Diverse Data Sets Can Yield Reliable Information through Mechanistic Modeling: Salicylic Acid Clearance.

This is a practical example of a powerful research strategy: putting together data from studies covering a diversity of conditions can yield a scientifically sound grasp of the phenomenon when the individual observations failed to provide definitive understanding. The rationale is that defining a realistic, quantitative, explanatory hypothesis for the whole set of studies, brings about a "consilience" of the often competing hypotheses considered for individual data sets. An internally consistent conjecture linking multiple data sets simultaneously provides stronger evidence on the characteristics of a system than does analysis of individual data sets limited to narrow ranges of conditions. Our example examines three very different data sets on the clearance of salicylic acid from humans: a high concentration set from aspirin overdoses; a set with medium concentrations from a research study on the influences of the route of administration and of sex on the clearance kinetics, and a set on low dose aspirin for cardiovascular health. Three models were tested: (1) a first order reaction, (2) a Michaelis-Menten (M-M) approach, and (3) an enzyme kinetic model with forward and backward reactions. The reaction rates found from model 1 were distinctly different for the three data sets, having no commonality. The M-M model 2 fitted each of the three data sets but gave a reliable estimates of the Michaelis constant only for the medium level data (Km = 24±5.4 mg/L); analyzing the three data sets together with model 2 gave Km = 18±2.6 mg/L. (Estimating parameters using larger numbers of data points in an optimization increases the degrees of freedom, constraining the range of the estimates). Using the enzyme kinetic model (3) increased the number of free parameters but nevertheless improved the goodness of fit to the combined data sets, giving tighter constraints, and a lower estimated Km = 14.6±2.9 mg/L, demonstrating that fitting diverse data sets with a single model improves confidence in the results. This modeling effort is also an example of reproducible science available at html://www.physiome.org/jsim/models/webmodel/NSR/SalicylicAcidClearance.

Left ventricular finite element model bounded by a systemic circulation model.

A series of models were developed in which a circulatory system model was coupled to an existing series of finite element (FE) models of the left ventricle (LV). The circulatory models were used to provide realistic boundary conditions for the LV models. This was developed for the JSim analysis package and was composed of a systemic arterial, capillary, and venous system in a closed loop with a varying elastance LV and left atria to provide the driving pressures and flows matching those of the FE model. Three coupled models were developed, a normal LV under normotensive aortic loading (116/80 mm Hg), a mild hypertension (137/89 mm Hg) model, and a moderate hypertension model (165/100 mm Hg). The initial step in the modeling analysis was that the circulation was optimized to the end-diastolic pressure and volume values of the LV model. The cardiac FE models were then optimized to the systolic pressure/volume characteristics of the steady-state JSim circulatory model solution. Comparison of the stress predictions for the three models indicated that the mild hypertensive case produced a 21% increase in the average fiber stress levels, and the moderate hypertension case had a 36% increase in average stress. The circulatory work increased by 18% and 43% over that of the control for the mild and moderate hypertensive cases, respectively.

The spectra and periodograms of anti-correlated discrete fractional Gaussian noise.

Discrete fractional Gaussian noise (dFGN) has been proposed as a model for interpreting a wide variety of physiological data. The form of actual spectra of dFGN for frequencies near zero varies as f(1-2H), where 0 < H < 1 is the Hurst coefficient; however, this form for the spectra need not be a good approximation at other frequencies. When H approaches zero, dFGN spectra exhibit the 1 - 2H power-law behavior only over a range of low frequencies that is vanishingly small. When dealing with a time series of finite length drawn from a dFGN process with unknown H, practitioners must deal with estimated spectra in lieu of actual spectra. The most basic spectral estimator is the periodogram. The expected value of the periodogram for dFGN with small H also exhibits non-power-law behavior. At the lowest Fourier frequencies associated with a time series of N values sampled from a dFGN process, the expected value of the periodogram for H approaching zero varies as f(0) rather than f(1-2H). For finite N and small H, the expected value of the periodogram can in fact exhibit a local power-law behavior with a spectral exponent of 1 - 2H at only two distinct frequencies.

Physiological time series: distinguishing fractal noises from motions.

