0.5 mmHg) when the buffer is gassed with 95% O_{2} to give an arterial Po_{2} of over 600 mmHg. The addition of red blood cells to give a hematocrit of only 5% prevents the regional hypoxia. Also, results from model simulations at high flows suggest that the regional hypoxia may be prevented in buffer-perfused hearts at flows of 15 ml · min^{–1} · ml tissue^{–1} and higher. It is not correct to assume that the usual Langendorff preparation perfused with buffer equilibrated with 95% oxygen is adequately oxygenated, for flows in the range of ≤10 ml · min^{–1} · ml tissue^{–1}.

## DISCLOSURES

This study was supported in part by Whitaker Foundation Research Grant RG-00-0220 and by National Center for Research Resources Grant NCRR-1243.

## Footnotes

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