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

When a bolus containing a nonpermeating indicator and an indicator which permeates the endothelial cell membrane by a saturable process is injected into the blood flowing into the lung, the instantaneous extraction ratio curves measured in the pulmonary venous outflow are asymmetric with respect to the nonpermeating indicator curve. If the bolus contains a sufficient quantity of the permeating indicator that the capillary concentration begins to saturate the transfort mechanism, the extraction ratio curves are concave upward as well. The purpose of this study was to determine whether a mathematical model which represents endothelial extraction by Michaelis-Menten kinetics could explain the time variation in the instantaneous extraction ratio curves. The venous concentration curves were assumed to be the result of the endothelial transfort and distributed capillary input and transit times. In addition, we evaluated a method for estimating the kinetic parameters (K_m and V_max) of the saturable transfort process in such an organ. The results of simulations indicate that the important features of the data can be reproduced by the model, and that useful estimates of the kinetic parameters will be obtained from linear multiple regression analysis of the venous concentration curves if the standard deviation of the capillary input time distribution is not less than that of the capillary transit time distribution.     

Impulse-response function of splanchnic circulation with model-independent constraints: theory and experimental validation

Modeling physiological processes using tracer kinetic methods requires knowledge of the time course of the tracer concentration in blood supplying the organ. For liver studies, however, inaccessibility of the portal vein makes direct measurement of the hepatic dual-input function impossible in humans. We want to develop a method to predict the portal venous time-activity curve from measurements of an arterial time-activity curve. An impulse-response function based on a continuous distribution of washout constants is developed and validated for the gut. Experiments with simultaneous blood sampling in aorta and portal vein were made in 13 anesthetized pigs following inhalation of intravascular [15O]CO or injections of diffusible 3-O-[11C]methylglucose (MG). The parameters of the impulse-response function have a physiological interpretation in terms of the distribution of washout constants and are mathematically equivalent to the mean transit time (T) and standard deviation of transit times. The results include estimates of mean transit times from the aorta to the portal vein in pigs: T = 0.35 +/- 0.05 min for CO and 1.7 +/- 0.1 min for MG. The prediction of the portal venous time-activity curve benefits from constraining the regression fits by parameters estimated independently. This is strong evidence for the physiological relevance of the impulse-response function, which includes asymptotically, and thereby justifies kinetically, a useful and simple power law. Similarity between our parameter estimates in pigs and parameter estimates in normal humans suggests that the proposed model can be adapted for use in humans.     

Blood flow in branching circulatory systems during rest and activity

On the basis of simple physical considerations the blood flow in a branching circulatory system is studied. The case of two groups of parallel vessels is treated. The vessels of the same group are supposed to be identical. The resistance of each group is determined by the resistance of each vessel in the group and by the number of vessels in the group. From the dependence of the resistance of each vessel on its radius an expression is obtained for the blood flow through each group of vessels in terms of the numbers and sizes of the vessels in each group. The number of open vessels in an organ and the radius of each of those vessels are assumed to depend on the metabolic rate of that organ. The relations so obtained, together with the expression above, are applied to derive the blood flow through an organ as a function of the metabolic rate of that organ. It is indicated that the relations obtained might describe the shifting of blood from one organ to another if the activity of one of them changes. A way is pointed out to treat neural regulation of this phenomenon.     

The presence of substance P-immunoreactive nerve fibres in the organum vasculosum laminae terminalis of the Mongolian gerbil (Meriones unguiculatus)

Summary Serial frontal and sagittal sections through the organum vasculosum laminae terminalis (OVLT) of the male Mongolian gerbil (Meriones unguiculatus) were incubated with an antibody against substance P and stained by the indirect peroxidase anti-peroxidase technique. A high density of nerve fibres with large-sized terminals exhibiting substance P-immunoreactivity was observed in the OVLT in close relation to the blood vessels both in the internal and the external zone of the organ. In addition, a few fibres penetrated into the ependyma covering the organ. These results indicate that substance P is released into the bloodstream from the OVLT and in addition might influence the ependymal cells bordering the organ towards the third ventricle.This study was supported in part by the Carlsberg Foundation     

