Moments of physiological transit time distributions and the time course of drug disposition in the body

Characterizing the tissue distribution kinetics of drugs by physiological and physico-chemical parameters and using a circulatory model the time course of blood concentration after intravenous injection is predicted for linear pharmacokinetic systems. The interrelationships between the first three (zero to second) moments of the distribution functions of organ transfer times, circulation times and residence times of drug molecules in the body are described. Utilizing literature data the model is applied to the analysis of lidocain kinetics in humans.     

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).     

Physiological flow model for drug elimination interactions in the rat

Drug elimination interactions in the rat are modelled based on physiological blood flow rates and organ weights. A previous model has been substantially improved by the addition of a compartment representing the skin and the interactions are computed using Michaelis-Menten kinetics for competitive inhibition in the shared pathways. Furthermore, the results of repetitive dosing may also be simulated. The programs, which are extensively annotated and user oriented, are illustrated on the results of an acute warfarin—BSP interaction experiment in rats.     

A physiologically-based flow network model for hepatic drug elimination III: 2D/3D DLA lobule models

Background

One of the major issues in current pharmaceutical development is potential hepatotoxicity and drug-induced liver damage. This is due to the unique metabolic processes performed in the liver to prevent accumulation of a wide range of chemicals in the blood. Recently, we developed a physiologically-based lattice model to address the transport and metabolism of drugs in the liver lobule (liver functional unit).

Method

In this paper, we extend our idealized model to consider structural and spatial variability in two and three dimensions. We introduce a hexagonal-based model with one input (portal vein) and six outputs (hepatic veins) to represent a typical liver lobule. To capture even more realistic structures, we implement a novel sequential diffusion-limited aggregation (DLA) method to construct a morphological sinusoid network in the lobule. A 3D model constructed with stacks of multiple 2D sinusoid realizations is explored to study the effects of 3D structural variations. The role of liver zonation on drug metabolism in the lobule is also addressed, based on flow-based predicted steady-state O₂ profiles used as a zonation indicator.

Results

With this model, we analyze predicted drug concentration levels observed exiting the lobule with their detailed distribution inside the lobule, and compare with our earlier idealized models. In 2D, due to randomness of the sinusoidal structure, individual hepatic veins respond differently (i.e. at different times) to injected drug. In 3D, however, the variation of response to the injected drug is observed to be less extreme. Also, the production curves show more diffusive behavior in 3D than in 2D.

Conclusion

Although, the individual producing ports respond differently, the average lobule production summed over all hepatic veins is more diffuse. Thus the net effect of all these variations makes the overall response smoother. We also show that, in 3D, the effect of zonation on drug production characteristics appears quite small. Our new biophysical structural analysis of a physiologically-based 3D lobule can therefore form the basis for a quantitative assessment of liver function and performance both in health and disease

     

On testing dependence between time to failure and cause of failure when causes of failure are missing

The hypothesis of independence between the failure time and the cause of failure is studied by using the conditional probabilities of failure due to a specific cause given that there is no failure up to certain fixed time. In practice, there are situations when the failure times are available for all units but the causes of failures might be missing for some units. We propose tests based on U-statistics to test for independence of the failure time and the cause of failure in the competing risks model when all the causes of failure cannot be observed. The asymptotic distribution is normal in each case. Simulation studies look at power comparisons for the proposed tests for two families of distributions. The one-sided and the two-sided tests based on Kendall type statistic perform exceedingly well in detecting departures from independence.     

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.     

Fixation probabilities when generation times are variable: the burst death model

Estimating the fixation probability of a beneficial mutation has a rich history in theoretical population genetics. Typically, to attain mathematical tractability, we assume that generation times are fixed, while the number of offspring per individual is stochastic. However, fixation probabilities are extremely sensitive to these assumptions regarding life history. In this article, we compute the fixation probability for a "burst-death" life-history model. The model assumes that generation times are exponentially distributed, but the number of offspring per individual is constant. We estimate the fixation probability for populations of constant size and for populations that grow exponentially between periodic population bottlenecks. We find that the fixation probability is, in general, substantially lower in the burst-death model than in classical models. We also note striking qualitative differences between the fates of beneficial mutations that increase burst size and mutations that increase the burst rate. In particular, once the burst size is sufficiently large relative to the wild type, the burst-death model predicts that fixation probability depends only on burst rate.     

