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.     

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.     

Transit time dispersion in pulmonary and systemic circulation: effects of cardiac output and solute diffusivity

We present an in vivo method for analyzing the distribution kinetics of physiological markers into their respective distribution volumes utilizing information provided by the relative dispersion of transit times. Arterial concentration-time curves of markers of the vascular space [indocyanine green (ICG)], extracellular fluid (inulin), and total body water (antipyrine) measured in awake dogs under control conditions and during phenylephrine or isoproterenol infusion were analyzed by a recirculatory model to estimate the relative dispersions of transit times across the systemic and pulmonary circulation. The transit time dispersion in the systemic circulation was used to calculate the whole body distribution clearance, and an interpretation is given in terms of a lumped organ model of blood-tissue exchange. As predicted by theory, this relative dispersion increased linearly with cardiac output, with a slope that was inversely related to solute diffusivity. The relative dispersion of the flow-limited indicator antipyrine exceeded that of ICG (as a measure of intravascular mixing) only slightly and was consistent with a diffusional equilibration time in the extravascular space of approximately 10 min, except during phenylephrine infusion, which led to an anomalously high relative dispersion. A change in cardiac output did not alter the heterogeneity of capillary transit times of ICG. The results support the view that the relative dispersions of transit times in the systemic and pulmonary circulation estimated from solute disposition data in vivo are useful measures of whole body distribution kinetics of indicators and endogenous substances. This is the first model that explains the effect of flow and capillary permeability on whole body distribution of solutes without assuming well-mixed compartments.     

A note on the rôle of generalized inverse Gaussian distributions of circulatory transit times in pharmacokinetics

Based on a stochastic pharmacokinetical model (which mirrors topological properties of the circulatory system) it is shown by reinterpreting results of Wise (1974) that if the transit times of circulating drug molecules have a generalized inverse Gaussian distribution the corresponding residence times are gamma distributed. The condition that the probability of elimination of a drug molecule in a single circulatory passage is sufficiently small appears to be valid for most drugs. Thus theoretical evidence is given for fitting blood concentration-time curves following bolus injection of a single dose by power functions of time.     

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.     

Transit time dispersion in the pulmonary arterial tree

Knowledge of the contributions of arterialand venous transit time dispersion to the pulmonary vascular transittime distribution is important for understanding lung function and forinterpreting various kinds of data containing information aboutpulmonary function. Thus, to determine the dispersion of blood transittimes occurring within the pulmonary arterial and venous trees, imagesof a bolus of contrast medium passing through the vasculature ofpump-perfused dog lung lobes were acquired by using an X-ray microfocalangiography system. Time-absorbance curves from the lobar artery andvein and from selected locations within the intrapulmonary arterial tree were measured from the images. Overall dispersion within the lunglobe was determined from the difference in the first and second moments(mean transit time and variance, respectively) of the inlet arterialand outlet venous time-absorbance curves. Moments at selected locationswithin the arterial tree were also calculated and compared with thoseof the lobar artery curve. Transit times for the arterial pathwaysupstream from the smallest measured arteries (200-µm diameter) wereless than ~20% of the total lung lobe mean transit time. Transittime variance among these arterial pathways (interpathway dispersion)was less than ~5% of the total variance imparted on the bolus as itpassed through the lung lobe. On average, the dispersion that occurredalong a given pathway (intrapathway dispersion) was negligible. Similar results were obtained for the venous tree. Taken together, the resultssuggest that most of the variation in transit time in theintrapulmonary vasculature occurs within the pulmonary capillary bedrather than in conducting arteries or veins.

     

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.     

Pulmonary arterial transit times

To begin to characterize the pulmonary arterial transport function we rapidly injected a bolus containing a radiopaque dye and a fluorescence dye into the right atrium of anesthetized dogs. The concentrations of the dye indicators were measured in the main pulmonary artery (fluoroscopically) and in a subpleural pulmonary arteriole (by fluorescence microscopy). The resulting concentration vs. time curves were subjected to numerical deconvolution and moment analysis to determine how the bolus was dispersed as it traveled through the arteriole stream tube from the main pulmonary artery to the arteriole. The mean transit time and standard deviation of the transport function from the main pulmonary artery to the arterioles studied averaged 1.94 and 1.23 s, respectively, and the relative dispersion (ratio of standard deviation to mean transit time) was approximately 64%. This relative dispersion is at least as large as those reported for the whole dog lung, indicating that relative to their respective mean transit times the dispersion upstream from the arterioles is comparable to that taking place in capillaries and/or veins. The standard deviations of the transport functions were proportional to their mean transit times. Thus the relative dispersion from the main pulmonary artery to the various arterioles studied was fairly consistent. However, there were variations in mean transit time even between closely adjacent arterioles, suggesting that variations in mean transit times between arteriole stream tubes also contribute to the dispersion in the pulmonary arterial tree.     

