A mathematical analysis of the influence of perfusion heterogeneity on indicator extraction

Heterogeneities in myocardial flow and metabolism: exacerbation with abnormal excitation

Because regional myocardial blood flows are remarkably heterogeneous-with a 6- to 10-fold range of flows in normal hearts-and because the spatial profiles of the flows are stable over long periods and over a range of conditions, the relation between flows and other physiologic functions has been explored. Local fatty acid uptake and oxygen consumption are almost linearly related to the flows. Coronary network structure and hydrodynamic resistances give suitable flow heterogeneity but are thought to be a response to local needs rather than being causative. Presumably the cause is the need for adenosine triphosphate (ATP) synthesis locally, and therefore flows, substrate delivery, and oxygen utilization are driven primarily by local rates of ATP hydrolysis, mainly by contractile proteins. This hypothesis is by no means fully tested. Data on pacing dog hearts from different sites, on patients with left bundle branch block, and on unloading transplanted rat hearts, all point in the same direction: unloading ventricular muscle leads to diminished flow and exaggeratedly diminished glucose uptake. The mechanism is likely to be that discovered by Taegtmeyer and colleagues, namely, the expression of fetal genes in regions where the muscle is unloaded and particular metabolic enzymes and transporters are downregulated.

The Cardiome Project. An integrated view of cardiac metabolism and regional mechanical function

The goal, to develop a functioning three-dimensional computational model of the excitation, metabolism and contraction of the heart within three years, is one of the beginnings for the Cardiome Project. Our first stage will not be likely to provide highly accurate prediction of physiological behavior in general, but will be focussed so that it is adequate for at least three specific purposes: response to regional flow reduction, response to heart rate changes, and response to increased metabolic drive. We would like to make the model visualizable by three-dimensional viewing, with cross-sectional and transparency viewing approaches, illustrate the fiber directions, the arteries, the deformation with contraction and images of regional functions such as oxygen consumption, preejection strain, or lactate concentration. The display techniques developed by Hunter et al. and by McCulloch et al. would be excellent for such demonstration and teaching purposes, and should be attractive enough for public display. The Physiome Project is underway now, with growing government and private support. Now we are going from the era of molecular biology, led by the Genome Project, into a new era of integrative biology. The goal is to understand biology so deeply and so broadly that predictions about interventions can be made. Methods of experimentation and of diagnosis are critical to acquiring the data, and therefore in making the prediction, and so all aspects of our Society's efforts and interests are relevant to undertaking this grand challenge.

Blood flows and metabolic components of the cardiome

This is a plan for the first stage of The Cardiome Project. The cardiome is the representation, in quantitative, testable form, of the functioning of the normal heart and its responses to intervention. The goal is to integrate the efforts of many years into a comprehensive understandable scheme. Past efforts have spanned the fields of transport within blood vessels, the distributions of regional coronary blood flows, permeation processes through capillary and cell walls, mediated cell membrane transport, extra- and intracellular diffusion, cardiac electrophysiology, the uptake and metabolism of the prime substrates (fatty acid and glucose), the metabolism of the purine nucleosides and nucleotides (mainly adenosine and ATP), the regulation of the ionic currents and of excitation-contraction coupling and finally the regulation of contraction. The central theme is to define the coronary flows and metabolic components of a computer model that will become a part of a three-dimensional heart with appropriate fibre shortening and volume ejection. The components are: (a) coronary flow distributions with appropriate heterogeneity, (b) metabolism of the substrates for energy production, (c) ATP, PCr and energy metabolism and (d) calcium metabolism as it relates to excitation-contraction coupling. The modeling should provide: (1) appropriate responses to regional ischemia induced by constriction of a coronary artery, including tissue contractility loss and aneurysmal dilation of the ischemic region; (2) physiological responses to rate changes such as treppe and changes in metabolic demand and (3) changes in local metabolic needs secondary to changes in the site of pacing stimulation and shortening inactivation or stretch activation of contraction.

Computationally efficient algorithms for convection-permeation-diffusion models for blood-tissue exchange

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.

