Estimation of the diagnostic threshold accounting for decision costs and sampling uncertainty

Medical diagnostic tests are used to classify subjects as non-diseased or diseased. The classification rule usually consists of classifying subjects using the values of a continuous marker that is dichotomised by means of a threshold. Here, the optimum threshold estimate is found by minimising a cost function that accounts for both decision costs and sampling uncertainty. The cost function is optimised either analytically in a normal distribution setting or empirically in a free-distribution setting when the underlying probability distributions of diseased and non-diseased subjects are unknown. Inference of the threshold estimates is based on approximate analytically standard errors and bootstrap-based approaches. The performance of the proposed methodology is assessed by means of a simulation study, and the sample size required for a given confidence interval precision and sample size ratio is also calculated. Finally, a case example based on previously published data concerning the diagnosis of Alzheimer's patients is provided in order to illustrate the procedure.     

Age structure in predator-prey systems. I. A general model and a specific example

To study the effects of age structure in predator-prey systems, a general, analytically tractable model is formulated and solved. We demonstrate the usefulness of the model in a study of a specific system of two mites. We show that to maintain stable equilibrium between the herbaceous (pest) mite and the predacious mite, the nonintuitive strategy of reducing the growth rate of the predator may be necessary. The modelling technique allows a determination of the magnitude of the effect of age structure on stability.     

The dynamics of continuous microbial culture described by cell age distribution and concentration of one substrate

A mathematical model has been considered in which the known equation of McKendrick and Von Foerster for cell age distribution is combined with that for substrate concentration. The dependence of cell division rate on cell age has been taken as a step function. The interrelation between culture parameters describing the substrate consumption and cell division has been found. The shape of cell age distribution as well as the values of substrate and cell concentrations in steady and transient states have been investigated. Stationary regimes at the initial culture state synchronized by ages have been found to be established as damped oscillations and age waves. Under definite conditions the transition from one steady growth regime to another includes sharp single-time age synchronization of the culture.     

Risk prediction of complex diseases from family history and known susceptibility loci, with applications for cancer screening

Risk prediction based on genomic profiles has raised a lot of attention recently. However, family history is usually ignored in genetic risk prediction. In this study we proposed a statistical framework for risk prediction given an individual's genotype profile and family history. Genotype information about the relatives can also be incorporated. We allow risk prediction given the current age and follow-up period and consider competing risks of mortality. The framework allows easy extension to any family size and structure. In addition, the predicted risk at any percentile and the risk distribution graphs can be computed analytically. We applied the method to risk prediction for breast and prostate cancers by using known susceptibility loci from genome-wide association studies. For breast cancer, in the population the 10-year risk at age 50 ranged from 1.1% at the 5th percentile to 4.7% at the 95th percentile. If we consider the average 10-year risk at age 50 (2.39%) as the threshold for screening, the screening age ranged from 62 at the 20th percentile to 38 at the 95th percentile (and some never reach the threshold). For women with one affected first-degree relative, the 10-year risks ranged from 2.6% (at the 5th percentile) to 8.1% (at the 95th percentile). For prostate cancer, the corresponding 10-year risks at age 60 varied from 1.8% to 14.9% in the population and from 4.2% to 23.2% in those with an affected first-degree relative. We suggest that for some diseases genetic testing that incorporates family history can stratify people into diverse risk categories and might be useful in targeted prevention and screening.     

SIMS: computation of a smooth invariant molecular surface.

SIMS, a new method of calculating a smooth invariant molecular dot surface, is presented. The SIMS method generates the smooth molecular surface by rolling two probe spheres. A solvent probe sphere is rolled over the molecule and produces a Richards-Connolly molecular surface (MS), which envelops the solvent-excluded volume of the molecule. In deep crevices, Connolly's method of calculating the MS has two deficiencies. First, it produces self-intersecting parts of the molecular surface, which must be removed to obtain the correct MS. Second, the correct MS is not smooth, i.e., the direction of the normal vector of the MS is not continuous, and some points of the MS are singular. We present an exact method for removing self-intersecting parts and smoothing the singular regions of the MS. The singular MS is smoothed by rolling a smoothing probe sphere over the inward side of the singular MS. The MS in the vicinity of singularities is replaced with the reentrant surface of the smoothing probe sphere. The smoothing method does not disturb the topology of a singular MS, and the smooth MS is a better approximation of the dielectric border between high dielectric solvent and the low dielectric molecular interior. The SIMS method generates a smooth molecular dot surface, which has a quasi-uniform dot distribution in two orthogonal directions on the molecular surface, which is invariant with molecular rotation and stable under changes in the molecular conformation, and which can be used in a variety of implicit methods of modeling solvent effects. The SIMS program is faster than the Connolly MS program, and in a matter of seconds generates a smooth dot MS of a 200-residue protein. The program is available from the authors on request (see http:@femto.med.unc.edu/SIMS).     