New Mathematical Modeling Approach for Predicting Microbial Inactivation by High Hydrostatic Pressure

A new primary model based on a thermodynamically consistent first-order kinetic approach was constructed to describe non-log-linear inactivation kinetics of pressure-treated bacteria. The model assumes a first-order process in which the specific inactivation rate changes inversely with the square root of time. The model gave reasonable fits to experimental data over six to seven orders of magnitude. It was also tested on 138 published data sets and provided good fits in about 70% of cases in which the shape of the curve followed the typical convex upward form. In the remainder of published examples, curves contained additional shoulder regions or extended tail regions. Curves with shoulders could be accommodated by including an additional time delay parameter and curves with tails shoulders could be accommodated by omitting points in the tail beyond the point at which survival levels remained more or less constant. The model parameters varied regularly with pressure, which may reflect a genuine mechanistic basis for the model. This property also allowed the calculation of (a) parameters analogous to the decimal reduction time D and z, the temperature increase needed to change the D value by a factor of 10, in thermal processing, and hence the processing conditions needed to attain a desired level of inactivation; and (b) the apparent thermodynamic volumes of activation associated with the lethal events. The hypothesis that inactivation rates changed as a function of the square root of time would be consistent with a diffusion-limited process.     

A biphasic model for sinusoidal liver perfusion remodeling after outflow obstruction

Liver resection can lead to focal outflow obstruction due to transection of hepatic veins. Outflow obstruction may cause additional damage to the small remnant liver. Drainage of the obstructed territories is reestablished via dilatation of sinusoids. Subsequently, sinusoidal canals are formed draining the blood from the obstructed territory to the neighboring unobstructed territories. We raised the phenomenological hypothesis that the blood pressure gradient is the main driving force for the formation of sinusoidal vascular canals. We generated a biphasic mechanical model to describe this vascular remodeling process in relation to the variable pressure gradient. Therefore, we introduced a transverse isotropic permeability relation as well as an evolutional optimization rule to describe the relationship between pressure gradient and the direction of the sinusoidal blood flow in the fluid phase. As a next step, we developed a framework for the calculation concept including the representation of the governing weak formulations. Then, we examined a representative numerical example with simulation of the blood flow under both conditions, the physiological situation as well as after outflow obstruction. Doing so, we were able to reproduce numerically the experimentally observed process of reestablishing hepatic venous drainage via redirection of blood flow and formation of new vascular structures in respect to the fluid flow. The calculated results support the hypothesis that the reorientation of blood flow mainly depends on the pressure gradient. Further investigations are needed to determine the micromechanical influences on the reorientation of the sinusoids.     

Determination of pulmonary mean transit time and cardiac output using a one-dimensional model

In this work, we show that a one-dimensional model of the blood flow across the lungs can reproduce the evolution of a bolus versus the time. Solving the differential equation governing the bolus concentration in the framework of this model, we determine the solution which fulfills Gaussian initial boundary conditions. An effective parameter related to the ratio of a diffusion coefficient to the square of the mean speed of the flow is defined. The determination of its numerical values following a semi-empirical approach enables us to know accurately the mean transit time and the cardiac output. The results have been compared to other methods, and were found in good agreement. Such an approach could be of interest in all studies where the knowledge of flow—including micro-circulation—is needed.     

Modeling neural activity using the generalized inverse Gaussian distribution

Spike trains from neurons are often used to make inferences about the underlying processes that generate the spikes. Random walks or diffusions are commonly used to model these processes; in such models, a spike corresponds to the first passage of the diffusion to a boundary, or firing threshold. An important first step in such a study is to fit families of densities to the trains' interspike interval histograms; the estimated parameters, and the families' goodness of fit can then provide information about the process leading to the spikes. In this paper, we propose the generalized inverse Gaussian family because its members arise as first passage time distributions of certain diffusions to a constant boundary. We provide some theoretical support for the use of these diffusions in neural firing models. We compare this family with the lognormal family, using spike trains from retinal ganglion cells of goldfish, and simulations from an integrate-and-fire and a dynamical model for generating spikes. We show that the generalized inverse Gaussian family is closer to the true model in all these cases. Received: 16 September 1996 / Accepted in revised form: 2 July 1997     

Methods for assessing hepatic distending pressure and changes in hepatic capacitance in pigs

The equilibrium pressure obtained during simultaneous occlusion of hepatic vascular inflow and outflow was taken as the reference estimate of hepatic vascular distending pressure (P(hd)). P(hd) at baseline was 1.1 +/- 0.2 (mean +/- SE) mmHg higher than hepatic vein pressure (P(hv)) and 0.7 +/- 0.3 mmHg lower than portal vein pressure (P(pv)). Norepinephrine (NE) infusion increased P(hd) by 1. 5 +/- 0.5 mmHg and P(pv) by 3.7 +/- 0.6 mmHg but did not significantly increase P(hv). Hepatic lobar vein pressure (P(hlv)) measured by a micromanometer tipped 2-Fr catheter closely resembled P(hd) both at baseline and during NE-infusion. Dynamic pressure-volume (PV) curves were constructed from continuous measurements of P(hv) and hepatic blood volume increases (estimated by sonomicrometry) during brief occlusions of hepatic vascular outflow and compared with static PV curves constructed from P(hd) determinations at five different hepatic volumes. Estimates of hepatic vascular compliance and changes in unstressed blood volume from the two methods were in close agreement with hepatic compliance averaging 32 +/- 2 ml. mmHg(-1). kg liver(-1). NE infusion reduced unstressed blood volume by 110 +/- 38 ml/kg liver but did not alter compliance. In conclusion, P(hlv) reflects hepatic distending pressure, and the construction of dynamic PV curves is a fast and valid method for assessing hepatic compliance and changes in unstressed blood volume.     