Numerical Simulation of Passage of a Neutrophil through a Rectangular Channel with a Moderate Constriction

The authors have previously presented a mathematical model to predict transit time of a neutrophil through an alveolar capillary segment which was modeled as an axisymmetric arc-shaped constriction settled in a cylindrical straight pipe to investigate the influence of entrance curvature of a capillary on passage of the cell. The axially asymmetric cross section of a capillary also influences the transit time because it requires three-dimensional deformation of a cell when it passes through the capillary and could lead to plasma leakage between the cell surface and the capillary wall. In this study, a rectangular channel was introduced, the side walls of which were moderately constricted, as a representative of axially asymmetric capillaries. Dependence of transit time of a neutrophil passing through the constriction on the constriction geometry, i.e., channel height, throat width and curvature radius of the constriction, was numerically investigated, the transit time being compared with that through the axisymmetric model. It was found that the transit time is dominated by the throat hydraulic diameter and curvature radius of the constriction and that the throat aspect ratio little affects the transit time with a certain limitation, indicating that if an appropriate curvature radius is chosen, such a rectangular channel model can be substituted for an axisymmetric capillary model having the same throat hydraulic diameter in terms of the transit time by choosing an appropriate curvature radius. Thus, microchannels fabricated by the photolithography technique, whose cross section is generally rectangular, are expected to be applicable to in vitro model experiments of neutrophil retention and passage in the alveolar capillaries.     

Effects of stiffness and volume on the transit time of an erythrocyte through a slit

By using a fully coupled fluid–cell interaction model, we numerically simulate the dynamic process of a red blood cell passing through a slit driven by an incoming flow. The model is achieved by combining a multiscale model of the composite cell membrane with a boundary element fluid dynamics model based on the Stokes flow assumption. Our concentration is on the correlation between the transit time (the time it takes to finish the whole translocation process) and different conditions (flow speed, cell orientation, cell stiffness, cell volume, etc.) that are involved. According to the numerical prediction (with some exceptions), the transit time rises as the cell is stiffened. It is also highly sensitive to volume increase inside the cell. In general, even slightly swollen cells (i.e., the internal volume is increased while the surface area of the cell kept unchanged) travel dramatically slower through the slit. For these cells, there is also an increased chance of blockage.     

The variance of transit times of indicators through a vascular bed

A new method is developed of determining the variance of the times of transit of an indicator through a vascular bed (or through any perfused organ) from the time-course of the arterial-venous concentration difference, observed between two steady states. The method is illustrated using previously published data on arterial and hepatic venous radioactivities following a single injection of ¹³¹I-labelled serum albumin in man.     

Extension of the theory of the multiple-indicator dilution technique to metabolic systems with an arbitrary number of rate constants

The multiple indicator dilution technique consists in the instantaneous injection of a mixture of tracers into the arterial perfusate flow of a catheterized or isolated perfused organ, followed by the analysis of the effluent perfusate. The theory of this technique, which has hitherto been developed for cases where metabolism of a tracer is confined to sequestration described by a single rate constant, is extended in this paper to include an arbitrary number of metabolic rate constants. Partial differential equations with constant coefficients describing the events in a single capillary are derived by applying conventional compartmental analysis to infinitesimally small sections of the capillary. Methods for solving such systems in the time as well as in the frequency domain are developed. From the solutions, the impulse response of the whole organ is evaluated assuming variable capillary and uniform large-vessel transit times. In addition, an efficient method using much less computer time was developed, based on the approximation of the distribution of the capillary transit times by a sum of exponentials. Evaluation of moments (recoveries and mean transit times) is also treated. The results are applied to an example from hepatic lactate metabolism.     

Dietary fibres,fibre analogues,and glucose tolerance: importance of viscosity.

