Size distribution curves of senile plaques in patients with Alzheimer’s disease: do plaques grow according to the log-normal model?

The size frequency distributions of diffuse, primitive and cored senile plaques (SP) were studied in single sections of the temporal lobe from 10 patients with Alzheimer’s disease (AD). The size distribution curves were unimodal and positively skewed. The size distribution curve of the diffuse plaques was shifted towards larger plaques while those of the neuritic and cored plaques were shifted towards smaller plaques. The neuritic/diffuse plaque ratio was maximal in the 11 – 30 micron size class and the cored/ diffuse plaque ratio in the 21 – 30 micron size class. The size distribution curves of the three types of plaque deviated significantly from a log-normal distribution. Distributions expressed on a logarithmic scale were ‘leptokurtic’, i.e. with excess of observations near the mean. These results suggest that SP in AD grow to within a more restricted size range than predicted from a log-normal model. In addition, there appear to be differences in the patterns of growth of diffuse, primitive and cored plaques. If neuritic and cored plaques develop from earlier diffuse plaques, then smaller diffuse plaques are more likely to be converted to mature plaques.

Statnote 35: are the data log-normal?

In many of the Statnotes described in this series, the statistical tests assume the data are a random sample from a normal distribution These Statnotes include most of the familiar statistical tests such as the ‘t’ test, analysis of variance (ANOVA), and Pearson’s correlation coefficient (‘r’). Nevertheless, many variables exhibit a more or less ‘skewed’ distribution. A skewed distribution is asymmetrical and the mean is displaced either to the left (positive skew) or to the right (negative skew). If the mean of the distribution is low, the degree of variation large, and when values can only be positive, a positively skewed distribution is usually the result. Many distributions have potentially a low mean and high variance including that of the abundance of bacterial species on plants, the latent period of an infectious disease, and the sensitivity of certain fungi to fungicides. These positively skewed distributions are often fitted successfully by a variant of the normal distribution called the log-normal distribution. This statnote describes fitting the log-normal distribution with reference to two scenarios: (1) the frequency distribution of bacterial numbers isolated from cloths in a domestic environment and (2), the sizes of lichenised ‘areolae’ growing on the hypothalus of Rhizocarpon geographicum (L.) DC.

Size frequency distribution of the β-amyloid (aβ) deposits in dementia with Lewy bodies with associated Alzheimer’s disease pathology

The objective is to study beta-amyloid (Abeta) deposition in dementia with Lewy bodies (DLB) with Alzheimer's disease (AD) pathology (DLB/AD). The size frequency distributions of the Abeta deposits were studied and fitted by log-normal and power-law models. Patients were ten clinically and pathologically diagnosed DLB/AD cases. Size distributions had a single peak and were positively skewed and similar to those described in AD and Down's syndrome. Size distributions had smaller means in DLB/AD than in AD. Log-normal and power-law models were fitted to the size distributions of the classic and diffuse deposits, respectively. Size distributions of Abeta deposits were similar in DLB/AD and AD. Size distributions of the diffuse deposits were fitted by a power-law model suggesting that aggregation/disaggregation of Abeta was the predominant factor, whereas the classic deposits were fitted by a log-normal distribution suggesting that surface diffusion was important in the pathogenesis of the classic deposits.

Do β-amyloid (Aβ) deposits in patients with Alzheimer's disease and Down's syndrome grow according to the log-normal model?

The size frequency distributions of diffuse, primitive and classic β- amyloid (Aβ) deposits were studied in single sections of cortical tissue from patients with Alzheimer's disease (AD) and Down's syndrome (DS) and compared with those predicted by the log-normal model. In a sample of brain regions, these size distributions were compared with those obtained by serial reconstruction through the tissue and the data used to adjust the size distributions obtained in single sections. The adjusted size distributions of the diffuse, primitive and classic deposits deviated significantly from a log-normal model in AD and DS, the greatest deviations from the model being observed in AD. More Aβ deposits were observed close to the mean and fewer in the larger size classes than predicted by the model. Hence, the growth of Aβ deposits in AD and DS does not strictly follow the log-normal model, deposits growing to within a more restricted size range than predicted. However, Aβ deposits grow to a larger size in DS compared with AD which may reflect differences in the mechanism of Aβ formation.