The structural organization of mouse chromatin as a function of age.

Chromatin is organized into a repeating structure (nucleosome) made up of proteins and DNA. Micrococcal nuclease and DNAase I have been used to probe this structure in nuclear populations from three tissues (liver, brain, and heart) of the inbred mouse strain C57BL at different ages. For those parameters examined, for each tissue, chromatin contained essentially the same features of nucleosomal organization, regardless of the age of the mouse. Thus, the rate and extent of nuclease digestion and the size of the DNA repeat unit and nucleosome core are not significantly different as a function of age. However, the accessibility of internucleosomal DNA to micrococcal nuclease, as determined by measuring the DNA size distribution after nuclease cutting, may be partially limited in chromatin of brain (but not liver or heart) of older animals. These results indicate that there are no gross, age-related changes in the conformational state or organization of chromatin in these tissues. The results do not exclude smaller alterations in chromatin that might occur with age, which the current methodology might not be sensitive enough to detect.     

Towards robust evolutionary inference with integral projection models

Integral projection models (IPMs) are extremely flexible tools for ecological and evolutionary inference. IPMs track the distribution of phenotype in populations through time, using functions describing phenotype‐dependent development, inheritance, survival and fecundity. For evolutionary inference, two important features of any model are the ability to (i) characterize relationships among traits (including values of the same traits across ages) within individuals, and (ii) characterize similarity between individuals and their descendants. In IPM analyses, the former depends on regressions of observed trait values at each age on values at the previous age (development functions), and the latter on regressions of offspring values at birth on parent values as adults (inheritance functions). We show analytically that development functions, characterized this way, will typically underestimate covariances of trait values across ages, due to compounding of regression to the mean across projection steps. Similarly, we show that inheritance, characterized this way, is inconsistent with a modern understanding of inheritance, and underestimates the degree to which relatives are phenotypically similar. Additionally, we show that the use of a constant biometric inheritance function, particularly with a constant intercept, is incompatible with evolution. Consequently, current implementations of IPMs will predict little or no phenotypic evolution, purely as artefacts of their construction. We present alternative approaches to constructing development and inheritance functions, based on a quantitative genetic approach, and show analytically and through an empirical example on a population of bighorn sheep how they can potentially recover patterns that are critical to evolutionary inference.     

Pathology and physiology of microbes. III. Criteria for assessing functional states

The study of different characteristics of the vital activity of microorganisms, made on type B Clostridium perfringens taken as an example, has disclosed the difficulty of evaluating the functional state of microorganisms only on the basis of the traditionally used characteristics of the population activity: the specific growth rate, the toxicity of the culture fluid and the antigenic properties of the culture. The economic and metabolic coefficients have been found to be the most suitable criteria for such evaluation; the state of the maximum physiological activity of the population is characterized by the maximum values of the economic coefficient and the minimum values of the metabolic coefficient. The minimum values of the economic coefficient and the maximum values of the metabolic coefficient are characteristic of the pathological state of the population.     

The Impact of Heterogeneous Thresholds on Social Contagion with Multiple Initiators

The threshold model is a simple but classic model of contagion spreading in complex social systems. To capture the complex nature of social influencing we investigate numerically and analytically the transition in the behavior of threshold-limited cascades in the presence of multiple initiators as the distribution of thresholds is varied between the two extreme cases of identical thresholds and a uniform distribution. We accomplish this by employing a truncated normal distribution of the nodes’ thresholds and observe a non-monotonic change in the cascade size as we vary the standard deviation. Further, for a sufficiently large spread in the threshold distribution, the tipping-point behavior of the social influencing process disappears and is replaced by a smooth crossover governed by the size of initiator set. We demonstrate that for a given size of the initiator set, there is a specific variance of the threshold distribution for which an opinion spreads optimally. Furthermore, in the case of synthetic graphs we show that the spread asymptotically becomes independent of the system size, and that global cascades can arise just by the addition of a single node to the initiator set.     

