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PubMed2005

A unified cosolvency model for calculating solute solubility in mixed solvents.

Organic solvents are amongst the most powerful solubilization agents for a large number of water-insoluble drugs. A number of equations has been reported for mathematical representation of solute solubility in mixed solvents. The question is then posed--is there a mathematical difference between these models? To address this point, it has been demonstrated that all cosolvency models could be made equivalent by using algebraic manipulations. In order to familiarize the readers with the available cosolvency models, they are briefly reviewed. The models can be divided into two mathematical categories, i.e. linear and non-linear models. The linear models include: the log-linear, extended Hildebrand solubility approach, excess free energy equations, combined nearly ideal binary solvent/Redlich-Kister equation and Margule equations which can be converted to a general single model which expresses the logarithm of mole fraction solubility of a solute as a power series of volume fraction of the cosolvent. The non-linear models include the mixture response surface methods, two step solvation model and modified Wilson model which can be converted to a non-linear general form. Also, it has been shown that both the general single model and a non-linear general model are mathematically identical. To show the applicability of the models on real experimental data, 35 data sets have been collected from the literature. Both linear and nonlinear models produced comparable accuracies when an equal number of constant terms was employed in numerical analyses.

Models, Chemical↗
PubMedDate not supplied

Longitudinal principal components and non-linear regression models of early childhood growth.

The Jenss curve is a non-linear regression model which describes quite well the longitudinal length or weight measurements of an individual child from three months to six years. It is contrasted with modelling growth by longitudinal principal components analysis, an approach developed by Kent which is not restricted to any particular curve type or age range, and which can be thought of as providing the best fit among linear models having kappa parameters. Thus, in addition to being a model itself, it can be used to evaluate the success of other models. It is shown how this method differs from classical principal components. Data from Longitudinal Studies of Child Health and Development are used to explore these issues.

Child↗
PubMed2001

Linear equality constraints in the general linear mixed model.

Scientists may wish to analyze correlated outcome data with constraints among the responses. For example, piecewise linear regression in a longitudinal data analysis can require use of a general linear mixed model combined with linear parameter constraints. Although well developed for standard univariate models, there are no general results that allow a data analyst to specify a mixed model equation in conjunction with a set of constraints on the parameters. We resolve the difficulty by precisely describing conditions that allow specifying linear parameter constraints that insure the validity of estimates and tests in a general linear mixed model. The recommended approach requires only straightforward and noniterative calculations to implement. We illustrate the convenience and advantages of the methods with a comparison of cognitive developmental patterns in a study of individuals from infancy to early adulthood for children from low-income families.

Adolescent↗
PubMed1997

Derivation of the linear-logistic model and Cox's proportional hazard model from a canonical system description.

The linear-logistic regression model and Cox's proportional hazard model are widely used in epidemiology. Their successful application leaves no doubt that they are accurate reflections of observed disease processes and their associated risks or incidence rates. In spite of their prominence, it is not a priori evident why these models work. This article presents a derivation of the two models from the framework of canonical modeling. It begins with a general description of the dynamics between risk sources and disease development, formulates this description in the canonical representation of an S-system, and shows how the linear-logistic model and Cox's proportional hazard model follow naturally from this representation. The article interprets the model parameters in terms of epidemiological concepts as well as in terms of general systems theory and explains the assumptions and limitations generally accepted in the application of these epidemiological models.

Communicable Diseases↗
PubMed1984

Omaha childhood blood lead and environmental lead: a linear total exposure model.

The majority of experimental and population studies of blood lead (PbB) and environmental lead, including the Omaha study, have utilized the Goldsmith-Hexter log-log or power function model. Comparison was made of the log-log model and a linear model of total exposure to describe the Omaha Study of 1074 PbBs from children ages 1-18 years as related to air (PbA), soil (PbS), and housedust (PbHD) lead. The data fit of the linear model was statistically equivalent to the power model and the predicted curves were biologically more plausible. The linear model avoids the mathematical limitations of the power model which predicts PbB zero at PbA zero. From the Omaha data, this model, ln PbB = ln (beta 0 + B1 PbA + B2 PbS + beta 3 PbHD) predicts that PbB increases 1.92 micrograms/dl as PbA increases 1.0 microgram/m3. Since PbS and PbHD increase with PbA, however, the increases in total exposure predict a PbB increase of 4-5 micrograms/dl as PbA increases 1.0 microgram/m3.

