Overview of the processes of delivery: flow, transmembrane transport, reaction, and retention

The processes by which substrates reach the site at which they are consumed include convection, permeation and diffusion, and chemical reaction. Convective transport in the blood delivers material to the exchange region, mainly the capillaries, distributing material throughout the organ. The penetration of the capillary wall provides entry into the interstitium, through which diffusion over short distances brings solute to the membranes of the parenchymal cells in the heart, the sarcolemma of cardiac myocytes. For transport across the cell membrane, passive, facilitated, and active transport mechanisms may all be used. These all must precede intracellular reaction. Intracellular binding or chemical transformation tend to prevent tracer from leaving the cells; consequently, retention is associated with metabolism. The simplest mechanism of retention is dilution in the intracellular space, which is the largest part of most tissues. Freely diffusible substances wash out rapidly, illustrating that dilution alone does not prolong retention greatly and is unrelated to metabolism. Solute binding, as to a special receptor, tends to prolong retention, but is again unrelated to its metabolism. If metabolism cannot be predicted from retention, then how can it be assessed in vivo?

In the context of “metabolic imaging,” the goal is to obtain, by external detection, a measure of a metabolic event. By “event,” we usually mean that a substrate undergoes a chemical reaction producing a metabolite. There are hindrances to applying this direct approach: positron-emission tomographic imaging does not distinguish the tracer-labeled substrate from the tracer-labeled metabolite. Nuclear magnetic resonance (NMR) images with fine spatial resolution cannot currently be generated from specific chemical compounds (e.g., various phosphates), although their relative abundances of compounds can be obtained via spectroscopy. For positron-emission tomography, the key is in assessment of the kinetics of uptake, retention, and washout. With NMR, the kinetics of unnatural substances (shift reagents, ¹³C-labeled compounds, etc.) can likewise be studied; for native solutes in cycling situations, gating is a key in observing transformation of phosphate from one form to another; spectral analysis is useful in static or slowly changing situations. The question is how to use any of a variety of NMR probing styles to distinguish the normal from abnormal form in a spectrum obtained from a sufficiently well-defined, adequately small region.

For substances carried by blood, the kinetic processes leading to intracellular reaction in the heart are (1) delivery by flow, (2) transport across the capillary wall either between or across endothelial cells, (3) diffusion in the interstitium, and (4) transsarcolemmal transport, passive or carrier mediated.

Each substrate requires individual attention with respect to the identification of the transport mechanisms. Differences in molecular size, hydrophobic or hydrophilic nature, charge, binding affinities (to blood constituents, interstitium, cell surfaces), the abundance of specific transporters in the membrane, and their availability in the presence of competing nontracer substrate concentrations all have influences on kinetics.

Delivery via capillary flow

The myocardium is richly supplied with capillaries. There is about 1 capillary per muscle cell; they average 5 µm in diameter and are 17 to 20 µm apart. A photograph of silicon elastomer–filled capillaries in myocardium in which the cells are rendered transparent (figure 1) gives the impression that the capillary volume is high, but their actual volume is about 3.3% or 0.035 ml g⁻¹ of myocardium.¹

FIGURE 1.

Capillaries in dog myocardium. The heart was perfused with a white silicone elastomer (Microfil, Canton Biomedical Products, Boulder) and cleared by dehydration with ethanol and ethanol replacement with methylsalicylate. Capillaries average 5 µm in diameter, and number in cross section 3000 to 3300/mm², or about 1/myocyte. From an experiment by Bassingthwaighte et al.¹