Many physiological signals appear fractal, in having self-similarity over a large range of their power spectral densities. They are analogous to one of two classes of discretely sampled pure fractal time signals, fractional Gaussian noise (fGn) or fractional Brownian motion (fBm). The fGn series are the successive differences between elements of a fBm series; they are stationary and are completely characterized by two parameters, sigma2, the variance, and H, the Hurst coefficient. Such efficient characterization of physiological signals is valuable since H defines the autocorrelation and the fractal dimension of the time series. Estimation of H from Fourier analysis is inaccurate, so more robust methods are needed. Dispersional analysis (Disp) is good for noise signals while bridge detrended scaled windowed variance analysis (bdSWV) is good for motion signals. Signals whose slopes of their power spectral densities lie near the border between fGn and fBm are difficult to classify. A new method using signal summation conversion (SSC), wherein an fGn is converted to an fBm or an fBm to a summed fBm and bdSWV then applied, greatly improves the classification and the reliability of H, the estimates of H, for the times series. Applying these methods to laser-Doppler blood cell perfusion signals obtained from the brain cortex of anesthetized rats gave H of 0.24+/-0.02 (SD, n=8) and defined the signal as a fractional Brownian motion. The implication is that the flow signal is the summation (motion) of a set of local velocities from neighboring vessels that are negatively correlated, as if induced by local resistance fluctuations.

Stochastic fractal behavior in concentration fluctuation and fluorescence correlation spectroscopy.

Fluctuations in the concentration of Brownian particles in one and two dimensions, or any reasonable measurement of the concentration such as in fluorescence correlation spectroscopy, is shown to be a stochastic fractal with a long tail. Being singular at omega = 0, the power spectrum of the fluctuation S(omega) approximately omega-1/2 for diffusion in one dimension, approximately log omega in two dimensions, but non-singular in three dimensions. This discovery provides one simple physical mechanism for possible long-memory fractal behavior, and its implications to various biological processes are discussed.

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.

Evaluation of the dispersional analysis method for fractal time series.

Fractal signals can be characterized by their fractal dimension plus some measure of their variance at a given level of resolution. The Hurst exponent, H, is < 0.5 for rough anticorrelated series, > 0.5 for positively correlated series, and = 0.5 for random, white noise series. Several methods are available: dispersional analysis, Hurst rescaled range analysis, autocorrelation measures, and power special analysis. Short data sets are notoriously difficult to characterize; research to define the limitations of the various methods is incomplete. This numerical study of fractional Brownian noise focuses on determining the limitations of the dispersional analysis method, in particular, assessing the effects of signal length and of added noise on the estimate of the Hurst coefficient, H, (which ranges from 0 to 1 and is 2 - D, where D is the fractal dimension). There are three general conclusions: (i) pure fractal signals of length greater than 256 points give estimates of H that are biased but have standard deviations less than 0.1; (ii) the estimates of H tend to be biased toward H = 0.5 at both high H (> 0.8) and low H (< 0.5), and biases are greater for short time series than for long; and (iii) the addition of Gaussian noise (H = 0.5) degrades the signals: for those with negative correlation (H < 0.5) the degradation is great, the noise has only mild degrading effects on signals with H > 0.6, and the method is particularly robust for signals with high H and long series, where even 100% noise added has only a few percent effect on the estimate of H. Dispersional analysis can be regarded as a strong method for characterizing biological or natural time series, which generally show long-range positive correlation.

Regional myocardial flow and capillary permeability-surface area products are nearly proportional.

Analyses of data on the transcapillary exchange and cellular uptake in the normal heart have generally been based on the assumption that local membrane conductances and volumes of distribution are everywhere the same. The question is whether such an assumption is justified in view of the marked (sixfold) heterogeneity of local blood flows per gram tissue. The method was to estimate both flow and capillary membrane permeability-surface area products (PS) locally in the heart. For each of five dogs running on a sloped treadmill, the deposition of tracer microspheres and of [131I]iodophenylpentadecanoic acid (IPPA), after left atrial injection, was determined in 256 pieces of left ventricular myocardium by killing the animals at approximately 100 s after radiotracer injection. A hydraulic occluder stopped the flow to a portion of the myocardium supplied by the left circumflex coronary artery 30 s before tracer injection. Regional flows ranged from 0.1 to 7.0 ml.g-1.min-1. IPPA extractions ranged from 20 to 49%. Using the known flows, we assumed the applicability of an axially distributed blood-tissue exchange model to estimate the PS for the capillary (PSc) and the parenchymal cell. It was impossible to explain the data if the PSc values for membrane transport were uniform throughout the organ. Rather, the only reasonable descriptors of the data required that local PSc values increase with local flow, almost in proportion. Current methods of analysis using data based on deposition methods need to be revised to take into account the near proportionality of PS to flow for at least some substrates.

Evaluating rescaled ranged analysis for time series.