Intracellular freezing of human granulocytes

Human granulocyte suspensions were exposed to controlled freezing regimens on a cryomicroscope, and the incidence of intracellular freezing was measured as a function of cooling rate and extracellular nucleation temperature. The presence of intracellular ice was assessed by analysis of serially recorded images of the freeze-thaw process and by correlation with measured patterns of change in the cell volume. For granulocytes suspended in autologous plasma, a threshold was described for intracellular freezing as an empirical function of cooling rate (B) and extracellular nucleation temperature (Tn): B (degrees C/min) = 1.1 Tn (degrees C) + 12.3.     

Adaptive dynamics of cooperation may prevent the coexistence of defectors and cooperators and even cause extinction

It has recently been demonstrated that ecological feedback mechanisms can facilitate the emergence and maintenance of cooperation in public goods interactions: the replicator dynamics of defectors and cooperators can result, for example, in the ecological coexistence of cooperators and defectors. Here we show that these results change dramatically if cooperation strategy is not fixed but instead is a continuously varying trait under natural selection. For low values of the factor with which the value of resources is multiplied before they are shared among all participants, evolution will always favour lower cooperation strategies until the population falls below an Allee threshold and goes extinct, thus evolutionary suicide occurs. For higher values of the factor, there exists a unique evolutionarily singular strategy, which is convergence stable. Because the fitness function is linear with respect to the strategy of the mutant, this singular strategy is neutral against mutant invasions. This neutrality disappears if a nonlinear functional response in receiving benefits is assumed. For strictly concave functional responses, singular strategies become uninvadable. Evolutionary branching, which could result in the evolutionary emergence of cooperators and defectors, can occur only with locally convex functional responses, but we illustrate that it can also result in coevolutionary extinction.     

Instability and excitation propagation in a catalytic reaction model. II. Distributed system model]

The regime of excitation propagation in monomeric model of active medium is considered. Conditions exist when the phase plane falls into two regions in both of which the equations are linear and their automodel solutions are found analytically. After connecting the solutions a transcendental algebraic equation is obtained, from which the values of excitation propagation rate can be found. The trigger wave has one rate value the propagating impulse--two (physically realised--one). There is a parameter region with four rates (three physically realised ones).     

Population dynamics in variable environments. II. Correlated environments,sensitivity analysis and dynamics

Mean and mean square number are studied for age-structured populations with serially correlated temporally fluctuating vital rates. Results are that (1) Moments of population number can be used effectively to analyse growth rates of the coefficient of variation and an approximate median population number. (2) Analytical approximations to the growth rates of moments reveal dynamic consequences of covarying phenotypic traits and of temporal correlation along environmental sequences. (3) Dynamic properties can be explicitly related to the static sensitivity of an average vital rate matrix. (4) The use of (1), (2) and (3) allows an extension of many applications of static vital rate theory to dynamics with fluctuating rates.     

Ventilatory anaerobic threshold in healthy children. Age and sex differences

The ventilatory anaerobic threshold (VAT) during graded exercise was defined as the oxygen uptake (VO2) immediately below the exercise intensity at which pulmonary ventilation increased disproportionally relative to VO2. Since VAT is considered to be a sensitive and noninvasive measure for evaluating cardiorespiratory endurance performance, the purpose of the present study was to determine normal values in children. We examined 257 healthy children (140 boys and 117 girls) varying in age from 5.7 to 18.5 years, during treadmill exercise. The data were analyzed in relation to sex and age. In boys the lowest VO2max (ml X min-1 X kg-1) was found in the youngest age group (5-6 year). In girls, on the other hand, no significant increase occurred with age. For VAT, expressed as ml O2 X min-1 X kg-1 or as a percent of VO2max, a significant decrease was found in boys and girls with age. This suggests an increase in lactacid anaerobic capacity during growth. In contrast to observations in adults, only low correlations were found between VO2max and VAT (r = 0.28 in boys and r = 0.52 in girls), which suggests that the development of the underlying physiological mechanism does not occur at the same rate in growing children. These data provide normal values for VAT that can be used for clinical exercise testing in the pediatric age group.     