Factors determining the size frequency distribution of β-amyloid (Aβ) deposits in Alzheimer's disease

The size frequency distributions of discrete β-amyloid (Aβ) deposits were studied in single sections of the temporal lobe from patients with Alzheimer's disease. The size distributions were unimodal and positively skewed. In 18/25 (72%) tissues examined, a log normal distribution was a good fit to the data. This suggests that the abundances of deposit sizes are distributed randomly on a log scale about a mean value. Three hypotheses were proposed to account for the data: (1) sectioning in a single plane, (2) growth and disappearance of Aβ deposits, and (3) the origin of Aβ deposits from clusters of neuronal cell bodies. Size distributions obtained by serial reconstruction through the tissue were similar to those observed in single sections, which would not support the first hypothesis. The log normal distribution of Aβ deposit size suggests a model in which the rate of growth of a deposit is proportional to its volume. However, mean deposit size and the ratio of large to small deposits were not positively correlated with patient age or disease duration. The frequency distribution of Aβ deposits which were closely associated with 0, 1, 2, 3, or more neuronal cell bodies deviated significantly from a log normal distribution, which would not support the neuronal origin hypothesis. On the basis of the present data, growth and resolution of Aβ deposits would appear to be the most likely explanation for the log normal size distributions.

Income distribution dependence of poverty measure:a theoretical analysis

Using a modified deprivation (or poverty) function, in this paper, we theoretically study the changes in poverty with respect to the 'global' mean and variance of the income distribution using Indian survey data. We show that when the income obeys a log-normal distribution, a rising mean income generally indicates a reduction in poverty while an increase in the variance of the income distribution increases poverty. This altruistic view for a developing economy, however, is not tenable anymore once the poverty index is found to follow a pareto distribution. Here although a rising mean income indicates a reduction in poverty, due to the presence of an inflexion point in the poverty function, there is a critical value of the variance below which poverty decreases with increasing variance while beyond this value, poverty undergoes a steep increase followed by a decrease with respect to higher variance. Identifying this inflexion point as the poverty line, we show that the pareto poverty function satisfies all three standard axioms of a poverty index [N.C. Kakwani, Econometrica 43 (1980) 437; A.K. Sen, Econometrica 44 (1976) 219] whereas the log-normal distribution falls short of this requisite. Following these results, we make quantitative predictions to correlate a developing with a developed economy. © 2006 Elsevier B.V. All rights reserved.

Size frequency distribution of prion protein (PrP) aggregates in variant Creutzfeldt-Jakob disease (vCJD)

The frequency distribution of aggregate size of the diffuse and florid-type prion protein (PrP) plaques was studied in various brain regions in cases of variant Creutzfeldt-Jakob disease (vCJD). The size distributions were unimodal and positively skewed and resembled those of β-amyloid (Aβ) deposits in Alzheimer's disease (AD) and Down's syndrome (DS). The frequency distributions of the PrP aggregates were log-normal in shape, but there were deviations from the expected number of plaques in specific size classes. More diffuse plaques were observed in the modal size class and fewer in the larger size classes than expected and more florid plaques were present in the larger size classes compared with the log-normal model. It was concluded that the growth of the PrP aggregates in vCJD does not strictly follow a log-normal model, diffuse plaques growing to within a more restricted size range and florid plaques to larger sizes than predicted. © Springer-Verlag 2005.

Size frequency distribution of beta-amyloid (Abeta) deposits in dementia with Lewy bodies

In Alzheimer's disease (AD) and Down's syndrome (DS), the size frequency distribution of the beta-amyloid (Abeta) deposits can be described by a log-normal model and may indictae the growth of the deposits. This study determined the size frequency distribution of the Abeta deposits in the temporal lobe in 8 casaes of dementia with Lewy bodies (DLB) with associated AD pathology (DLB/AD. The size distributions of Abeta deposits were unimodal and positively skewed; the mean size of deposi and the degree of skew varying with deposit type and brain region. Size distributions of the primitive deposits had lower means and were less skewed compared with the diffuse and classic deposits. In addition, size distributions in the hippocampus and parahippocampal gyrus (PHG) had larger means and a greater degree of skew compared with other cortical gyri. All size distributions deviated significantly from a log-normal model. There were more Abeta deposits than expected in the smaller size classes and fewer than expected near the mean and in the larger size classes. The data suggest thatthe pattern of growth of the Abeta deposits in DLB/AD depends both on deposit morphology and brain area. In addition, Abeta deposits in DLB appear to grow to within a more restricted size range than predicted and hence, to have less potential for growth compared with cases of 'pure' AD and DS.

Modelling the growth of prion protein aggregates in the brain in variant Creutzfeldt-Jakob disease

Deposition of insoluble prion protein (PrP) in the brain in the form of protein aggregates or deposits is characteristic of the ‘transmissible spongiform encephalopathies’ (TSEs). Understanding the growth and development of these PrP aggregates is important both in attempting to the elucidate of the pathogenesis of prion disease and in the development of treatments designed to prevent or inhibit the spread of prion pathology within the brain. Aggregation and disaggregation of proteins and the diffusion of substances into the developing aggregates (surface diffusion) are important factors in the development of protein aggregates. Mathematical models suggest that if aggregation/disaggregation or surface diffusion is the predominant factor, the size frequency distribution of the resulting protein aggregates in the brain should be described by either a power-law or a log-normal model respectively. This study tested this hypothesis for two different types of PrP deposit, viz., the diffuse and florid-type PrP deposits in patients with variant Creutzfeldt-Jakob disease (vCJD). The size distributions of the florid and diffuse plaques were fitted by a power-law function in 100% and 42% of brain areas studied respectively. By contrast, the size distributions of both types of plaque deviated significantly from a log-normal model in all brain areas. Hence, protein aggregation and disaggregation may be the predominant factor in the development of the florid plaques. A more complex combination of factors appears to be involved in the pathogenesis of the diffuse plaques. These results may be useful in the design of treatments to inhibit the development of protein aggregates in vCJD.