An alternative model for bivariate random-effects meta-analysis when the within-study correlations are unknown

Multivariate meta-analysis models can be used to synthesize multiple, correlated endpoints such as overall and disease-free survival. A hierarchical framework for multivariate random-effects meta-analysis includes both within-study and between-study correlation. The within-study correlations are assumed known, but they are usually unavailable, which limits the multivariate approach in practice. In this paper, we consider synthesis of 2 correlated endpoints and propose an alternative model for bivariate random-effects meta-analysis (BRMA). This model maintains the individual weighting of each study in the analysis but includes only one overall correlation parameter, rho, which removes the need to know the within-study correlations. Further, the only data needed to fit the model are those required for a separate univariate random-effects meta-analysis (URMA) of each endpoint, currently the common approach in practice. This makes the alternative model immediately applicable to a wide variety of evidence synthesis situations, including studies of prognosis and surrogate outcomes. We examine the performance of the alternative model through analytic assessment, a realistic simulation study, and application to data sets from the literature. Our results show that, unless rho is very close to 1 or -1, the alternative model produces appropriate pooled estimates with little bias that (i) are very similar to those from a fully hierarchical BRMA model where the within-study correlations are known and (ii) have better statistical properties than those from separate URMAs, especially given missing data. The alternative model is also less prone to estimation at parameter space boundaries than the fully hierarchical model and thus may be preferred even when the within-study correlations are known. It also suitably estimates a function of the pooled estimates and their correlation; however, it only provides an approximate indication of the between-study variation. The alternative model greatly facilitates the utilization of correlation in meta-analysis and should allow an increased application of BRMA in practice.     

Variance of prediction error with mixed model equations when relationships are ignored

Summary Formulas are presented to illustrate the calculation of correct variances of prediction error (PEV) and the correlation between true and predicted values (r_TI) when the incorrect variance-covariance matrix for the random effects is used in mixed-model equations (MME). The example with progeny records of highly related and inbred sires showed that PEV were underestimated from the diagonals of the inverse of the coefficient matrix of the MME when sires were assumed unrelated and not inbred and were overestimated when relationships among sires were calculated with Henderson's simple rules for the inverse of the numerator relationship matrix, A^-1, which do not consider inbreeding. When Quaas' rules for A^-1, which do consider inbreeding, are used, the correct PEV are obtained. In the example, calculations of r_TI from the diagonals of the inverse of the coefficient matrix were too large when relationships and inbreeding were ignored and were obviously wrong when the approximation to the numerator relationship matrix, A, was based on the simple rules for calculating A^-1. If the correct A is used in the MME, the calculation of r_TI may be incorrect if inbreeding of the evaluated individual is not considered. If inbreeding is known, adjustment for inbreeding is easy for calculation of r_TI.Published as paper no. 9947, Journal Ser, Nebraska Agric Res Div, University of Nebraska, Lincoln, Neb.     

Study of pulse transit time oscillations during obstructive sleep apnoea by using a distributed model

The study of arterial compliance is useful in understanding the geometrical and mechanical properties of a systemic arterial tree. Numerous mathematical models have shown their potential in relating the physical phenomena in the arterial tree to properties of the wall itself. However, limited model is available that describes the pulse transit time (PTT) oscillations of a sleeping child during tidal breathing and obstructive sleep apnoea (OSA). Data from 20 children (17 male; aged 6.4 +/- 4.1 yr) whom were recruited for overnight polysomnography (PSG) were used. A modified windkessel model with related physiological parameters was utilised to describe PTT fluctuations due to the cardiovascular system during sleep. Verification with the recorded PSG data showed similar trends with the model for both types of respiratory events. For tidal breathing, undamped PTT oscillations of 3.89 s were predicted by the model while actual data yielded a mean value of 3.72 +/- 0.79 s. Conversely, under-damping PTT responses were expected based on the model for OSA. The model estimated a Q factor of 4.23 and actual mean data were 3.86 +/- 0.64. Hence, the findings herein suggest that the proposed model has the potential to illustrate tidal breathing and OSA events in sleeping children.     