Delivery is via flow of blood — red blood cells and plasma. Intracapillary hematocrits are lower than large-vessel hematocrits, but the ratio of erythrocyte flow to plasma flow through the organ is the same as in the aorta. This may also be true for individual capillaries. Duling et al.² observed that blood pulled through single skeletal muscle capillaries has the same hematocrit as that in the aorta. These statements are wholly compatible, the explanation being that red cell velocities are greater than plasma velocities. Indicator dilution curves obtained simultaneously after arterial injection of labeled red blood cells and plasma show that the red blood cells reach the venous outflow as much as 5% to 10% sooner than they reach plasma.³ The separation occurs only in vessels of less than 300 µm diameter.⁴ where erythrocyte velocities may he more than 150% of plasma velocity. These details are mentioned to provide a framework for the subtleties of exchange kinetics: 98% of the O₂ is carried in erythrocytes, but most substrates are carried solely in plasma. For O₂, a red cell marker such as ⁵¹Cr-hemoglobin is the appropriate intravascular reference tracer to use to measure the kinetics of tissue uptake and retention. For glucose, a large plasma protein is the appropriate intravascular reference. When a solute of interest is carried by both red cells and plasma, then both references are needed; in addition one must account for the exchange between them,⁵ preferably taking into account the differences in velocity.⁶

Dispersion in the vascular space is rapid and mixing in the whole body blood volume takes 5 to 15 min. When one records, by external detection, the time course of arrival of tracer into an organ or, by outflow detection, the dilution curve for an intravascular tracer, one observes considerable dispersion within a few seconds. Even a precise brief injection does not change this. The factors contributing to intravascular dispersion are (1) dispersion at the injection site by the force of the injection, (2) velocity profiles within vessels (faster centrally and slow at the wall), (3) some cross-stream mixing due to red cell rotation, (4) eddies at bifurcations, (5) variations in path length through parallel capillary beds, (6) fluctuations in flows with the cardiac cycle, and (7) regional variations in flows, even in normal organs. In the heart, velocity profiles and path length variations are important, but the dominant factor is the seventh on the list, regional variation in flows.

In normal awake baboons, the relative dispersion of regional flows per unit mass of myocardium (RD = standard deviation/mean) is about 32%,⁷ which implies, as seen in figure 2, a five- to six-fold range of flows from the lowest to the highest. The results obtained with microspheres are not substantially different from those obtained with a “molecular microsphere,” iododesmethylimipramine (IDMI), which is carried in plasma and over 99% extracted and deposited during transcapillary passage, ⁸ so that the breadth of the distribution can only be attributed to regional flow variation and not to the idiosyncrasies of microsphere rheology in branching networks.

FIGURE 2.

Distribution of flows in myocardium of left ventricle. Composite of microsphere distributions in 13 awake baboons. Each heart provided four to six flow distributions for a total of 53 distributions (some at rest, others during exercise and heating), which summarize 8800 estimates of regional flow. Data from King et al.⁷

The recognition of “underperfused regions” or “ischemic” regions is rendered difficult in the presence of this normal degree of heterogeneity. If regional flows were normally uniform, then a one-third or one-half reduction in flow to a region might be suspect. The basis for the normal heterogeneity is not known, but it is reasonable to suspect that flow is related to local metabolic needs. This being so, then regions of low flow cannot be considered necessarily to be ischemic, and in the normal heart presumably are not. Accordingly, ischemia is defined by a flow that is low relative to demand. For detection, one might use at least two indicators, one for flow and one for substrate or oxygen utilization. Such refinement may be useful in assessing the potentiality for cell recovery with reperfusion.

Intravascular convection and dispersion must be accounted for in even the simplest model. The major phenomena can be lumped into two groups, as follows: (1) The processes of intravascular dispersion are large in individual arteries and veins, the standard deviation divided by the mean being about 18%. Computationally this is a small problem since the system is mathematically linear and the arterial and venous processes can be convoluted together to provide an overall description of the dispersive processes in large vessel.⁹ (2) Each model must incorporate a set of several capillary-tissue units, the individual flows per unit volume tissue of which cover the range existent in the organ. Although axial dispersion within individual capillaries, spreading out a bolus within the stream, is not as great as in large vessels, the differences in transit times from unit to unit result in very substantial dispersion when summed of the outflow. Recruitment (opening additional vessels) and demobilization or derecruitment (closing down vessels), and twinkling of individual units (fluctuating or intermittent flow) add to the complexity. In skeletal muscle there is, with exercise, recruitment of previously “nonflowing” capillaries. Recruitment of cardiac capillaries is quantitatively less obvious: a part of “recruitment” may simply be some increase in flow in previously “slow-flowing” capillaries rather than an opening up of unused capillaries.

