Endothelial cells lining myocardial capillaries not only impede transport of blood solutes to the contractile cells but uptake and release substrates in competition to myocytes. To distinguish myocyte exchanges, one can use models to characterize such events using the multiple indicator dilution method. For analyzing tracer kinetic experiments one tries to account for all of the available observations simultaneously in order to impose maximal constraint on the model parameters. Where one can’t make observations in every tissue region in every experiment, one uses data from previous studies to constrain the parameters being evaluated. We avoid compartmental models for parameterization of data: they give biased estimates because the stirred tank analogy creates a sharp discontinuity between arterial concentration and that in the capillary. Axially distributed blood-tissue exchange models, using partial instead of ordinary differential equations account for the exchange and reaction events and for smoothly graded intracapillary concentrations. There are standing arteriovenous gradients for any substance that is consumed or formed in the tissue, Experiments using bolus injections of ¹⁵O-oxygen into the coronary inflow in blood-perfused rabbit hearts indicates that oxygen is flow-limited in its blood-tissue exchange. This means there are no effective barriers to exchange in the radial direction, that is, membrane permeation and radial diffusion are so fast that there are no significant gradients perpendicular to the axis of convection. Flow alone controls the rate of solute movement through the organ. These observations contradict the idea that the endothelium and the sarcolemma are barriers for oxygen transport. In dogs and humans, using positron emission tomography with a single breath of ¹⁵O-oxygen gas as input, the time-activity curves in regions of interest (ROI) in the myocardium are analyzed via a multipath model accounting for the reaction, oxygen to water. Fitting the composite signal gives separation of the ¹⁵O-oxygen and ¹⁵O-water signals; the fractional conversion times the flow times the inflow oxygen content gives the local oxygen utilization rate, rMRO₂. Estimates of rMRO₂ are roughly proportional to rMBF, regional myocardial blood flow. Flow and metabolism are both spatial fractals, with positive near-neighbour correlation. Diffusional exchanges among nearby regions smoothes the variation somewhat. Bassingthwaighte & Beard (1995) constructed a three-dimensional mathematical model of the coronary arterial network of the pig heart from the morphometric data of Kassab et al. (1993) so that it had the same segment lengths, diameters and connectivity properties as the real network. Assuming steady flow through the network, residue washout and outflow concentration-time curves were simulated for an impulse injection of tracer into the arterial inflow. Washout curves for tracer were power law type, with exponents the same as were found experimentally, 2.2 ± 0.3 for the water washout experiments versus 2.0 for the reconstructed networks. Also, the regional flow distribution showed fractal correlation structure with the same fractal dimension as found in animal studies. While the spatial fractal nature of the flow distribution is explicable in terms of the morphometry of the network, this does not explain why the normal heart shows an 8- to 10-fold range of flows in all species studied and, further, why this heterogeneity is basically stable. Tracer studies on purine nucleoside exchanges show rates of endothelial uptake and transformation similar to those in myocytes: endothelial cells assuredly influence delivery of adenosine to both smooth and cardiac muscle cells. During hypoxia endothelial cells can have a role in purine salvage, taking up adenosine and inosine released by nyocytes in the transient and returning purine as hypoxanthine (Schwartz et al. 1999). This may be particularly important when purine nucleotides are released, losing their high-energy phosphates to interstitial phosphatases. NMR data from isolated rabbit hearts at low perfusion levels (Gustafson & Kroll, 1998) help to reveal the kinetics of these interactions. With a second underperfusion after a brief recovery, much less adenosine and inosine reaches the coronary effluent, the result being attributable to a combination of events including down-regulation of 5’-nucleotidase in myocytes as well as the interplay between the two cell types. Our modelling therefore must include these cell-to-cell interactions to explain the multiple data sets of different types.