NMR analysis of carbohydrates with model-free spectral densities: the dispersion range revisited

Over the past decade molecular mechanics and molecular dynamics studies have demonstrated considerable flexibility for carbohydrates. In order to interpret the corresponding NMR parameters, which correspond to a time-averaged or 'virtual' conformer, it is necessary to simulate the experimental data using the averaged geometrical representation obtained with molecular modelling methods. This structural information can be transformed into theoretical NMR data using empirical Karplus-type equations for the scalar coupling constants and the appropriate formalism for the relaxation parameters. In the case of relaxation data, the 'model-free' spectral densities have been widely used in order to account for the internal motions in sugars. Several studies have been conducted with truncated model-free spectral densities based on the assumption that internal motion is very fast with respect to overall tumbling. In this report we present experimental and theoretical evidence that suggests that this approach is not justified. Indeed, recent results show that even in the case of moderate-sized carbohydrates internal motions are occurring on the same timescale as molecular reorientation. Simulations of relaxation parameters (NOESY volumes, proton cross-relaxation rates, carbon T1 and nOe values) in the dispersion range (0.1<Tc<5 ns) show that rates of internal motion can be fairly precisely defined with respect to overall tumbling. Experimental data for a variety of oligosaccharides clearly indicate similar timescales for internal and overall motion. This revised version was published online in August 2006 with corrections to the Cover Date.     

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.     

On the selective elimination of Y-bearing sperm

The potential use of antibodies that selectively recognize either X-bearing or Y-bearing sperm is self-evident. Thus our attention was directed to the fact that under optimal conditions, H-Y antibody lyses 50% of mouse spermatozoa. Accordingly, we asked whether expression of H-Y antigen is haploid in spermatozoa from XY male mice heterozygous for the autosomal dominantSxr gene, for if H-Y expression were haploid, H-Y antibody would be expected to kill 75% of spermatozoa derived from these XY,Sxr/- males. However, maximal lysis remained at the 50% level, which indicates that haploid expression of H-Y antigen and the potential immunoselection of Y-(or X-) bearing spermatozoa are unlikely.     

Optimal control of tumor size used to maximize survival time when cells are resistant to chemotherapy.

The high failure rates encountered in the chemotherapy of some cancers suggest that drug resistance is a common phenomenon. In the current study, the tumor burden during therapy is used to slow the growth of the drug-resistant cells, thereby maximizing the survival time of the host. Three types of tumor growth model are investigated--Gompertz, logistic, and exponential. For each model, feedback controls are constructed that specify the optimal tumor mass as a function of the size of the resistant subpopulation. For exponential and logistic tumor growth, the tumor burden during therapy is shown to have little impact upon survival time. When the tumor is in Gompertz growth, therapies maintaining a large tumor burden double and sometimes triple the survival time under aggressive therapies. Aggressive therapies aim for a rapid reduction in the sensitive cell subpopulation. These conclusions are not dependent upon the values of the model constants that determine the mass of resistant cells. Since treatments maintaining a high tumor burden are optimal for Gompertz tumor growth and close to optimal for exponential and logistic tumor growth, it may no longer be necessary to know the growth characteristics of a tumor to schedule anticancer drugs.     

On the validity of Freud's dream interpretations

In this article I defend Freud's method of dream interpretation against those who criticize it as involving a fallacy-namely, the reverse causal fallacy-and those who criticize it as permitting many interpretations, indeed any that the interpreter wants to put on the dream. The first criticism misconstrues the logic of the interpretative process: it does not involve an unjustified reversal of causal relations, but rather a legitimate attempt at an inference to the best explanation. The judgement of whether or not a particular interpretation is the best explanation depends on the details of the case in question. I outline the kinds of probabilities involved in making the judgement. My account also helps to cash out the metaphors of the jigsaw and crossword puzzles that Freudians have used in response to the 'many interpretations' objection. However, in defending Freud's method of dream interpretation, I do not thereby defend his theory of dreams, which cannot be justified by his interpretations alone.     

On the validity of free hormone measurements

The influence of equilibrium perturbation on the measurement of free or non-protein-bound hormones in serum or plasma is described. The free analyte concentration in the sample will alter after dilution or addition of binding reagents. The extent of this perturbation is not constant. It depends on the binding properties of the endogenous carrier proteins which vary from sample to sample and from sample to reference standard. As examples, the different changes of the free thyroxine and free cortisol concentrations after dilution are demonstrated both experimentally and by theoretical modeling of the hormone-protein distribution. It is shown how this systematical error can be minimized in the design of free hormone assays.