The total dispersion between inflow and outflow is more than the sum of these, and is due to the summation of the outflows from regions that are arrayed in parellel but have differing pathway mean transit times.

Transport across the capillary wall

The types of processes generally available for substrate, ion, or hormone transport in the heart are illustrated by those available for fatty acids (figure 3). The aqueous channel or cleft between endothelial cells (route 1) functionally about 10 nm wide, and therefore readily admits small molecules (glucose or free fatty acid), but strongly hinders passage of molecules the size of albumin or larger. Since fatty acids in plasma are 99 9% bound to albumin,¹⁰ there must be rapid dissociation of the complex to allow enough transport to account for the observed extractions of 30% to 50%.

FIGURE 3.

Possible mechanisms of transport of fatty acids across the capillary wall and cell membrane. Routes are (1) diffusion between endothelial cells, (2) lateral diffusion in membrane, (3) and (5) passive diffusion across cell membrane and inside cell, and (4) and (6) carrier-mediated transport across membranes.

The other routes of transport across the capillary wall involve the plasmalemma of the endothelial cell (routes 2, 3, and 4). For highly lipid-soluble substances, which may include fatty acids, the route is merely dissolution in and diffusion across the luminal plasmalemma, followed by diffusion into the endothelial cell cytoplasm and across it to the abluminal plasmalemma, where dissolution and diffusion again allow transfer into the interstitial fluid (route 3). The basement membrane offers no hindrance to molecular passage other than macromolecules.

A second membrane transport mechanism is carrier-mediated transport, either active or facilitated.¹¹ (Both are diagramed as route 4 in the illustration.) In uncomplicated carrier-transport the solute molecule combines with the carrier and is then translocated by one or another mechanism to the opposite side of the membrane, whereupon the solute molecule is released on the cytoplasmic side. The same process goes on at the luminal and abluminal surfaces. Such special transporters show the classic features of saturation at high substrate concentrations, inhibition by particular competitors or blockers, and countertransport, features that will be discussed below with respect to transsarcolemmal transport. Special transporters have been envisaged variously as carriers (of the ferry boat style) or integral membrane proteins that rotate or that surround a gated channel; all can be described by the same equations for solute transport.

An unlikely mechanism is lateral diffusion (a passive process; route 2), by which a lipid-soluble molecule dissolved in the luminal plasmalemma diffuses amongst the membrane constituents around the lateral border of the cell lining of the interendothelial cellular cleft to the interstitial surface, where it can dissolve in interstitial fluid. This route, hypothesized by Scow et al.,¹² is not likely to provide rapid transport because there are tight junctions between cells that prevent lateral fluidic movement of plasmalemmal constituents. Perhaps the situation is best suited for fatty acids, especially if the solubility in the membrane for fatty acids is high enough to compete with attachment to albumin, a situation that would facilitate transcapillary extraction as well as transport.

Endothelial cells and capillary walls differ greatly from organ to organ. In brain capillaries the endothelial cells are joined at their lateral borders by tight junctions that permit no passage of hydrophillic solutes; transport occurs only via passage through the plasmalemma. In the heart the clefts are wide enough (10 nm) to permit passage of molecules up to the size of albumin, although the steric hindrance is increasingly strong.¹³ Steric hindrance is a combination of size restriction to solute entry into a channel and friction with the wall. Evidence for it is that ratios of permeabilities of a larger to a small molecule are less than ratios of free diffusion coefficients. The hindrance is greater the larger the molecule, as illustrated in figure 4, which shows the reflection coefficient as a measure of hindrance. While there are a very few larger channels in small venules that may play a role in protein transport, these are unimportant for substrate transport.

FIGURE 4.