Kinetic analysis of blood-brain barrier transport of D-glucose in man: quantitative evaluation in the presence of tracer backflux and capillary heterogeneity

The present study deals with the analysis of double-indicator curves for blood-brain barrier studies. Two mathematical models which provide for the estimation of backflux of tracer from brain to blood in conjunction with heterogeneity of the cerebral capillary and large-vessel transit times were used for the analysis of D-glucose transport on the basis of cerebral venous outflow curves. The two models, non-mixed and well mixed, arise from differing assumptions regarding the effective region surrounding the capillary lumen. An approximate solution for the well-mixed model was developed to increase computation speed. Fourteen D-glucose outflow curves and their reference curves were obtained from nine patients and subsequently analyzed by the two models. Further, in five patients data were obtained under different physiological conditions: normal, decreased, and increased cerebral blood flow rates. The results support the appropriateness of the well-mixed model and heterogeneity of the cerebral capillary transit times. The median value for the average extraction was 0.18 and the median distribution space was 0.14. The latter value is similar to the brain extracellular space that has been estimated by other methods. The extraction values calculated from the peak of the venous outflow curves were significantly smaller than the whole-brain average extraction values estimated with the well-mixed model (0.157 vs 0.178, P less than 0.0005). In summary: (a) capillary heterogeneity is present in the human brain and changes with cerebral blood flow; (b) after crossing the blood-brain barrier, D-glucose distributes in the brain extracellular fluid; and (c) the extraction curve is significantly influenced by backflux.

Blood-tissue exchange via transport and transformation by capillary endothelial cells

The escape of solutes from the blood during passage along capillaries in 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 convection-permeation 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 parenchymal 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.

Transport of radioactive deoxyglucose by the isolated perfused human placental cotyledon as measured by the paired-tracer dilution method. Inconsistency of the transfer rates across the two trophoblast borders with the simultaneously measured rate of transplacental transfer

The uptake of [3H]2-deoxy-D-glucose ([3H]2DG) by the maternal and fetal borders of the trophoblast was measured in the dually perfused human placental cotyledon by use of the paired-tracer dilution method. The uptake was estimated from the maximum extraction of the tracer. The transcellular component of transplacental transport of [3H]2DG was measured simultaneously. From the rates of uptake by the two trophoblast borders the rate of transcellular transport was predicted. The rate predicted was significantly lower than the rate observed. This was taken to indicate that the paired-tracer dilution method based on the maximum extraction underestimated the rates of uptake.

Extraction of radioactive rubidium and D-glucose from the artificially perfused intervillous space of the human placenta

Extraction (Et) of 86Rb and [14C]D-glucose from the artificially perfused intervillous space of the human placenta was measured using [3H]L-glucose as a reference tracer. E. of 86Rb increased slowly from initial values near zero to a late maximum, which indicates that Et was greatly influenced by heterogeneity of indicator transit times through the intervillous space. The ascending part of the plot of -1n(I-Et) against time (t) of 86Rb was approximately linear. In each experiment the time corresponding to zero extraction was estimated by linear extrapolation of the plot. The mean of the times obtained in the individual experiments corresponded to the most frequent transit time of the indicators through the system outside the placenta. These observations suggest that 86Rb is taken up by the trophoblast from the entire space perfused. Under such conditions the rate of the trophoblast uptake can be estimated from the slope of the above plot. Unlike that of 86Rb, Et of [14C]D-glucose increased rapidly to a relatively steady level. This time course of Et may result from combined effects of transit time heterogeneity and rapid back-flux of the tracer.

Through the microcirculatory maze with machete, molecule, and minicomputer (1986 Alza lecture)

This is a personal historical essay on meanderings through the jungle of the microcirculatory swamp. Because one pretends that the wandering was purposefully exploratory, a few guideposts are placed at positions where one could discern blaze-marks from earlier wanderers, and the path cut a little wider along some of the routes that may be enjoyed by investigators wanting to put their blazes along more distant paths. Naturally, one starts by coming up the broad rivers, then branching into the little streams. Each of us chooses to seek a different "mother lode," up a different stream.

Model identification and estimation of organ-function parameters using radioactive tracers and the impulse-response function

Examination of the input-output events in functioning organs by the use of the impulse-response function (IRF) for a radioactive tracer is gaining more and more ground in nuclear medicine. This study summarizes the development of deconvolution analysis, laying special stress on the 'model-free' approach. System linearity and time invariance are discussed, and means of eliminating noise in IRFs originating from the input and organ-time-activity curves are outlined. Typical IRFs are illustrated by flow diagrams, time-domain curves, and their representation by Laplace transforms. The cases of nondiffusible and diffusible tracers as well as parenchymally extracted and transported substances are discussed. Methods for the derivation of models and for the calculation of physiologically important parameters from the IRFs are suggested.