To define the type of dietary fibre of fibre analogue with the greatest potential use in diabetic treatment, groups of four to six volunteers underwent 50-g glucose tolerance tests (GTT) with and without the addition of either guar, pectin, gum tragacanth, methylcellulose, wheat bran, or cholestyramine equivalent to 12 g fibre. The addition of each substance significantly reduced blood glucose concentration at one or more points during the GTT and generally reduced serum insulin concentrations. The greatest flattening of the glucose response was seen with guar, but this effect was abolished when hydrolysed non-viscous guar was used. The reduction in the mean peak rise in blood glucose concentration for each substance correlated positively with its viscosity (r = 0.926; P less than 0.01), as did delay in mouth-to-caecum transit time (r = 0.885; P less than 0.02). Viscous types of dietary fibre are therefore most likely to be therapeutically useful in modifying postprandial hyperglycaemia.     

Transient phenomena in capillary exchange

The case of a vessel, which supplies a region through which it passes with some substance, is considered for the situation in which the permeability is the limiting factor. Diffusion parallel to the vessel is neglected. The substance may, however, be consumed proportional to its concentration in the inner or outer region. A solution is given for the case in which the input is an arbitrary function of time. It is suggested that the results may be applied in some cases to data on the injection of substances into blood vessels, or they may be applied to the transient effects in the case of vapors or gases passing through the respiratory passages.     

A COMPARTMENTAL ANALYSIS OF CIRCULATORY LYMPHOCYTES IN THE SPLEEN

A tentative model describing the passage of circulatory lymphocytes through the spleen is formulated in accord with known anatomical features. In order to preserve isomorphism between the model and the splenic system, the model is formulated in compartmental form and its design allows alternative routes and modes of lymphocyte transit to be considered. The simultaneous differential equations arising from the model are solved using an analogue computer which also provides the means whereby the performance of the model may be compared with suitable dynamic data drawn from literature. This not only allows the selection of a particular configuration of the model in preference to its alternatives, but also allows the numerical determination of certain unknown parameters. In the case of the rat spleen, best agreement between model and experimental data is obtained when between 10 and 25% of the total lymphocyte flux in the model spleen passes through the marginal zone where the average dwell time of the lymphocytes is about 50 min. The white pulp receives a lymphocyte flux from the marginal zone amounting to about 10% of the total splenic flux and the white pulp lymphocytes are sequestered for a period of 4–6 hr before release to the venous circulation. The red pulp receives 90% of the total splenic flux but the majority of lymphocytes find transit through the red pulp in less than 5 min. The remaining flux of lymphocytes, amounting to 10% of the splenic input, is delayed in transit through the red pulp by 2–3 hr before release to the venous circulation.     

A compartmental analysis of circulatory lymphocytes in the spleen.

A tentative model describing the passage of circulatory lymphocytes through the spleen is formulated in accord with known anatomical features. In order to preserve isomorphism between the model and the splenic system, the model is formulated in compartmental form and its design allows alternative routes and modes of lymphocyte transit to be considered. The simultaneous differential equations arising from the model are solved using an analogue computer which also provides the means whereby the performance of the model may be compared with suitable dynamic data drawn from literature. This not only allows the selection of a particular configuration of the model in preference to its alternatives, but also allows the numerical determination of certain unknown parameters. In the case of the rat spleen, best agreement between model and experimental data is obtained when between 10 and 25% of the total lymphocyte flux in the model spleen passes through the marginal zone where the average dwell time of the lymphocytes is about 50 min. The white pulp receives a lymphocyte flux from the marginal zone amounting to about 10% of the total splenic flux and the white pulp lymphocytes are sequestered for a period of 4-6 hr before release to the venous circulation. The red pulp receives 90% of the total splenic flux but the majority of lymphocytes find transit through the red pulp in less than 5 min. The remaining flux of lymphocytes, amounting to 10% of the splenic input, is delayed in transit through the red pulp by 2-3 hr before release to the venous circulation.     