Steric hindrance in passage through aqueous channels between endothelial cells. As the reflection coefficient approaches unity, the permeability to the molecule goes to zero. Thin line represents calculations of Bean²⁸; thick line represents improved computation by Lightfoot et al.,²⁹ with points from Lewellen³⁰ and interpolated dashed line.

Endothelial cells differ in their ability to transport solutes across their membranes. Pulmonary endothelial cells extract serotonin at high rates, whereas serotonin appears to traverse the myocardial capillary wall by passive diffusion through the clefts. Likewise, while brain capillaries show high transport rates for glucose across the endothelial cells, the endothelial cells in the heart appear inactive with respect to d-glucose, its transport rate across the capillary wall being the same as for l-glucose, solely through the aqueous clefts.¹⁴ In contrast, adenosine is transported rapidly into endothelial cells.¹⁵

Other features of endothelial cells that may be of importance in transport and regulation are the communications between the cells.¹⁶ These are presumably through the regions of gap junctions where specific proteins surround aqueous channels allowing passage of small molecules from one endothelial cell to its neighbor. In fact, in arterioles there is some indication that there are junctions between endothelial cells and smooth muscle cells. Occasionally in the heart there are regions where the endothelial cells lie extremely close to myocytes, certainly within 50 to 100 Å, but I know of no demonstration of gap junctions linking the two cell types.

Interstitial diffusion

For the purpose of kinetic modeling, radial diffusion from the capillary outward toward the parenchymal cells is considered to be so fast that delays are inconsequential. Since the minimum distances between the capillary basement membrane and the myocyte is less than 1 to 2 µm, this sounds reasonable. The maximum is half the intercapillary distances of 17 to 20 µm. For sucrose to distribute 90% of the way toward equilibrium over this maximal distance would take about 0.1 sec, but for an average distance of 3 µm, the diffusional relaxation time is only about 10 msec. Since this diffusion time is short compared with times for transport across the capillary wall or the sarcolemma, one is usually justified in considering the region to be radially equilibrated.

In contrast, axial gradients in the interstitium may be quite large for any substance being consumed by the cells, and diffusional relaxation times axially might be about 1000 sec. This is long compared with transmembrane transport times, so axial gradients must be included in the modeling.

Transsarcolemmal transport

The only options here are passive transport via dissolution in the membrane (route 5) or passage via specialized transport mechanisms (route 6). For ions, the conductivity through special channels depends on the voltage across the membrane and the time history of voltage changes. Channels for sodium, potassium, and calcium show kinetics of the Hodgkin-Huxley type.^{17, 18} The intracellular concentration of sodium is kept low by the sodium pump (the sodium-potassium ATPase). Low intracellular calcium is maintained via the sodium-calcium exchanger and, to a lesser extent, by a calcium ATPase. Foreign ions such as rubidium and thallium can use the channels for K⁺ and Na⁺.

For substrates of metabolic interest such as glucose there are also specialized transporters. Their overall kinetic description¹¹ is the same when describing a carrier moving by diffusion back and forth across the membrane (the historic analog), an integral protein that rotates within the membrane (an unlikely possibility), or an integral protein that undergoes a conformational change to move a substrate molecule from one side to the other. The characteristics are as follows:

1. “Saturation ” or maximal transport rate at high concentrations. The maximum rate is limited by the number of transport rates and by the flux at each.

2. Competition. In tracer studies, high levels of nontracer native substrate inhibit the tracer transport by competing for the site of attachment to the transporter. Other molecules with features similar to the substrate molecule also compete for attachment to the transporter, or may block transport.

3. Countertransport. When there is a high concentration of substrate on only the cis side of the membrane and the mobility of the free transporter is less than that of the transporter-substrate complex, then the rate of reavailability of transporter on the cis side of the membrane is the rate-limiting step. Then, increasing the trans concentration of another substrate that combines with the transporter to give a more mobile form speeds up the reavailability of transporter from the trans to the cis side and, by making more transporter available in the cis side, enhances flux from cis to trans. In the heart, 3-O-methyl-glucose and glucose can serve as countertransporters for each other.¹⁹ The fact that countertransport is facilitating indicates that transmembrane conductance for the transporter-substrate complex is greater than that of the free transporter.