Four-state models and regulation of contraction of smooth muscle. I. Physical considerations, stability, and solutions.

Cyclic four-state models are frequently used in biology to represent a variety of molecular behaviors. A common experimental strategy to test such models is to follow the behavior of the real system after some of the rate constants are changed in a stepwise manner. We analyze the mathematical behavior of a simple example of such a model applicable to the regulation of contraction of smooth muscle, but our results apply in general to any linear, cyclic four-state model. We discuss detailed balance and requirements for linearity. We find that the only way to have sustained oscillations is for the rate constants of the model to be themselves oscillatory. We state conditions for decaying oscillations and find that in models that do not follow strictly first-order kinetics and do not satisfy detailed balance, these conditions can hold. We show analytically that the response of any state to step changes in the rate constants is the sum of three weighted exponentials plus a constant term, the steady-state value. We provide explicit expressions for the time dependence of all state variables. We discuss a simple way to use these results to obtain numerical solutions in cases where the rate constants change in an arbitrary way.     

Spike tranfer in statistical neuron assemblies. II. Neuron--nonlinear spike source]

The mathematical model of the neuron function is known to rely on space summing of excitement. The spikes contribute to the inner state of the neuron the farther from cell soma the synapses are located. The difference between excitatory and inhibitory effect results in spike firing if only neural firing threshold is achieved. The values of spike flux have been estimated on the basis of the model of CA3 sector of the Hippocampus and were found to be 15 divided by 35 imp/s.     

Quantitative description of cell cycle kinetics under chemotherapy utilizing flow cytometry.

A discrete time cell cycle kinetics model is developed to account for the effects of cytotoxic chemotherapy, particularly including the existence of cells destined to die. A model structure is determined from related experiments, leaving key parameter values undetermined. These values are found by determining the best least squares fit of the predicted to the observed DNA distribution data at a series of time intervals. The numerical methods include separable least squares, linear inequality constrained least squares and the Gauss--Newton method. This approach is applied to an experiment in which the Ehrlich ascites tumour was given a single dose of bleomycin. The results include several different parameters, including the age response function and a time series of cell age and DNA distributions, which can be used as a basis for further treatment.     

The faithful copy neuron

Theoretical and experimental evidence is presented for the presence in nervous tissue of neurons whose firing rate faithfully follow their input stimulus. Such neurons are shown to deliver their spikes with minimum dissipation per spike. This optimal performance is likely accomplished by use of local circuitry that adjusts conductances to match input currents so that the neuron operates near the threshold for firing. This results in an unusual mechanism for neuronal firing that uses background noise to achieve the desired firing rate. This framework takes place dynamically, and the present deliberations apply under time varying conditions. It is shown that an analytically explicit probability distribution function, which depends on one dimensionless parameter, can account for the interspike interval statistics under general time varying conditions. An innovative analysis based on the unsteady firing rate fits data to the appropriate probability distribution function.     

The stochastic evolution of catalysts in spatially resolved molecular systems.

A fully stochastic chemical modelling technique is derived which describes the influence of spatial separation and discrete population size on the evolutionary stability of coupled amplification in biopolymers. The model is analytically tractable for an infinite-dimensional space (simplex geometry), which also provides insight into evolution in normal Euclidean space. The results are compared with stochastic simulations describing the co-evolution of combinatorial families of molecular sequences both in the simplex geometry and in lower (one, two and three) space dimensions. They demonstrate analytically the generic limits which exploitation place on co-evolving multi-component amplification systems. In particular, there is an optimal diffusion (or migration) coefficient for cooperative amplification and minimal and maximal threshold values for stable cooperation. Over a bounded range of diffusion rates, the model also exhibits stable limit cycles. Furthermore, the co-operatively coupled system has a maximum tolerable error rate at intermediate rates of diffusion. A tractable model is thereby established which demonstrates that spatial effects can stabilize catalytic biological information. The analytic behaviour in infinite-dimensional simplex space is seen to provide a reasonable guide to the spatial dependence of the error threshold in physical space. Nanoscale possibilities for the evolution of catalysis on the basis of the model are outlined. We denote the modelling technique by PRESS, Probability Reduced Evolution of Spatially-discrete Species.