Gompertz Research Articles

The entropy-transformed Gompertz distribution: Distributional insights and cross-disciplinary utilizations

A novel two-parameter continuous model titled the entropy-transformed Gompertz (ETGPZ) distribution has been developed via the entropy transform. A new framework has been investigated and found to meet the criteria of the probability function. By significantly improving the functional shape and having the ability to model the most likely form of the hazard rate function, this new modification has increased the adaptability of the typical distribution. Some of its core characteristics, such as its statistical and computational features, are clearly presented. A thorough simulation analysis has been done to examine the final behavior of maximum likelihood estimators while estimating model parameters. We assess the performance and practical applicability of the ETGPZ distribution using eight real datasets from engineering and biomedical fields. The results demonstrate that the ETGPZ outperforms the baseline Gompertz (GPZ) distribution, highlighting its superiority and broader potential for various applications.

Analysis of the inflection points of height-diameter models

The inflection point is an important feature of sigmoidal height-diameter (H-D) models. It is often cited as one of the properties favoring sigmoidal model forms. However, there are very few studies analyzing the inflection points of H-D models. The goals of this study were to theoretically and empirically examine the behaviors of inflection points of six common H-D models with a regional dataset. The six models were the Wykoff (WYK), Schumacher (SCH), Curtis (CUR), Hossfeld IV (HOS), von Bertalanffy-Richards (VBR), and Gompertz (GPZ) models. The models were first fitted in their base forms with tree species as random effects and were then expanded to include functional traits and spatial distribution. The distributions of the estimated inflection points were similar between the two-parameter models WYK, SCH, and CUR, but were different between the three-parameter models HOS, VBR, and GPZ. GPZ produced some of the largest inflection points. HOS and VBR produced concave H-D curves without inflection points for 12.7% and 39.7% of the tree species. Evergreen species or decreasing shade tolerance resulted in larger inflection points. The trends in the estimated inflection points of HOS and VBR were entirely opposite across the landscape. Furthermore, HOS could produce concave H-D curves for portions of the landscape. Based on the studied behaviors, the choice between two-parameter models may not matter. We recommend comparing several three-parameter model forms for consistency in estimated inflection points before deciding on one. Believing sigmoidal models to have inflection points does not necessarily mean that they will produce fitted curves with one. Our study highlights the need to integrate analysis of inflection points into modeling H-D relationships.

Allometry and Individual Growth of the Temperate Pacific Sardine (Sardinops sagax) Stock in the Southern California Current System

We analyzed individual growth and body mass allometry of the temperate stock of Sardinops sagax from samples collected on commercial fleet during 2005 to 2014 in the southern California Current System. Ages were estimated on otolith growth ring reading. Age (years) and standard length (SL; mm) data were fitted using four growth models: von Bertalanffy (VBM), Gompertz (GM), logistic (LM), and Schnute (SM) models. Parameters of each model were calculated by the maximum likelihood method using Akaike information criterion and Akaike’s weight (WAICi) to select the model with the greatest plausibility. Additionally, the interannual growth performance (φ′) variation was described. Body mass allometry was established from the relationship between total body mass (TM; g) and standard length. The sardine samples ranged in size from 114 to 226 mm SL and in weight from 20 to 192 g TM; ages varied from 0.5 to 6.0 years. The allometric parameter (b) was estimated to be 3.155 (3.154–3.156), showing a hyper-allometry growth type. On the basis of raw data, we found that the LM model fit the original data better (WAICi ~35%). When models were fitted to data bolstered with simulated values, the VBM model obtained the greatest weight (WAICi = 60.9%), but the LM model had null plausibility. Consequently, we determined that the Pacific sardine growth pattern was best described by the VBM model, which appears to be more robust against biased data. The φ′ showed interannual variation (2.13–2.51), with an average of φ′ = 2.25. An apparent inverse relationship was observed between φ′ and the Oceanic Niño Index (ONI) or the Pacific Decadal Oscillation (PDO) and the apparent positive relationship between Upwelling Index (UI) and φ’.

Implementation of sigmoidal models with different functional forms to estimate length at 50% maturity: A case study of the Pacific red snapper Lutjanus peru

To estimate mean size at sexual maturity (L50), a particular model is usually selected arbitrarily, and this decision does not consider the functional form (FF) of both the model (FFM) and the observed data (FFD). In this study, the performance of different sigmoidal models was evaluated by using data sources with different FF. To illustrate this, five sigmoidal models—Gompertz (GoM), logistic (LoM), Hill (HiM), Weibull (WeM), and gamma (GaM)—were used to estimate L50 for each sex and for the combined sexes of Lutjanus peru captured in the Eastern Central Pacific. To avoid redundancy, the models were selected based on different statistical properties: parameters in common L50; the same number of parameters (2); non-nested; different FF including symmetric or asymmetric; and asymptotic maturity equal to 1. Each model was fit to minimize the negative log likelihood by using a binomial probabilistic density function. Bias-corrected non-parametric confidence intervals CI were estimated for the parameters and predictions. The model average (AvM) and their CI were calculated by using the multi-model inference approach. According to the FFD, the best-fitting model was different for females (WeM), males (HiM), and the combined sexes (GaM). The comparison among the models revealed three features: i) the CI values overlapped among the data sets; ii) the estimate of L50 showed a gradient from right-skewed to left-skewed FF, which increases its value by ca. 7 cm; and iii) according to the selected model, it was possible to identify a marked variability in the computed L50. According to the multi-model approach, AvM was estimated for: i) females, L50 = 42.3 cm standard length LS computed from the WeM and LoM; ii) males, L50 = 37.4 cm LS computed from the GoM and WeM; and iii) the combined sexes, L50 = 39.5 cm LS estimated from the GaM as the unique model. The results show the importance of considering the FFD when L50 is estimated. By following this procedure, the AvM could be more informative regarding L50 estimates. This study proposes the implementation of sigmoidal models with different FF so that they can be fit to different FFD; this procedure exhibits the potential uncertainty linked to the candidate models.

