A mathematical model of indicator extraction by the pulmonary endothelium via saturation kinetics

Modeling serotonin uptake in the lung shows endothelial transporters dominate over cleft permeation

A four-region (capillary plasma, endothelium, interstitial fluid, cell) multipath model was configured to describe the kinetics of blood-tissue exchange for small solutes in the lung, accounting for regional flow heterogeneity, permeation of cell membranes and through interendothelial clefts, and intracellular reactions. Serotonin uptake data from the Multiple indicator dilution "bolus sweep" experiments of Rickaby and coworkers (Rickaby DA, Linehan JH, Bronikowski TA, Dawson CA. J Appl Physiol 51: 405-414, 1981; Rickaby DA, Dawson CA, and Linehan JH. J Appl Physiol 56: 1170-1177, 1984) and Malcorps et al. (Malcorps CM, Dawson CA, Linehan JH, Bronikowski TA, Rickaby DA, Herman AG, Will JA. J Appl Physiol 57: 720-730, 1984) were analyzed to distinguish facilitated transport into the endothelial cells (EC) and the inhibition of tracer transport by nontracer serotonin in the bolus of injectate from the free uninhibited permeation through the clefts into the interstitial fluid space. The permeability-surface area products (PS) for serotonin via the inter-EC clefts were ~0.3 ml·g⁻¹·min⁻¹, low compared with the transporter-mediated maximum PS of 13 ml·g⁻¹·min⁻¹ (with Km = ~0.3 μM and Vmax = ~4 nmol·g⁻¹·min⁻¹). The estimates of serotonin PS values for EC transporters from their multiple data sets were similar and were influenced only modestly by accounting for the cleft permeability in parallel. The cleft PS estimates in these Ringer-perfused lungs are less than half of those for anesthetized dogs (Yipintsoi T. Circ Res 39: 523-531, 1976) with normal hematocrits, but are compatible with passive noncarrier-mediated transport observed later in the same laboratory (Dawson CA, Linehan JH, Rickaby DA, Bronikowski TA. Ann Biomed Eng 15: 217-227, 1987; Peeters FAM, Bronikowski TA, Dawson CA, Linehan JH, Bult H, Herman AG. J Appl Physiol 66: 2328-2337, 1989) The identification and quantitation of the cleft pathway conductance from these studies affirms the importance of the cleft permeation.

Detection of changes in lung tissue properties with multiple-indicator dilution

We evaluated the potential utility of a group of indicators, each of which targets a particular tissue property, as indicators in the multiple-indicator dilution method to detect and to identify abnormalities in lung tissue properties resulting from lung injury models. We measured the pulmonary venous outflow concentration vs. time curves of [14C]diazepam, 3HOH, [14C]phenylethylamine, and a vascular reference indicator following their bolus injection into the pulmonary artery of isolated perfused rabbit lungs under different experimental conditions, resulting in changes in the lung tissue composition. The conditions included granulomatous inflammation, induced by the intravenous injection of complete Freund's adjuvant (CFA), and intratracheal fluid instillation, each of which resulted in similar increases in lung wet weight. Each of these conditions resulted in a unique pattern among the concentration vs. time outflow curves of the indicators studied. The patterns were quantified by using mathematical models describing the pulmonary disposition of each of the indicators studied. A unique model parameter vector was obtained for each condition, demonstrating the ability to detect and to identify changes in lung tissue properties by using the appropriate group of indicators in the multiple-indicator dilution method. One change that was particularly interesting was a CFA-induced change in the disposition of diazepam, suggestive of a substantial increase in peripheral-type benzodiazepine receptors in the inflamed lungs.

Transit time dispersion in the pulmonary arterial tree

Knowledge of the contributions of arterial and venous transit time dispersion to the pulmonary vascular transit time distribution is important for understanding lung function and for interpreting various kinds of data containing information about pulmonary function. Thus, to determine the dispersion of blood transit times occurring within the pulmonary arterial and venous trees, images of a bolus of contrast medium passing through the vasculature of pump-perfused dog lung lobes were acquired by using an X-ray microfocal angiography system. Time-absorbance curves from the lobar artery and vein and from selected locations within the intrapulmonary arterial tree were measured from the images. Overall dispersion within the lung lobe was determined from the difference in the first and second moments (mean transit time and variance, respectively) of the inlet arterial and outlet venous time-absorbance curves. Moments at selected locations within the arterial tree were also calculated and compared with those of the lobar artery curve. Transit times for the arterial pathways upstream from the smallest measured arteries (200-micron diameter) were less than approximately 20% of the total lung lobe mean transit time. Transit time variance among these arterial pathways (interpathway dispersion) was less than approximately 5% of the total variance imparted on the bolus as it passed through the lung lobe. On average, the dispersion that occurred along a given pathway (intrapathway dispersion) was negligible. Similar results were obtained for the venous tree. Taken together, the results suggest that most of the variation in transit time in the intrapulmonary vasculature occurs within the pulmonary capillary bed rather than in conducting arteries or veins.