Photosynthesis-irradiance response at physiological level: a mechanistic model

A mechanistic model is developed to present the photosynthetic response of phytoplankton to irradiance at the physiological level. The model is operated on photosynthetic units (PSU), and each PSU is assumed to have two states: reactive and activated. Light absorption that drives a reactive PSU into the activated state results from the effective absorption of the PSU. Transitions between the two states are asymmetrical in rate. A PSU in the reactive state becomes activated much faster than it recovers from the activated state to the reactive one. The turnover time for an activated PSU to transit into the reactive one is defined by the turnover time of the electron transport chain. The present model yields a photosynthesis-irradiance curve (PE-curve) in a hyperbola, which is described by three physiological parameters: effective cross-section (sigmaPSII), turnover time of electron transport chain (tau) and number of PSUs (N). The PE-curve has an initial slope of sigmaPSII x N, a half-saturated irradiance of 1/(tau sigmaPSII), and a maximal photosynthetic rate of N/tau at the saturated irradiance. The PE-curve from the present model is comparable to the empirical function based on the target theory described by the Poisson distribution.     

A compartmental capillary, convolution integration model to investigate nutrient transport and metabolism in vivo from paired indicator/nutrient dilution curves.

Thirty-three paired indicator/nutrient dilution curves across the mammary glands of four cows were obtained after rapid injection of para-aminohippuric acid (PAH) plus glucose into the external iliac artery. For the measurement of extracellular volume and kinetics of nutrient uptake from indicator dilution curves, several models of solute dispersion and disappearance have been proposed. The Crone-Renkin models of exchange in a single capillary assume negligible washout of solutes from the extracellular space and do not describe entire dilution curves. The Goresky models include a distribution of capillary transit times to generate whole system outflow profiles but require two indicators to parametize extracellular behavior. A compartmental capillary, convolution integration model is proposed that uses one indicator to account for the extracellular behavior of the nutrient after a paired indicator/nutrient injection. With the use of an iterative approach to least squares, unique solutions for nonexchanging vessel transit time t(mu) and its variance sigma were obtained from all 33 PAH curves. The average of heterogeneous vascular transit times was approximated as 2sigma = 8.5 s. The remainder of indicator dispersion was considered to be due to washout from a well-mixed compartment representing extracellular space that had an estimated volume of 5.5 liters or 24% of mammary gland weight. More than 99% of the variation in the time course of venous PAH concentration after rapid injection into the arterial supply of the mammary glands was explained in an unbiased manner by partitioning the organ into a heterogeneous nonexchanging vessel subsystem and a well-mixed compartmental capillary subsystem.     

Transient mass transfer processes during the perfusion of a biological organ with a cryophylactic agent solution

An analytical study was performed for the mass transfer processes which occurs during a typical CPA introduction protocol in a biological organ. In such a protocol the concentration of CPA in the perfusate changes linearly with time to a maximal value and is then kept at that value for an additional period of time. Numerical solutions to the Kedem-Katchalsky equations for mass transfer processes in an organ modeled by a large number of typical Krogh tissue units were found. The solutions indicate that several phenomena possibly harmful to the organ occur in the tissue as a function of H, the rate of increase in the CPA concentration.     

Residence time distribution of diazepam in the isolated perfused rat liver. Analysis with the axial dispersion model.

The application of the axial dispersion model to diazepam hepatic elimination was evaluated using data obtained for impulse-response experiments with diazepam in the single-pass isolated perfused rat liver preparation. The transient form of the two-compartment dispersion model was applied to the output concentration versus time profile of diazepam after bolus input of a radiolabelled tracer into the hepatic portal vein (n = 4), providing DN and CLint estimates of 0.251 +/- 0.093 and 135 +/- 59 ml min-1, respectively. In contrast, the one-compartment form of the axial dispersion model, which assumes instantaneous transversal distribution of substance to the accessible spaces within the liver, could not adequately describe the residence time distribution (RTD) of diazepam. Furthermore, the magnitude of DN, a stochastic parameter which characterizes the axial spreading of solutes during transit through the liver, is similar to that determined for non-eliminated substances such as erythrocytes, albumin, sucrose and water. These findings suggest that the dispersion of diazepam in the perfused rat liver is determined primarily by the architecture of the hepatic microvasculature.     