Interconnected-tubes model of hepatic elimination: steady-state considerations.

In the interconnected-tubes model of hepatic transport and elimination, intermixing between sinusoids was modelled by the continuous interchange of solutes between a set of parallel tubes. In the case of strongly interconnected tubes and for bolus input of solute, a zeroth-order approximation led to the governing equation of the dispersion model. The dispersion number was expressed for the first time in terms of its main physiological determinants: heterogeneity of flow and density of interconnections. The interconnected-tubes model is now applied to steady-state hepatic extraction. In the limit of strong interconnections, the expression for output concentrations is predicted to be similar in form to those predicted by the distributed model for a narrow distribution of elimination rates over sinusoids, and by the dispersion model in the limit of a small dispersion number D(N). More generally, the equations for the predicted output concentrations can be expressed in terms of a dimensionless 'heterogeneity number'H(N), which characterizes the combined effects of variations in enzyme distribution and flow rates between different sinusoids, together with the effects of interconnections between sinusoids. A comparative analysis of the equations for the dispersion and heterogeneity numbers shows that the value of H(N)can be less than, greater than or equal to the value of D(N)for a correlation between distributions of velocities and elimination rates over sinusoids, anticorrelation between them, and when all sinusoids have the same elimination rate, respectively. Simple model systems are used to illustrate the determinants of H(N)and D(N).     

A Two-Stage Model for In Vivo Assessment of Brain Tumor Perfusion and Abnormal Vascular Structure Using Arterial Spin Labeling

The ability to assess brain tumor perfusion and abnormalities in the vascular structure in vivo could provide significant benefits in terms of lesion diagnosis and assessment of treatment response. Arterial spin labeling (ASL) has emerged as an increasingly viable methodology for non-invasive assessment of perfusion. Although kinetic models have been developed to describe perfusion in healthy tissue, the dynamic behaviour of the ASL signal in the brain tumor environment has not been extensively studied. We show here that dynamic ASL data acquired in brain tumors displays an increased level of ‘biphasic’ behaviour, compared to that seen in healthy tissue. A new two-stage model is presented which more accurately describes this behaviour, and provides measurements of perfusion, pre-capillary blood volume fraction and transit time, and capillary bolus arrival time. These biomarkers offer a novel contrast in the tumor and surrounding tissue, and provide a means for measuring tumor perfusion and vascular structural abnormalities in a fully non-invasive manner.     

Estimating and comparing thermal performance curves

I show how one can estimate the shape of a thermal performance curve using information theory. This approach ranks plausible models by their Akaike information criterion (AIC), which is a measure of a model's ability to describe the data discounted by the model's complexity. I analyze previously published data to demonstrate how one applies this approach to describe a thermal performance curve. This exemplary analysis produced two interesting results. First, a model with a very high r² (a modified Gaussian function) appeared to overfit the data. Second, the model favored by information theory (a Gaussian function) has been used widely in optimality studies of thermal performance curves. Finally, I discuss the choice between regression and ANOVA when comparing thermal performance curves and highlight a superior method called template mode of variation. Much progress can be made by abandoning traditional methods for a method that combines information theory with template mode of variation.     

Splanchnic vascular capacitance and positive end-expiratory pressure in dogs

We have investigated the effect of positive end-expiratory pressure ventilation (PEEP) on regional splanchnic vascular capacitance. In 12 anesthetized dogs hepatic and splenic blood volumes were assessed by sonomicrometry. Vascular pressure-diameter curves were defined by obstructing hepatic outflow. With 10 and 15 cmH2O PEEP portal venous pressure increased 3.1 +/- 0.3 and 5.1 +/- 0.4 mmHg (P less than 0.001) while hepatic venous pressure increased 4.9 +/- 0.4 and 7.3 +/- 0.4 mmHg (P less than 0.001), respectively. Hepatic blood volume increased (P less than 0.01) 3.8 +/- 0.9 and 6.3 +/- 1.4 ml/kg body wt while splenic volume decreased (P less than 0.01) 0.8 +/- 0.2 and 1.3 +/- 0.2 ml/kg body wt. The changes were similar with closed abdomen. The slope of the hepatic vascular pressure-diameter curves decreased with PEEP (P less than 0.01), possibly reflecting reduced vascular compliance. There was an increase (P less than 0.01) in unstressed hepatic vascular volume. The slope of the splenic pressure-diameter curves was unchanged, but there was a significant (P less than 0.05) decrease in unstressed diameter during PEEP. In conclusion, hepatic blood volume increased during PEEP. This was mainly a reflection of passive distension due to elevated venous pressures. The spleen expelled blood and thus prevented a further reduction in central blood volume.