Since affinity of a transporter to only one substrate is unlikely physiologically and chemically, countertransport should be kept clearly in mind. Countertransport of a substrate with a metabolite (e.g., adenosine in, inosine out) is useful.

Since sugars in the heart do demonstrate counter-transport facilitation, the two-sidedness of the membrane needs to be taken into account. Thus the expressions of Gjedde and Christiansen²⁰ are incomplete. They account for bidirectional transport but assume independence of transport in the two directions, which is to state implicitly the assumption that free-transporter permeability is infinitely high, and therefore that there is no countertransport. Countertransport demonstrates conservation of the transporter molecule, and that the transporter is not equally accessible from both sides of the membrane simultaneously.

The evidence for glucose transporters in the heart, and for countertransport, is clear, but for fatty acids the evidence is not yet in. Diffusion of fatty acids is the probable mechanism, but Schwab et al.²¹ identified a permease for pyruvate and lactate on the hepatocyte membrane and it appears that these show competition with octanoate. Rose and Goresky²² show one set of data in which it appears that raising the lactate level diminishes the rate of transsarcolemmal uptake of palmitate. These experiments, although unverified, suggest that lactate and fatty acids might share a common permease or carrier, but the relevant experiments testing competition, inhibition, and countertransport have not been done.

Intracellular reaction

From the point of view of retention of tracer within the cell it is the first intracellular set of reactions that are most important. If the sequestering or transformation reactions within the cell occur at the same rate as the transsarcolemmal transport then there will be equal probabilities for reescape from the cell and for intracellular reaction. This holds even if the rate-limiting intracellular reaction is not the first of the series, so long as those reactions preceding the rate-limiting one are freely reversible.

Integration of the concepts into a comprehensive model

The safest approach to developing a model is to make it as realistic as one reasonably can and then to reduce it, if necessary, to more manageable form. Making the model realistic only to reduce it sounds inefficient to say the least, but it has the singular advantage that there is then a realistic reference model the behavior of which sets a standard against which simpler models can be assessed for validity and accuracy over a wide range of situations. The process of model formulation is in itself one that requires choosing amongst alternatives and consideration of details or concepts that might otherwise be negleted. Description of the events in mathematic terms necessitates simplifications from the outset, while forcing the choices to be made deliberately rather than unconsciously. However, it has the advantage of allowing checks for mass balance and other fundamental characteristics.

Model formulation

Following the concepts expressed above, a reasonable structure for a capillary-tissue unit is shown in figure 5. The capillary is considered to have gradients from inflow to outflow, allowing for arteriovenous differences for nontracer substrates and for tracer transients. The equations are in the style of those for a two-region model as described by Bassingthwaighte²³ and for a three-region model by Rose et al.,²⁴ but now can be solved for four regions, or for five including the erythrocytes.²⁵ While analytic solutions are formally available when axial diffusion is ignored, they are not useful, being too cumbersome to be computed accurately or swiftly: for the problem illustrated by the four-region model (without red cells), solution requires calculating a double convolution integral containing sets of Bessel functions, each containing the roots of a cubic expression that must be solved iteratively

FIGURE 5.

A model for capillary-tissue exchange in the heart. F = flow; PS = permeability surface area; G = consumption or reaction; V = volume. pl = plasma; ec = endothelial cell (l. luminal or a, abluminal); isf = interstitial fluid; pc = parenchymal cell or myocyte.