Removal of adenosine from the rabbit pulmonary circulation, in vivo and in vitro

The contributions of pulmonary endothelial and blood cells to the removal of adenosine during a single passage through the rabbit pulmonary circulation were investigated. In the isolated, blood-free perfused rabbit lung in situ, single pass pulmonary removal of [3H]-adenosine injected into the pulmonary artery was accomplished by a low-affinity, saturable process yielding apparent Michaelis-Menten constants of Km = 498 +/- 36 microM and Vmax = 39 +/- 4 mumol/min per lung. Similar experiments in the intact, anesthetized animal, in vivo revealed a rapid, high-affinity removal of [3H]adenosine from plasma with apparent Michaelis-Menten constants of Km = 3.3 +/- 0.5 microM and Vmax = 6.5 +/- 2.4 mumol/min per lung. However, complete recovery of the injected adenosine was achieved upon lysing of cells from blood collected during a single transpulmonary passage of the substrate, indicating that blood cells were responsible for adenosine removal in vivo. The rate of disappearance of adenosine from plasma, observed by incubating the substrate with whole rabbit blood in vitro, was comparable to that found in vivo. We conclude that although, in the absence of circulating blood, rabbit lung is able to extract adenosine in vitro, this mechanism is of little significance in vivo where blood cells appear to be primarily responsible for such removal.

Microcirculatory mass transfer

The dynamics of Krogh tissue cylinders and related structures are critically reviewed to determine the roles of underlying transport and reaction processes, and the interactions between them. Emphasis is put on the gaining of insight through efficient scaling procedures, and discussion is organized about the time constants characteristic of these structures and the processes occurring in them. The basis of discussion is a new analytic solution technique which provides a formal description of indefinitely large arrays of parallel interacting elements and which includes both axial diffusion and uniform convection as well as first or zero order reaction within each element. The solution has the form of an expansion in the eigenfunctions of a non-self-adjoint differential operator. Comparison of model predictions with previously available results identifies the useful parameter ranges of analytic approximations and determines the accuracy of existing numerical procedures.

Changes in cardiac transcapillary exchange with metabolic coronary vasodilation in the intact dog

The effects of metabolic coronary vasodilation on transcapillary exchange in the heart were examined in anesthetized dogs by use of the multiple indicator dilution technique. Animals were studied under basal conditions and during coronary sinus pacing. To obviate adrenal medullary stimulation, catheters were placed in coronary artery and coronary sinus in a closed chest preparation. Plasma catecholamine concentrations were determined to provide an index of the level of sympathetic tone. Labeled albumin and sucrose were injected into the coronary artery, and outflow dilution curves were secured. Analysis of these, with a model incorporating throughput and returning components, and heterogeneity of capillary transit times, provided parameters reflecting flow, permeability-surface product for sucrose, and capillary heterogeneity. Coronary sinus pacing increased both heart rate and plasma norepinephrine values; in response, myocardial oxygen consumption increased, metabolic vasodilation occurred, and coronary flow increased. The capillary permeability-surface product for sucrose increased with the flow but tended to plateau at higher values, showing a saturation phenomenon. Capillary heterogeneity, present in control animals with low sympathetic tone, was grossly decreased during cardiac metabolic activation. The Crone-Renkin approximation formula for the permeability-surface product yielded values that were too low at low flows and values approaching those from the complete model at high flows. The findings indicate an integrated pattern of circulatory response to cardiac metabolic activation characterized by decreased resistance, increased flow, increased permeability-surface product, and reduced heterogeneity. The last two effects amplify the capacity of increased flow to deliver substrates to heart muscle cells.

Uptake of inorganic phosphate by the maternal border of the guinea pig placenta

Mechanism of uptake of inorganic phosphate (Pi) by the maternal border of the dually perfused guinea pig placenta was studied using the paired-tracer dilution technique with 32P-phosphate and 14C-sucrose being the tracers. Placental uptake of radioactive phosphate increased when the concentration of Pi in the perfusion fluid was reduced, and it decreased during anoxia, in presence of CN or during perfusion with low-Na or Na-free fluids. Iodoacetate was without effect. These observations are consistent with placental uptake of Pi being effected by a carrier mediated process dependent on external Na and, partly, on placental metabolism. Unidirectional flux of Pi from the maternal vascular space into the cell compartment of the placenta, estimated from the values of instantaneous extraction of 32P, correlated significantly with foetal weight. The flux per unit weight of the foetus was 17.0 +/- 1.0 nmol X min-1 g-1.

Capillary permeability of heterogeneous organs: a parsimonious interpretation of indicator diffusion data

1. Calculations of capillary permeability of perfused organs from indicator diffusion data are reviewed, and reconsidered in terms of two kinds of organ heterogeneity of capillary extraction. 2. New expressions are derived for calculating the organ permeability-surface area product PS. In the simplest circumstances.