Adolescent↗
PubMed1995

A non-linear mathematical model for the in vivo evaluation of the RES phagocytic function.

A new non-linear mathematical model was constructed in order to perform in vivo quantification of the RES phagocytic function. This method is based on the same technical facilities as used for the routine liver-spleen scintigraphy with radiocolloids [1, 2]. But kinetic modeling of dynamic Tc-99m-sulfur colloid data produced estimations of the functional RE-parameters: the clearance rate of the colloidal particles, the rate of phagocytosis, and the RES functional volume, which can not be obtained by classical approaches. This non-linear model was designed on the basis of the principal characteristics of particulate material interaction with macrophages (attachment, phagocytosis, digestion) [3, 4, 5]. The theoretically examined behavior of this in vivo mathematical model corresponds with the experimental behavior of the RES. The mathematical expression of the dynamics is the system of non-linear differential equations with constant coefficients that have no analytical solution. Fitting of the normalized heart blood time-activity curve was obtained to identify the unknown model parameters via non-linear regression. For this purpose general interactive PASCAL procedure IDPAR for a PDP-11/34 computer was used (an IBM PC version is also available). Two to three iterations were needed to estimate the set of unknown parameters for any patient study (1-1.5 min). A very good fitting was obtained between experimental and model curves in every case of different pathologies (error of the approximation is about 2-3%). Studies were performed using an in vivo bolus injection of 3.6 mg/80 kg commercially available colloid KOREN labeled with 3m-Ci 99m-Tc (analog of TCK-1). Our method was used to determine the RES functional parameters for patient groups with different levels of the RES dysfunction. Obtained results illustrate the possibilities of our technique to quantitatively estimate not only great pathology (portal cirrhosis), but also small changes of the RE-function (case of hyperlipidemia and ulcer gaster). In all patient groups marked changes of Tc-99m-sulfur colloid turnover were observed. In general, tracer clearance from the circulation was decreased, and the rate of phagocytosis and the RES volume were diminished compared with controls. The effect of a reduction of phagocytosis increases when the RES dysfunction becomes stronger. It can be shown that a non-parametric Wilcoxon-Mann-Whitney test gives a significant difference (P95%) for these patient groups. Further, we represent the possibility of using the model for monitoring changes of the RES-function parameters during and after therapy. The quantitative test of the RES function can significantly enhance the diagnosis and management of different diseases. Serial colloidal studies may document changes in the RES-function for the tumors, cirrhosis, hyperlipidemia, reticulosis, hepatitis, thrombosis, infection, AIDS, burn injury, shock and trauma patients. The technique may be useful for the different RES investigations with laboratory animals. Created computer software can be used as a tool for kinetic models, simulation, and unknown parameters identification.

Humans↗
PubMed1992

A comparison of linear and exponential models for drug expiry estimation.

While a large portion of pharmaceutical stability data is known to follow an exponential model decay, linear modeling of this data for expiry estimation is the norm. Expiry predictions based on linear and exponential fits to stability data were made to estimate the bias due to the linear fitting. It was found that within the usual expiry limits on drug potency, the difference between the model fits is relatively trivial. In cases of loss of potency greater than 15%, small assay variability, or great disparity between the length of the study and the time of expiry prediction, however, there is a nontrivial difference in the predictions and the exponential model is preferable.

Drug Compounding↗
PubMed1995

Residual plots for the censored data linear regression model.

To be consistent, censored data linear regression estimators typically require a correctly specified linear regression function and independent and identically distributed errors. For uncensored data one can assess these model assumptions informally by examining plots of the residuals against the independent variables or fitted values. In this paper I propose plots for censored data analogous to these uncensored data residual plots. One can use such plots in the same way as their uncensored data counterparts for checking model assumptions; if the model assumptions are correct, then the plots should exhibit a random scatter. I show that the proposed plots are useful in selecting a linear regression model for the Stanford heart transplant data.

Heart Transplantation↗
PubMedDate not supplied

Assessing the performance of neural encoding models in the presence of noise.