Rescaled range analysis is a means of characterizing a time series or a one-dimensional (1-D) spatial signal that provides simultaneously a measure of variance and of the long-term correlation or "memory," The trend-corrected method is based on the statistical self-similarity in the signal: in the standard approach one measures the ratio R/S on the range R of the sum of the deviations from the local mean divided by the standard deviation S from the mean. For fractal signals R/S is a power law function of the length tau of each segment of the set of segments into which the data set has been divided. Over a wide range of tau's the relationship is: R/S = a tau H, where kappa is a scalar and the H is the Hurst exponent. (For a 1-D signal f(t), the exponent H = 2 - D, with D being the fractal dimension.) The method has been tested extensively on fractional Brownian signals of known H to determine its accuracy, bias, and limitations. R/S tends to give biased estimates of H, too low for H > 0.72, and too high for H < 0.72. Hurst analysis without trend correction differs by finding the range R of accumulation of differences from the global mean over the total period of data accumulation, rather than from the mean over each tau. The trend-corrected method gives better estimates of H on Brownian fractal signals of known H when H > or = 0.5, that is, for signals with positive correlations between neighboring elements. Rescaled range analysis has poor convergence properties, requiring about 2,000 points for 5% accuracy and 200 for 10% accuracy. Empirical corrections to the estimates of H can be made by graphical interpolation to remove bias in the estimates. Hurst's 1951 conclusion that many natural phenomena exhibit not random but correlated time series is strongly affirmed.

A vascular transport operator.

A pulse or a sharp front in concentration of a tracer or a substrate in the blood within a vessel becomes dispersed while being transported along a vessel. Cross-stream mixing and pulsations in flow with the heartbeat cause the dispersion to be less than would occur with a parabolic velocity profile (Newtonian flow). These characteristics allow intravascular mass transport to be described well by a simple two-parameter differential operator, which is a one-dimensional representation of the rather complex real situation. The operator consists of two components in series, a pure delay and a fourth-order linear differential operator. The latter is merely two underdamped second-order operators in series, with fixed relationships between the natural frequencies and damping coefficients. The operator is useful because it provides a transport function with skewness and kurtosis suitable to intravascular transport where the mean velocity profile is blunter than in Newtonian parabolic flow. The parameters of the operator are its mean transit time, t, and its relative dispersion, RD, which is the standard deviation of the response impulse divided by t. The operator transport function describes blood transport through the human leg arterial system, where the RD values are approximately 15-20%.

Maintenance of corneal endothelial cell shape by prostaglandin E2: effects of EGF and indomethacin.

Confluent, cultured, rabbit corneal endothelial cells maintain a polygonal shape which is characteristic of these cells in vivo. When cultured in the presence of EGF (10 ng/ml) and/or indomethacin (1.0 microM), the endothelial cells have markedly different shapes at confluency. By morphometry, untreated cells are polygonal and have a maximum axis of 33 mu; EGF treatment causes a spindle-shaped elongation to 48 mu and indomethacin treatment causes a stellate-shaped elongation to 48 mu. There is a slight increase in cell density. When cells are cultured in the presence of both drugs, elongation is more pronounced to a fibroblastic appearing cell population, with maximum axes of 60 mu and more, but no additive increase in cell density. Continuity of cell borders is often lost. Corneal endothelial cells cultured in the presence of EGF, indomethacin, and PGE2 (0.5 microgram/ml) maintain their polygonal shape; PGF2 alpha is not effective at reversing the drugs' effects. Untreated and EGF-treated cells synthesize and release substantial quantities of PGE2 (2-4 ng/10(4) cells). Indomethacin completely inhibits PGE2 synthesis. It is concluded that PGE2 maintains the polygonal cell shape of the corneal endothelium in vitro and, perhaps, in vivo. The elongated forms of the cell may be related to migration and important in wound closure.

Rabbit corneal endothelial cells in vitro: effects of EGF.

Rabbit corneal endothelial cells were grown in tissue culture. Epidermal growth factor (EGF) increased the mitotic rate during the growth phase by 70% over control without affecting the plating efficiency. Within 48 hr of exposure to EGF, the endothelial cells became spindle-shaped. This morphological change was quantitated by morphometry; cells treated with EGF had a major axis 1.5 X larger than that of non-EGF treated cells. The spindle-shaped morphological change did not occur in response to other growth factors, was not related to cell density, and was reversible within 24 hr after removal of EGF from the media or subculture in the absence of EGF. The addition of 5-fluorouracil blocked cell division but did not affect the EGF-induced morphological change. The appearance of the endothelial cells following EGF stimulation is similar to migrating cells closing a wound in vivo.