Thomas A. Bronikowski

Model-free deconvolution techniques for estimating vascular transport functions

In this paper we present two methods which can be used to numerically deconvolve indicator dilution curves to obtain vascular transport functions. In the first method, direct algebraic deconvolution is made stable and practical by the damped least squares method. The second method involves a time-shift of the output curve which is based on the first and second moments of the input and output curves. This method is stable, computationally simple and can provide reasonable estimates of the transport function.

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

Abstract The permeability–surface-area product ( PS ) of an organ having heterogeneous capillary transit times can be determined by the sudden-injection, multiple-indicator dilution method from the relationship 1n[ C D ( t )/ C P ( t )]=( PS c / Q c ) τ ( t ), where C P and C D are the venous concentrations of the permeating and nonpermeating indicators, respectively, and S c / Q c is the capillary surface-area-to-volume ratio, if the capillary transit-time function τ ( t ) is known. Rose and Goresky [12] analyzed the properties of τ ( t ) assuming that capillary and conducting-vessel transit times were coupled so that conducting vessels with short transit times serve capillaries with short transit times and conducting vessels with long transit times serve capillaries with long transit times (flow coupling). In this study, we examine the properties of τ ( t ) assuming random coupling of conducting vessels and capillaries by representing the organ as a convolution of the capillary transit-time distribution h ( t ) and the conducting-vessel transit-time distribution C ( t ). The results indicate that in such an organ τ( t ) will be an increasing function bounded by the minimum and maximum capillary transit times and the duration of C ( t ). The analysis and simulations using typical h ( t ) and C ( t ) functions indicate that τ ( t ) can have similar properties regardless of whether flow coupling or random coupling of conducting and capillary vessels exists.

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

Abstract 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.

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.

On indicator dilution and perfusion heterogeneity: A stochastic model

Abstract Unidirectional extraction of a substrate S in the capillaries following the arterial injection of a bolus containing S and a reference tracer R is assumed to follow first-order kinetics. If C R and C S denote normalized venous effluent concentrations of R and S , respectively, let L ( t )=ln[ C R ( t )⧸ C S ( t )]. We derive a formula which expresses the experimental L(t) data in terms of the mean μ( t ) and variance of the transit times of those capillaries which are contributing indicators at each sample time t . We examine the information thus contained in the L data about capillary and noncapillary transit times under several kinematic assumptions. We show that if the capillary and noncapillary transit times are stochastically independent with frequency functions h c (t) and h av (t) , respectively, then the shapes of the graphs of L(t) and μ( t ) depend on the variances and skewnesses of h c (t) and h av (t) . Specifically, let r 2 be the ratio of the variance of h c ( t ) to the variance of h av ( t ), and let r 3 be the ratio of skewnesses in the same order. Then the graph of μ( t ) is concave downward if r 2 ⧸ r 3 > 1, concave upward if r 2 ⧸ r 3 r 2 ⧸ r 3 = 1. If the fraction of S extracted is not too large, L(t) has nearly the same shape as μ( t ), and therefore, L(t) contains information about h c ( t ) and h av ( t ).

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.

Limits on the continuous distribution of pulmonary vascular resistance versus compliance from outflow occlusion

Rapid occlusion of the venous outflow from a lung lobe perfused with constant flow causes the arterial and venous pressure to increase with time. Pressure values obtained by linear extrapolation of the nearly linear portions of the resulting arterial and venous pressure versus time curves back to the instant of occlusion provide data which can be used to place limits on the actual unknown distribution of resistance versus compliance. This has been proven deductively for the case of compartmental models having a small number of resistances and compliances. For compartmental models having a large number of resistances and compliances approaching a continuous distribution, an inductive approach was used in which a large number of simulations indicated that the actual continuous distribution is confined by the experimental data to the same limits as those for the small number of resistances and compliances.

A model of the vascular resistance and compliance distribution in a lung lobe.

The arterial and venous pressure curves obtained after occluding the venous outflow from a dog lung lobe perfused with constant flow contain information about the intralobar longitudinal distribution of vascular resistance (R) and compliance (C). To utilize this information, a lumped model consisting of four parallel Cs separated by three serial Rs was used. Solutions of the governing differential equations yield a nonlinear system of four algebraic equations in the seven unknowns and the measured data. Three of the equations form a linear subsystem in which the unknowns are the three Rs and the coefficients are functions of the four Cs. This is an underdetermined system, but when nonnegativity and boundedness constraints are adjoined, the solution set falls within a narrow band of distributions of cumulative R relative to cumulative C. The shape of this band changes when data are obtained from lobes influenced by various vasoactive stimuli revealing the changes in the longitudinal distribution of the vascular resistance relative to the vascular compliance.

Sites of Vasoactivity in the Pulmonary Circulation Evaluated Using a Low-Viscosity Bolus Method

Because of the importance of pulmonary capillary pressure in the fluid balance of the lungs and the propensity for various pulmonary vasomotor stimuli to cause constriction of pulmonary veins (Dawson, 1984), there has been considerable interest in methods for determining pulmonary capillary pressure and the arteriovenous sites of pulmonary vasoconstriction. A number of approaches have been used, and each approach has had advantages and disadvantages (Agostoni and Piiper, 1962; Bhattacharya and Staub, 1980; Brody et al., 1968; Bronikowski et al., 1985; Dawson et al., 1988; Gaar et al., 1967; Gable and Drake, 1978; Kadowitz et al., 1975; McDonald and Butler, 1967; Michel et al., 1984; Nagasaka et al., 1984; Piiper, 1970; Zhuang et al., 1983). Like several other methods, the low-viscosity bolus method has been an experimental method used in studies of pump-perfused lungs. In such studies it has the potential for providing some unique insights into the influence of vasomotion on the longitudinal distribution of pulmonary vascular resistance and intravascular pressure from pulmonary artery to pulmonary veins. The method, originally introduced by Piiper (1970), has been modified by Brody et al. (1968) and Grimm et al. (1977) and more recently by us (Dawson et al., 1988) in an attempt to improve resolution to take advantage of this potential.

Sites of Vasoactivity in the Pulmonary Circulation Evaluated Using Rapid Occlusion Methods

The transient pressure and flow data obtained following rapid occlusion of the arterial inflow and/or venous outflow from a lung lobe contain information about the arterial-to-venous distribution of vascular resistance (R) relative to the distribution of vascular compliance (C) (Bronikowski et al., 1985; Bshouty et al., 1987; Dawson et al, 1982; Hakim et al., 1979, 1982, 1983; Holloway et al., 1983; Rippe et al., 1987; Rock et al., 1985). Interest in this approach has, to a large extent, centered around the potential for following changes in microvascular pressure and determining the arterial or venous site of action of vasomotor stimuli. In the following discussion we will present some of our ideas on the information content of occlusion data as interpreted using mathematical models in an attempt to provide useful parameters descriptive of the pulmonary microvascular bed.