An analytical method is introduced for evaluating the performance of neural encoding models. The method addresses a critical question that arises during the course of the development and validation of encoding models: is a given model near optimal in terms of its accuracy in predicting the stimulus-elicited responses of a neural system, or can the predictive accuracy be improved significantly by further model development? The evaluation method is based on a derivation of the minimum mean-square error between actual responses and modeled responses. It is formulated as a comparison between the mean-square error of the candidate model and the theoretical minimum mean-square error attainable through an optimal model for the system. However, no a priori information about the nature of the optimal model is required. The theoretically minimum error is determined solely from the coherence function between pairs of system responses to repeated presentations of the same dynamic stimulus. Thus, the performance of the candidate model is judged against the performance of an optimal model rather than against that of an arbitrarily assumed model. Using this method. we evaluated a linear model for neural encoding by mechanosensory cells in the cricket cercal system. At low stimulus intensities, the best-fit linear model of encoding by single cells was found to be nearly optimal, even though the coherence between stimulus-response pairs (a commonly used measure of system linearity) was low. In this low-stimulus-intensity regime, the mean square error of the linear model was on the order of the power of the cell responses. In contrast, at higher stimulus intensities the linear model was not an accurate representation of neural encoding. even though the stimulus-response coherence was substantially higher than in the low-intensity regime.

Action Potentials↗
PubMed1996

A note on fitting a marginal model to mixed effects log-linear regression data via GEE.

Marginal generalized linear models for correlated data are often fit by generalized estimating equations (Liang, K.Y. and Zeger, S.L., 1986, Biometrika 73, 13-22), which requires the marginal expectation to be correctly specified while the covariance structure is allowed to be misspecified. This note is concerned with the correct specification of the marginal mean for data from a mixed log-linear regression model and the relationship between the true subject-specific parameters and the true marginal parameters.

Analysis of Variance↗
PubMed2004

Differential gene expression detection using penalized linear regression models: the improved SAM statistics.

UNLABELLED: Differential gene expression detection using microarrays has received lots of research interests recently. Many methods have been proposed, including variants of F-statistics, non-parametric approaches and empirical Bayesian methods etc. The SAM statistics has been shown to have good performance in empirical studies. SAM is more like an ad hoc shrinkage method. The idea is that for small sample microarray data, it is often useful to pool information across genes to improve efficiency. Under Bayesian framework Smyth formally derived the test statistics with shrinkage using the hierarchical models. In this paper we cast differential gene expression detection in the familiar framework of linear regression model. Commonly used test statistics correspond to using least squares to estimate the regression parameters. Based on the vast literature of research on linear models, we can naturally consider other alternatives. Here we explore the penalized linear regression. We propose the penalized t-/F-statistics for two-class microarray data based on [Formula: see text] penalty. We will show that the penalized test statistics intuitively makes sense and through applications we illustrate its good performance. AVAILABILITY: Supplementary information including program codes, more detailed analysis results and R functions for the proposed methods can be found at http://www.biostat.umn.edu/~baolin/research CONTACT: baolin@biostat.umn.edu SUPPLEMENTARY INFORMATION: http://www.biostat.umn.edu/~baolin/research.

Cell Line, Tumor↗
PubMed2003

Individual and school level effects of perceived harm, perceived availability, and community size on marijuana use among 12th-grade students: a random effects model.

A hierarchical linear model was used to estimate the individual and school level effects for marijuana use among a national sample of 12th-grade students. School effects were small in comparison to individual level effects, accounting for 2.9% of the variance in marijuana use. At the individual level, perceived harm, perceived availability, and their interaction were significant predictors, each of which varied randomly across schools. Among two school-level predictors, the normative environment for perceived harm was not significant, but normative perceived availability predicted level of marijuana use. The effect of perceived availability on marijuana use was stronger in larger, compared to smaller communities. Results are discussed in light of the use of random regression methods for identifying school-specific patterns of risk and protection for prevention planning.

Adolescent↗
PubMed2004

Residual analysis in linear regression models with an interval-censored covariate.

Residual analysis is a useful class of techniques for the evaluation of the goodness of a fitted model. Checking the underlying assumptions is important since most linear regression estimators require a correctly specified regression function and independent and identically distributed errors to be consistent. For uncensored data, the examination of the residuals of the fitted model is a standard tool for checking whether or not the underlying model assumptions hold. Such analysis has not been widely developed for censored data. Hillis (Statistics in Medicine 1995; 14:2023-2036) developed a residual plot for model checking when the response variable of a linear model is right-censored, and Gomez et al. (Statistics in Medicine 2003; 22:409-425) proposed residuals in models with interval-censored covariates. In this paper, we propose a new definition of residuals for linear models that incorporate interval-censored covariates. This definition can be also applied when the response variable is interval-censored. These new residuals are shown to perform better in model checking than other types of residuals in this context. We illustrate them with a data set from an AIDS clinical trial study.

