The escape of solutes from the blood during passage along capillaries hi heart and skeletal muscle occurs via diffusion through clefts between endothelial cells and, for some solutes, via adsorption to or transport across the luminal plasmalemma of the endothelial cell. To quantitate the rates of permeation via these two routes of transport across capillary wall, we have developed a linear model for transendothelial transport and Illustrated its suitability for the design and analysis of multiple simultaneous indicator dilution curves from an organ. Data should be obtained for at least three solutes: 1) an intravascular reference, albumin; 2) a solute transported by endothelial cells; and 3) another reference solute, of the same molecular size as solute 2, which neither binds nor traverses cell membranes. The capillary-tissue convectionpermeation model is spatially distributed and accounts for axial variation in concentrations, transport through and around endothelial cells, accumulation and consumption within them, exchange with the interstitium and parenchyma! cells, and heterogeneity of regional flows. The upslope of the dilution curves is highly sensitive to unidirectional rate of loss at the luminal endothelial surface. There is less sensitivity to transport across the antiluminal surface, except when endothelial retention is low. The model is useful for receptor kinetics using tracers during steady-state conditions and allows distinction between equilibrium binding and reaction rate limitations. Uptake rates at the luminal surface are readily estimated by fitting the model to the experimental dilution curves. For adenosine and fatty acids, endothelial transport accounts for 30-99% of the transcapillary extraction. (Circulation Research 1989;65:997-1020) T he amount of a solute transported via aqueous channels across the capillary wall in skeletal muscle or in heart is limited by the channel dimensions and the numbers of channels. The permeabilities for hydrophilic solutes are nearly in proportion to the free diffusion coefficients in water. Thus, if this were the only avenue of traversal of the barrier, the rates of transport for substrates and humoral agents would be quite low, as they are for L-glucose and for sucrose. This would imply that some substrates would be seriously barrier-limited in their ability to reach extravascular receptor sites or sites of metabolic utilization. It is common to find that facilitated or active transmembrane transport mechanisms are available to enhance transport rates. It is the purpose of this study to demonstrate a combination of experimental designs and analytical approaches whereby such

Analysis of data on tissue depositions obtained by positron tomographic or NMR imaging, or of multiple tracer outflow dilution curves, requires fitting data with models composed of aggregates of capillary-tissue units. These units account for heterogeneities of flows and multisolute exchanges between longitudinally distributed regions across capillary and cell barriers within an organ. Because the analytic solutions to the partial differential equations require convolution integration, solutions are obtained relatively efficiently by a fast numerical method. Our approach centers on the use of a sliding fluid element algorithm for capillary convection, with the time step set equal to the length step divided by the fluid velocity. Radial fluxes by permeation between plasma, interstitial fluid, and cells and axial diffusion exchanges within each time step are calculated analytically. The method enforces mass conservation unless there is regional consumption. Solution for a 2-barrier, 3-region model, accurate to within 0.5%, are 100 to 1000 times faster than the corresponding, purely analytic solution, and over 10,000 times for a 4-region model. Applications include multiple indicator dilution studies of kinetics of transcapillary exchange and positron emission tomographic studies of the mechanisms of substrate transport into cells of organs in vivo.

The deposition of microspheres in small tissue regions is not strictly flow dependent. In comparison with the soluble flow marker 2-iododesmethylimipramine (IDMI), deposition of 16.5-,um microspheres was mildly but systematically biased into high flow regions of rabbit hearts (Bassingthwaighte JB, Malone MA, Moffett T-C, King RB, Little SE, Link JM, Krohn KA. Am J Physiol 1987;253(Heart Circ Physiol 22):H184-H193). To examine the possibility of bias in larger hearts, a similar study was undertaken in sheep. 141Ce-and '03Ru-labeled 16.5 -,um microspheres in one syringe and 1251-and '311-DMI in another syringe were injected simultaneously into the left atrium of five open-chest sheep while obtaining reference blood samples from the femoral artery. In six other sheep, one microsphere type and one IDMI were used. Hearts were removed 1 minute after injection, cut into approximately 254 pieces averaging 217 mg, and regional deposition densities calculated for each tracer from the isotopic counts. Correlations in the five animals between the two differently labeled IDMIs and between the two microspheres were both .0.98. In all 11 sheep, scatter plots of microsphere deposition densities versus IDMI densities showed that differences between microspheres and IDMI had substantially more scatter (0.84

Nonlinear least squares optimization is used most often in fitting a complex model to a set of data. An ordinary nonlinear least squares optimizer assumes a constant variance for all the data points. This paper presents SENSOP, a weighted nonlinear least squares optimizer, which is designed for fitting a model to a set of data where the variance may or may not be constant. It uses a variant of the Levenberg-Marquardt method to calculate the direction and the length of the step change in the parameter vector. The method for estimating appropriate weighting functions applies generally to 1-dimensional signals and can be used for higher dimensional signals. Sets of multiple tracer outflow dilution curves present special problems because the data encompass three to four orders of magnitude; a fractional power function provides appropriate weighting giving success in parameter estimation despite the wide range.

We used the multiple-indicator-dilution technique to observe the capillary transport of adenosine in isolated Krebs-Henseleit-perfused guinea pig hearts. Tracer concentrations of radiolabeled albumin, sucrose, and adenosine were injected into the coronary inflow; outflow samples were collected for 10-25 s and analyzed by high-performance liquid chromatography (HPLC) and by gamma- and beta-counting. The albumin data define the intravascular transport characteristics; the sucrose data define permeation through interendothelial clefts and dilution in interstitial fluid (ISF). Parameters calculated from adenosine data include permeability-surface area products for endothelial cell uptake at the luminal and abluminal membranes and intraendothelial metabolism. We found that in situ endothelial cells avidly take up and metabolize adenosine. Tracer adenosine in the capillary lumen is twice as likely to enter an endothelial cell as it is to permeate the clefts. There was no adenosine in the arterial perfusate. Under control conditions, the steady-state venous adenosine concentration was 3.6 +/- 0.8 nM, which from the flow and the parameters estimated from the tracer data gave a calculated ISF concentration of 6.8 +/- 1.5 nM. During dipyridamole infusion (10 microM) at constant pressure, the cell permeabilities went essentially to zero, whereas the venous adenosine concentration increased to 44.0 +/- 12.6 nM, giving an estimated ISF concentration of 191 +/- 53 nM. With constant flow perfusion, venous concentration during dipyridamole infusion was 30.9 +/- 6.3 nM, and estimated ISF concentration was 88 +/- 20 mM. We conclude that in this preparation, at rest, the ISF adenosine concentration is about twice the venous concentration and the ISF adenosine concentration increases with dipyridamole administration.