Efficient numerical methods for nonlinear-facilitated transport and exchange in a blood-tissue exchange unit

The analysis of experimental data obtained by the multiple-indicator method requires complex mathematical models for which capillary blood-tissue exchange (BTEX) units are the building blocks. This study presents a new, nonlinear, two-region, axially distributed, single capillary, BTEX model. A facilitated transporter model is used to describe mass transfer between plasma and intracellular spaces. To provide fast and accurate solutions, numerical techniques suited to nonlinear convection-dominated problems are implemented. These techniques are the random choice method, an explicit Euler-Lagrange scheme, and the MacCormack method with and without flux correction. The accuracy of the numerical techniques is demonstrated, and their efficiencies are compared. The random choice, Euler-Lagrange and plain MacCormack method are the best numerical techniques for BTEX modeling. However, the random choice and Euler-Lagrange methods are preferred over the MacCormack method because they allow for the derivation of a heuristic criterion that makes the numerical methods stable without degrading their efficiency. Numerical solutions are also used to illustrate some nonlinear behaviors of the model and to show how the new BTEX model can be used to estimate parameters from experimental data.

An interpretation of 14C-urea and 14C-primidone extraction in isolated rabbit lungs

We measured the venous concentration versus time curves of 14C-urea and 14C-primidone after rapid bolus injections of a vascular reference indicator, fluorescein isothiocyanate dextran, and one of the two 14C-labeled indicators in isolated rabbit lungs perfused with Krebs-Ringer bicarbonate solution containing 4.5% bovine serum albumin at flow rates (F) of 6.67, 3.33, 1.67, and 0.83 ml/sec and with nearly constant microvascular pressure and total lung vascular volume. When we calculated the permeability-surface area product, PS, from the 14C-urea and 14C-primidone outflow curves using the Crone model, the estimates of the PS product were directly proportional to F. However, the fractional change in the PS with flow was different for the two indicators. We also estimated the PS from the same 14C-urea and 14C-primidone data using an alternative model that includes perfusion heterogeneity, estimated in a previous study, and flow-limited and barrier-limited extravascular volumes accessible to both urea and primidone. This model was able to fit the outflow curves of either 14C-urea or 14C-primidone at all four flows studied with one flow-independent PS for each indicator. The ability of the new model to explain the 14C-urea and 14C-primidone data with no flow-dependent change in PS suggests that a change in PS with F estimated using other models such as the Crone model is not sufficient for capillary surface area recruitment.

Modeling blood flow heterogeneity

It has been known for some time that regional blood flows within an organ are not uniform. Useful measures of heterogeneity of regional blood flows are the standard deviation and coefficient of variation or relative dispersion of the probability density function (PDF) of regional flows obtained from the regional concentrations of tracers that are deposited in proportion to blood flow. When a mathematical model is used to analyze dilution curves after tracer solute administration, for many solutes it is important to account for flow heterogeneity and the wide range of transit times through multiple pathways in parallel. Failure to do so leads to bias in the estimates of volumes of distribution and membrane conductances. Since in practice the number of paths used should be relatively small, the analysis is sensitive to the choice of the individual elements used to approximate the distribution of flows or transit times. Presented here is a method for modeling heterogeneous flow through an organ using a scheme that covers both the high flow and long transit time extremes of the flow distribution. With this method, numerical experiments are performed to determine the errors made in estimating parameters when flow heterogeneity is ignored, in both the absence and presence of noise. The magnitude of the errors in the estimates depends upon the system parameters, the amount of flow heterogeneity present, and whether the shape of the input function is known. In some cases, some parameters may be estimated to within 10% when heterogeneity is ignored (homogeneous model), but errors of 15-20% may result, even when the level of heterogeneity is modest. In repeated trials in the presence of 5% noise, the mean of the estimates was always closer to the true value with the heterogeneous model than when heterogeneity was ignored, but the distributions of the estimates from the homogeneous and heterogeneous models overlapped for some parameters when outflow dilution curves were analyzed. The separation between the distributions was further reduced when tissue content curves were analyzed. It is concluded that multipath models accounting for flow heterogeneity are a vehicle for assessing the effects of flow heterogeneity under the conditions applicable to specific laboratory protocols, that efforts should be made to assess the actual level of flow heterogeneity in the organ being studied, and that the errors in parameter estimates are generally smaller when the input function is known rather than estimated by deconvolution.