Use numerical model solutions rather than analytic solutions

The use of numerical solutions avoids these problems and improves the accuracy and speeds up the calculation by 10,000 to 10 million times or more if the data time course is long. The method of choice is to use a Lagrangian flow regimen, that is, a sliding fluid element in the capillary region, with the surrounding regions being stagnant.^{23, 25} The flow analog is essentially plug or piston flow, but is made more realistic with little additional computation by adding axial dispersion, which is equivalent to axial diffusion or Taylor turbulent diffusion. This approach assumes that radial diffusion within each region is rapid compared with the transmembrane transport processes, as has been verified for the heart.²⁶ The recent improvements in our numerical approach include use of analytic solutions for the radial exchanges at multiple positions along the capillary-tissue unit with each time step. Since the use of the local analytic result involves no more computation than does a first-order difference expression, it is just as fast, and being analytic allows the steps in axial position to be longer and fewer, thereby gaining yet more in speed.

Residue and outflow detection provide similar information

The outflow from a single capillary is not accessible; in standard animal studies coronary sinus catheterization is used to obtain the outflow dilution curves, but this is not so appealing for human studies and in any case does not give any regional information. A fundamental principle to keep in mind is that outflow and residue detection techniques give the same information in somewhat different form: the residual content is the amount that has entered the organ minus that which has escaped into the outflow. The potential advantage of high spatial resolution imaging is that one can analyze data such as the time course of the residue function for each of many small regions. The use of positron-emission monographic data acquisition in list mode and reconstruction with limited counts over short intervals results in a noisier image and noisy residue functions, but these can be rather accurately analyzed in spite of the noise with standard least squares optimizing techniques to fit the model solutions to the data.

Use the models to design the most informative experiments

To get a feel for the influences of the various parameters of the model on the forms of the model’s solutions, one may simply explore by changing values and observing the result. This is particularly fruitful when one can do it by observing graphic output on-line at a terminal. When the system is complex this has great value because the intuition of the investigator tends to fail easily on higher order systems.

A useful approach is to use sensitivity functions. A sensitivity function, S(t), is defined as the change in the model solution per unit change in the value of a parameter as a function of time. Accordingly, for a five-parameter model there are five sensitivity functions, each being a curve extending over the full duration of the model solution ²⁷ For a three-region model the sensitivity functions have their initial influences ordered identically to the order in which tracer injected into the inflow arrives at a barrier or enters a volume; that is, capillary permeability first and then interstitial volume, sarcolemmal permeability, cell volume, and finally intracellular reaction. The sensitivities have relative magnitudes depending on the absolute magnitudes: when capillary permeability is high then sensitivity to PSg is reduced and the sensitivity to cell permeability is increased. Likewise, sensitivity to intracellular consumption is increased when sarcolemmal permeability is increased.

The standard multiple indicator dilution technique, which has provided much of the background for the imaging techniques, makes use of reference tracers injected simultaneously for obtaining accurate parameter evaluation. An example is the use of an intravascular reference, tracer-labeled albumin, and an extracellular tracer, l-glucose, injected simultaneously with d-glucose, the transport of which is of interest.¹⁴ While with positron-emission tomography, unlike NMR, the time course of local residue functions of d-glucose can be observed, the time courses of the reference tracers cannot be simultaneously observed because one can record from only one tracer at a time with current technology. (There is a case to be made for considering the development of dual photon-emission computed tomography to obtain the desired simultaneity, but this would be at considerable sacrifice in spatial resolution.) With ¹¹C- or ¹⁵O-d-glucose without an l-glucose or albumin reference it is unlikely that sarcolemmal transport and reflux from cell to interstitium could be accurately distinguished from reaction.

Endothelial cells must be considered in both experiment design and the analysis. In the heart there appears to be so little uptake of glucose by the luminal surface of endothelial cells that they can safely be ignored, but this is not true of other solutes. How fatty acids are transported across endothelial cells, or between them, is not yet known. Adenosine is so stongly taken up by endothelial cells that very little intra-arterially injected adenosine actually reaches the myocytes. The data on adenosine handing by skeletal muscle illustrates another important point: tracer metabolites (inosine and hypoxanthine) emerge from the muscle with an initial peak that is delayed not more than a second behind the peak of the adenosine curve.¹⁵ Such rapid escape of metabolites indicates that transformation within endothelial cells may be more important than in myocytes. Multispecies modeling is required to handle the residue functions for the retained tracer since tracer exists in more than one form.