Liquid-liquid extraction in a pilot scale rotating disc contactor

A study of the hydrodynamics and mass transfer characteristics of a liquid-liquid extraction process in a 450 mm diameter, 4.30 m high Rotating Disc Contactor (R.D.C.) has been undertaken. The literature relating to this type of extractor and the relevant phenomena, such as droplet break-up and coalescence, drop mass transfer and axial mixing has been revjewed. Experiments were performed using the system C1airsol-350-acetone-water and the effects of drop size, drop size-distribution and dispersed phase hold-up on the performance of the R.D.C. established. The results obtained for the two-phase system C1airso1-water have been compared with published correlations: since most of these correlations are based on data obtained from laboratory scale R.D.C.'s, a wide divergence was found. The hydrodynamics data from this study have therefore been correlated to predict the drop size and the dispersed phase hold-up and agreement has been obtained with the experimental data to within +8% for the drop size and +9% for the dispersed phase hold-up. The correlations obtained were modified to include terms involving column dimensions and the data have been correlated with the results obtained from this study together with published data; agreement was generally within +17% for drop size and within +14% for the dispersed phase hold-up. The experimental drop size distributions obtained were in excellent agreement with the upper limit log-normal distributions which should therefore be used in preference to other distribution functions. In the calculation of the overall experimental mass transfer coefficient the mean driving force was determined from the concentration profile along the column using Simpson's Rule and a novel method was developed to calculate the overall theoretical mass transfer coefficient Kca1, involving the drop size distribution diagram to determine the volume percentage of stagnant, circulating and oscillating drops in the sample population. Individual mass transfer coefficients were determined for the corresponding droplet state using different single drop mass transfer models. Kca1 was then calculated as the fractional sum of these individual coefficients and their proportions in the drop sample population. Very good agreement was found between the experimental and theoretical overall mass transfer coefficients. Drop sizes under mass transfer conditions were strongly dependant upon the direction of mass transfer. Drop Sizes in the absence of mass transfer were generally larger than those with solute transfer from the continuous to the dispersed phase, but smaller than those with solute transfer in the opposite direction at corresponding phase flowrates and rotor speed. Under similar operating conditions hold-up was also affected by mass transfer; it was higher when solute transfered from the continuous to the dispersed phase and lower when direction was reversed compared with non-mass transfer operation.

Solvent extraction of phosphoric acid

This work investigated the purification of phosphoric acid using a suitable organic solvent, followed by re-extraction of the acid from the solvent using water. The work consisted of practical batch and continuous studies and the economics and design of a full scale plant, based on the experimental data. A comprehensive literature survey on the purification of wet process phosphoric acid by organic solvents is presented and the literature describing the design and operation of mixer-settlers has also been reviewed. In batch studies, the equilibrium and distribution curves for the systems water-phosphoric acid-solvent for Benzaldehyde, Cyclohexanol and Methylisobutylketone (MIBK) were determined together with hydrodynamic characteristics for both pure and impure systems. The settling time increased with acid concentration, but power input had no effect. Drop size was found to reduce with acid concentration and power input. For the continuous studies a novel horizontal mixer~settler cascade was designed, constructed and operated using pure and impure acid with MIBK as the solvent. The cascade incorporates three air turbine agitated, cylindrical 900 ml mixers, and three cylindrical 200 ml settlers with air-lift solvent interstage transfer. Mean drop size in the fully baffled mixer was correlated. Drop size distributions were log-normal and size decreased with acid concentration and power input and increased with dispersed phase hold-up. Phase inversion studies showed that the width of the ambivalent region depended upon rotor speed, hold-up and acid concentration. Settler characteristics were investigated by measuring wedge length. Distribution coefficients of impurities and acid were also investigated. The following optimum extraction conditions were found: initial acid concentration 63%, phase ratio of solvent to acid 1:1 (v/v), impeller speed recommended 900 r.p.m. In the washing step the maximum phase ratio of solvent to water was 8:1 (v/v). Work on phosphoric acid concentration involved constructing distillation equipment consisting of a 10& spherical still. A 100 T/d scale detailed process design including capital cost, operating cost and profitability was also completed. A profit model for phosphoric acid extraction was developed and maximised. Recommendations are made for both the application of the results to a practical design and for extensions of the study.