Anti-HIV Agents↗
PubMed1999

Efficient linear elastic models of soft tissues for real-time surgery simulation.

In this paper, we describe the basic components of a surgery simulator prototype developed at INRIA. We present two physical models which are well suited for surgery simulation. These models are based on linear elasticity theory and finite elements modeling. The former model can deforme large tetrahedral meshes in real-time but does not allow any topological changes. On the contrary, the latter biomechanical model can simulate the cutting and tearing of soft tissue but must have a limited number of vertices to run in real-time. We propose a method for combining these two approaches into a hybrid model which may allow real time deformations and cuttings of large enough anatomical structures.

Biomechanical Phenomena↗
PubMed1997

Linearity and normalization in simple cells of the macaque primary visual cortex.

Simple cells in the primary visual cortex often appear to compute a weighted sum of the light intensity distribution of the visual stimuli that fall on their receptive fields. A linear model of these cells has the advantage of simplicity and captures a number of basic aspects of cell function. It, however, fails to account for important response nonlinearities, such as the decrease in response gain and latency observed at high contrasts and the effects of masking by stimuli that fail to elicit responses when presented alone. To account for these nonlinearities we have proposed a normalization model, which extends the linear model to include mutual shunting inhibition among a large number of cortical cells. Shunting inhibition is divisive, and its effect in the model is to normalize the linear responses by a measure of stimulus energy. To test this model we performed extracellular recordings of simple cells in the primary visual cortex of anesthetized macaques. We presented large stimulus sets consisting of (1) drifting gratings of various orientations and spatiotemporal frequencies; (2) plaids composed of two drifting gratings; and (3) gratings masked by full-screen spatiotemporal white noise. We derived expressions for the model predictions and fitted them to the physiological data. Our results support the normalization model, which accounts for both the linear and the nonlinear properties of the cells. An alternative model, in which the linear responses are subject to a compressive nonlinearity, did not perform nearly as well.

Animals↗
PubMed2000

Dynamic optimization of a linear-quadratic model with incomplete repair and volume-dependent sensitivity and repopulation.

PURPOSE: The linear-quadratic model typically assumes that tumor sensitivity and repopulation are constant over the time course of radiotherapy. However, evidence suggests that the growth fraction increases and the cell-loss factor decreases as the tumor shrinks. We investigate whether this evolution in tumor geometry, as well as the irregular time intervals between fractions in conventional hyperfractionation schemes, can be exploited by fractionation schedules that employ time-varying fraction sizes. METHODS: We construct a mathematical model of a spherical tumor with a hypoxic core and a viable rim, which is most appropriate for a prevascular tumor, and is only a caricature of a vascularized tumor. This model is embedded into the traditional linear-quadratic model by assuming instantaneous reoxygenation. Dynamic programming is used to numerically compute the fractionation regimen that maximizes the tumor-control probability (TCP) subject to constraints on the biologically effective dose of the early and late tissues. RESULTS: In several numerical examples that employ five or 10 fractions per week on a 1-cm or 5-cm diameter tumor, optimally varying the fraction sizes increases the TCP significantly. The optimal regimen incorporates large Friday (afternoon, if 10 fractions per week) fractions that are escalated throughout the course of treatment, and larger afternoon fractions than morning fractions. CONCLUSION: Numerical results suggest that a significant increase in tumor cure can be achieved by allowing the fraction sizes to vary throughout the course of treatment. Several strategies deserve further investigation: using larger fractions before overnight and weekend breaks, and escalating the dose (particularly on Friday afternoons) throughout the course of treatment.

Algorithms↗
PubMed2002

Bayesian prediction of spatial count data using generalized linear mixed models.

Spatial weed count data are modeled and predicted using a generalized linear mixed model combined with a Bayesian approach and Markov chain Monte Carlo. Informative priors for a data set with sparse sampling are elicited using a previously collected data set with extensive sampling. Furthermore, we demonstrate that so-called Langevin-Hastings updates are useful for efficient simulation of the posterior distributions, and we discuss computational issues concerning prediction.

Agriculture↗