Kinetics of serotonin uptake in the intact lung

The pulmonary endothelium is capable of removing and metabolizing serotonin (5HT) carried in the venous blood. Thus the lungs can influence the arterial concentrations of 5HT. In addition, there is evidence that changes in the lung uptake of 5HT might portend more serious endothelial damage wherein the barrier function of the endothelium is compromised. This has been a stimulus for finding methods for evaluating these endothelial functions. These methods must be able to distinguish changes in whole organ function which result from changes in perfusion (e.g., cardiac output, redistribution of flow, etc.) from those resulting from changes in the function of the endothelial cells. When a bolus containing radio-labeled 5HT and an unmetabolizable indicator which is confined to the vascular space is injected into the pulmonary artery, the pulmonary venous or systemic arterial concentration curves contain information about both the convective transport and endothelial cell process involved. Some of this information can be interpreted quantitatively using a simple mathematical model.

Effects of changes in pulmonary perfusion and surface area on endothelial ACE activity

Measurement of transport or enzymatic processes associated with pulmonary endothelial cells could provide unique information regarding the physiologic function of the microcirculation as well as data describing the biochemical integrity of these cells in acute lung injury. Since an important goal of such measurements is to quantify the kinetics of reactions of substrates (indicators) with the endothelium, it would be highly advantageous to account for (convective) phenomena related to blood flow and its distribution which also influence whole organ metabolism. In the present report, we describe studies that have utilized a nonlinear model of organ metabolism to estimate Vmax, the apparent maximal velocity and Km, concentration at one-half Vmax of angiotensin-converting enzyme (ACE) activity. Our hypotheses have been that (a) there is sufficient data to calculate apparent kinetic constants from indicator-dilution outflow curves after injection into the pulmonary circulation of a radiolabelled synthetic substrate for ACE, and (b) Vmax for ACE is a property related to the amount of enzyme located on the endothelial cells, and as such, should be directly proportional to perfused surface area of an organ. In isolated perfused rabbit lungs, when flow was increased over a two-fold range, but surface area was unchanged, neither Vmax nor Km for ACE activity was significantly altered. When indicator-dilution measurements of pulmonary ACE activity were made in lambs from 1-171 days of age there was a progressive increase in Vmax with no significant change in Km as a function of age. The increase in Vmax was closely correlated with an independent measure of surface area, carbon monoxide diffusing capacity, and a morphometric measure of capillary endothelial cell surface area (stereology at electron microscopic level) made at post-mortem. These experimental observations in whole organs support our hypotheses and predictions regarding the metabolism or removal of substrates from the circulation by means of a saturable process.

Kinetics of uptake and metabolism by endothelial cell from indicator dilution data

Certain substrates are rapidly taken up and/or metabolized by pulmonary endothelial cells in a saturable process. When such a substrate and a reference indicator are included in a bolus which is injected into the blood flowing into the lung, the extraction ratio, E(t), curves measured in the pulmonary venous outflow are asymmetric with respect to the reference indicator curve. If a sufficient quantity of substrate is included in the bolus, the extraction curves are concave upward. The shapes of the E(t) curves contain information regarding the chemical-physical processes which govern the fate of the substrate during its single passage through the lung. To interpret the shapes, computer simulations are used to illustrate separately the effects of the uptake of substrate into the cell, the returning flux of the substrate from the cell, the saturation phenomena of the extraction process, and the perfusion heterogeneity of the capillaries. Lastly, a simple analytical method for estimating the organ kinetic parameters of the extraction process is presented.

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.

Captopril removal by rabbit lung in vivo

Removal of [14C]captopril by the lungs of anesthetized rabbits was measured by the multiple indicator dilution technique. After coinjection of indocyanine green (ICG) and [14C]captopril into the jugular vein of anesthetized rabbits, serial blood samples were collected from the carotid artery and each was analyzed for its content of both substances. Percent removal (R) of captopril after its initial injection of 10 nmoles captopril/kg (calculated at the peak of the ICG outflow curve) was 40.2 +/- 2.5 (S.E.M.) and was significantly greater than R after a second injection of 10 nmoles captopril/kg (20.1 +/- 2.4) 1 hr later. Removal of 70 nmoles captopril/kg (5.8 +/- 3.0 after first injection, 6.4 +/- 2.2 after second injection) was significantly lower than R of 10 nmoles captopril/kg. During a single pulmonary passage of either dose of captopril, R was inversely related to the calculated fractional concentration of intravascular captopril. Pulmonary metabolism of the angiotensin converting enzyme (ACE) substrate [3H]benzoyl-Phe-Ala-Pro [( 3H]BPAP) was 70.1 +/- 1.7% in the absence of captopril, and was reduced significantly to 27.4 +/- 2.4% by 10 nmoles captopril/kg and 7.6 +/- 0.2% by 6 mumoles BPAP/kg. BPAP (6.4 +/- 0.6 mumoles/kg) significantly reduced R of the first and second injections of 10 nmoles captopril/kg but this effect was selective, since BPAP did not reduce pulmonary removal of [14C]serotonin. These data indicate that pulmonary removal of captopril in vivo is saturable and may primarily reflect binding of the drug to pulmonary endothelial ACE.

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.