Reoxygenation and Split-Dose Response to Radiation in a Tumour Model with Krogh-Type Vascular Geometry

After a single dose of radiation, transient changes caused by cell death are likely to occur in the oxygenation of surviving cells. Since cell radiosensitivity increases with oxygen concentration, reoxygenation is expected to increase the sensitivity of the cell population to a successive irradiation. In previous papers we proposed a model of the response to treatment of tumour cords (cylindrical arrangements of tumour cells growing around a blood vessel of the tumour). The model included the motion of cells and oxygen diffusion and consumption. By assuming parallel and regularly spaced tumour vessels, as in the Krogh model of microcirculation, we extend our previous model to account for the action of irradiation and the damage repair process, and we study the time course of the oxygenation and the cellular response. By means of simulations of the response to a dose split in two equal fractions, we investigate the dependence of tumour response on the time interval between the fractions and on the main parameters of the system. The influence of reoxygenation on a therapeutic index that compares the effect of a split dose on the tumour and on the normal tissue is also investigated.     

Differential Mechanisms Associated with Vascular Disrupting Action of Electrochemotherapy: Intravital Microscopy on the Level of Single Normal and Tumor Blood Vessels

Electropermeabilization/electroporation (EP) provides a tool for the introduction of molecules into cells and tissues. In electrochemotherapy (ECT), cytotoxic drugs are introduced into cells in tumors, and nucleic acids are introduced into cells in gene electrotransfer. The normal and tumor tissue blood flow modifying effects of EP and the vascular disrupting effect of ECT in tumors have already been determined. However, differential effects between normal vs. tumor vessels, to ensure safety in the clinical application of ECT, have not been determined yet. Therefore, the aim of our study was to determine the effects of EP and ECT with bleomycin on the HT-29 human colon carcinoma tumor model and its surrounding blood vessels. The response of blood vessels to EP and ECT was monitored in real time, directly at the single blood vessel level, by in vivo optical imaging in a dorsal window chamber in SCID mice with 70 kDa fluorescently labeled dextrans. The response of tumor blood vessels to EP and ECT started to differ within the first hour. Both therapies induced a vascular lock, decreased functional vascular density (FVD) and increased the diameter of functional blood vessels within the tumor. The effects were more pronounced for ECT, which destroyed the tumor blood vessels within 24 h. Although the vasculature surrounding the tumor was affected by EP and ECT, it remained functional. The study confirms the current model of tumor blood flow modifying effects of EP and provides conclusive evidence that ECT is a vascular disrupting therapy with a specific effect on the tumor blood vessels.     

The effect on gastrointestinal motility after ingestion of raw bile of grass carp (Ctenopharyngodon idellus) in the conscious rats

The influence of raw grass carp bile on gastrointestinal motility was studied in conscious rats. It was found that the toxic bile prolonged the gastric emptying time, increased a marker substance traversed along the small intestine and accelerated the transit time along the colon. These findings might be explained by a very efficient protective response of the digestive system, and/or poisoning effect of the invading toxic substance in rats. Accumulation of fluid was found in the stomach, 20 min after ingestion of 0.3 ml of the toxic bile. Besides, ingestion of the raw bile also increased the urea nitrogen in blood, which may result from a loss of fluid volume through the gastrointestinal tract and (or) a direct action of the bile on the kidneys.     

Evaluation of dynamic response and biomechanical properties of isolated blood vessels

In this study we present the experimental and mathematical model for a precise assessment of isolated blood vessels dynamic response under a sudden change of blood pressure. Only the end points within the time interval of the considered dynamic response of the blood vessel, or so-called “alternate steady states” of the processes, were usually considered in various studies. These studies do not provide an insight how the process variables change between these alternate steady states. Isolated blood vessels (rat abdominal aorta) were used to determine how the process dynamics can be described in detailed quantitative terms by mathematical parameters. The experimental model and mathematical procedures presented in this study describe precisely (at a high sensitivity level) the time history of the pressure and the diameter change in between alternate steady states, when an abrupt change of blood pressure occurs at the vessel outlet. Also, the experimental model and mathematical procedures were used to determine changes in the stress–strain law, caused by the action of L-arginine. The presented experimental design and mathematical model can be used for assessment of isolated blood vessel dynamic responses under different stimuli, such as drug effects, electrostimulation etc.