Modelling growth of prion protein aggregates in the brain in variant Creutzfeldt-Jakob disease

Deposition of insoluble prion protein (PrP) in the brain in the form of protein aggregates or deposits is characteristic of the ‘transmissible spongiform encephalopathies’ (TSEs). Understanding the growth and development of PrP aggregates is important both in attempting to elucidate the pathogenesis of prion disease and in the development of treatments designed to inhibit the spread of prion pathology within the brain. Aggregation and disaggregation of proteins and the diffusion of substances into the developing aggregates (surface diffusion) are important factors in the development of protein deposits. Mathematical models suggest that if either aggregation/disaggregation or surface diffusion is the predominant factor, then the size frequency distribution of the resulting protein aggregates will be described by either a power-law or a log-normal model respectively. This study tested this hypothesis for two different populations of PrP deposit, viz., the diffuse and florid-type PrP deposits characteristic of patients with variant Creutzfeldt-Jakob disease (vCJD). The size distributions of the florid and diffuse deposits were fitted by a power-law function in 100% and 42% of brain areas studied respectively. By contrast, the size distributions of both types of aggregate deviated significantly from a log-normal model in all areas. Hence, protein aggregation and disaggregation may be the predominant factor in the development of the florid deposits. A more complex combination of factors appears to be involved in the pathogenesis of the diffuse deposits. These results may be useful in the design of treatments to inhibit the development of PrP aggregates in vCJD.

A quantitative study of abnormally enlarged neurons in cognitively normal brain and neurodegenerative disease

Abnormally enlarged neurons (AEN) occur in many neurodegenerative diseases. To define AEN more objectively, the frequency distribution of the ratio of greatest cell diameter(CD) to greatest nuclear diameter (ND) was studied in populations of cortical neurons in tissue sections of seven cognitively normal brains. The frequency distribution of CD/ND deviated from a normal distribution in 15 out of 18 populations of neurons studied and hence, the 95th percentile (95P) was used to define a limit of the CD/ND ratio excluding the5% most extreme observations. The 95P of the CD/ ND ratio varied from 2.0 to 3.0 in different cases and regions and a value of 95P = 3.0 was chosen to define the limit for normalneurons under non-pathological conditions. Based on the 95P = 3.0 criterion, the proportion of AEN with a CD/ND ≥ 3 varied from 2.6% in Alzheimer's disease (AD) to 20.3% in Pick's disease (PiD). The data suggest: (1) that a CL/ND ≥ 3.0 may be a useful morphological criterion for defining AEN, and (2) AEN were most numerous in PiD and corticobasal degeneration (CBD) and least abundant in AD and in dementia with Lewy bodies (DLB). © 2013 Dustri-Verlag Dr. K. Feistle.

Statnote 36: Do the data fit the Poisson distribution

In previous Statnotes, many of the statistical tests described rely on the assumption that the data are a random sample from a normal or Gaussian distribution. These include most of the tests in common usage such as the ‘t’ test ), the various types of analysis of variance (ANOVA), and Pearson’s correlation coefficient (‘r’) . In microbiology research, however, not all variables can be assumed to follow a normal distribution. Yeast populations, for example, are a notable feature of freshwater habitats, representatives of over 100 genera having been recorded . Most common are the ‘red yeasts’ such as Rhodotorula, Rhodosporidium, and Sporobolomyces and ‘black yeasts’ such as Aurobasidium pelculans, together with species of Candida. Despite the abundance of genera and species, the overall density of an individual species in freshwater is likely to be low and hence, samples taken from such a population will contain very low numbers of cells. A rare organism living in an aquatic environment may be distributed more or less at random in a volume of water and therefore, samples taken from such an environment may result in counts which are more likely to be distributed according to the Poisson than the normal distribution. The Poisson distribution was named after the French mathematician Siméon Poisson (1781-1840) and has many applications in biology, especially in describing rare or randomly distributed events, e.g., the number of mutations in a given sequence of DNA after exposure to a fixed amount of radiation or the number of cells infected by a virus given a fixed level of exposure. This Statnote describes how to fit the Poisson distribution to counts of yeast cells in samples taken from a freshwater lake.

Towards a framework for combining stochastic and deterministic descriptions of nonstationary financial time series

We present in this paper ideas to tackle the problem of analysing and forecasting nonstationary time series within the financial domain. Accepting the stochastic nature of the underlying data generator we assume that the evolution of the generator's parameters is restricted on a deterministic manifold. Therefore we propose methods for determining the characteristics of the time-localised distribution. Starting with the assumption of a static normal distribution we refine this hypothesis according to the empirical results obtained with the methods anc conclude with the indication of a dynamic non-Gaussian behaviour with varying dependency for the time series under consideration.