Bioenergy production from the organic fraction of municipal solid waste and sewage sludge using mesophilic anaerobic co-digestion: An experimental and kinetic modeling study

Following the rapid surge in global demand for energy as well as the rising generation of municipal solid waste (MSW) worldwide, much more attention has been drawn to sustainable waste management solutions to tackle both problems. The present study investigated the potential for biogas production from the anaerobic co-digestion (ACD) of the organic fraction of MSW (OFMSW) and sewage sludge after physical and chemical pretreatments at three inoculum-to-substrate (I/S) ratios (ISR: 2, 4, & 6) and three levels of total solids (TS: 3, 5, & 7%). A series of experiments were further carried out in laboratory-scale (1-L) single-stage digesters in batch mode at the temperature of 35 °C (viz. Under mesophilic conditions) for 50 days. The results demonstrated that the digestion produced the best outputs at TS = 5% and ISR = 2, whereby it offered the biogas yield of 0.697, the methane (CH4) production of 0.448 m3/Kg-volatile solid (VS)added, and the total VS (TVS) removal by 65%. Moreover, it was observed that most reactors with TS = 7% encountered the phase separation problem while augmenting the TS content from 3 to 5% increased the biogas yield and CH4 production. The kinetic study of the mesophilic ACD additionally showed that both models of modified Gompertz (GM) equation and logistic function (LF) provided a reasonably accurate description of biogas production, but the first-order (FO) model failed in this task. Furthermore, the amount of sludge (namely, ISR) in the digester enlarged and the models showed a better fit to the experimental data.

Genomic predictions of growth curves in Holstein dairy cattle based on parameter estimates from nonlinear models combined with different kernel functions

Availability of longitudinal body weight (BW) records allows the application of nonlinear models (NLINM) to predict phenotypic and genomic growth curves in dairy cattle. In this regard, we considered a data set including 31,722 BW records from 4,952 female Holstein cattle, during the period from birth (mo 0) to approximately age at first calving (mo 24). Parameters of the growth curves were estimated using 3 NLINM: the logistic (LOG), the Gompertz (GOM), and the Richards (RICH) functions. Residuals for the growth curve parameters from the NLINM applications were used as pseudo-phenotypes in the ongoing genomic analyses with different similarity matrices, including 2 genomic relationship matrices (G1 and G2), a combined pedigree and genomic relationship matrix (H), and 3 kernel matrices. The kernels were a weighted "alike by state" kernel function (K1), an exponential dissimilarity kernel (K2), and a Gaussian kernel (K3). On the basis of G1 and G2 matrices, genomic heritabilities for the growth curve parameters birth weight (W0), mature weight (Wm), and growth rate (k), and the shape parameter (m; only available from RICH) were moderate to large, in the range from 0.29 (m from RICH) to 0.46 (k from RICH). Fitting the similarity matrices based on kernel functions contributed to an increase of the ratio of the variance explained by the similarity matrix in relation to the total variance (compared with the heritability when modeling G1 or G2). Genetic correlations between W0, Wm, and k were always positive (>0.30), especially for the same growth curve parameters estimated from different NLINM (>0.90). The shape parameter m from RICH was negatively correlated with other growth curve parameters, from -0.29 to -0.95. In a next step, estimated genomic breeding values for growth curve parameters were input data for the respective NLINM, aiming to construct genomic growth curves. Prediction accuracies were correlations between genomic growth curves and genomic breeding values from random regression models for sires and female cattle. Considering all genotyped female cattle with pseudo-phenotypes, prediction accuracies were larger from RICH than from LOG and GOM. However, differences in prediction accuracies from the NLINM × similarity matrix combinations were quite small. Accordingly, in 5-fold cross-validations using heifer groups with masked phenotypes, very similar prediction accuracies across modeling approaches were identified. Especially for specific age months, genomic growth curve predictions were more accurate for sires than for female cattle, indicating that the relationships between animals in training and validation sets are more important than the selection of specific NLINM × similarity matrix combinations.

Skew-Reflected-Gompertz Information Quantifiers with Application to Sea Surface Temperature Records

The Skew-Reflected-Gompertz (SRG) distribution, introduced by Hosseinzadeh et al. (J. Comput. Appl. Math. (2019) 349, 132–141), produces two-piece asymmetric behavior of the Gompertz (GZ) distribution, which extends the positive to a whole dominion by an extra parameter. The SRG distribution also permits a better fit than its well-known classical competitors, namely the skew-normal and epsilon-skew-normal distributions, for data with a high presence of skewness. In this paper, we study information quantifiers such as Shannon and Rényi entropies, and Kullback–Leibler divergence in terms of exact expressions of GZ information measures. We find the asymptotic test useful to compare two SRG-distributed samples. Finally, as a real-world data example, we apply these results to South Pacific sea surface temperature records.

Modelling the growth of lactic acid bacteria at different temperatures

ABSTRACT Mathematical models are widely used to predict the shelf life of foods. Lactic acid bacteria (LAB), particularly Lactobacillus plantarum, Weissella viridescens and Lactobacillus sakei, are the main spoilage bacteria of refrigerated, vacuum-packed meat products, stored in modified atmosphere, and their growth determines the shelf life length of these products. The objective of this study was to model the growth of L. plantarum, W. viridescens and L. sakei under different isothermal cultivation conditions and establish secondary models to describe the effect of temperature on the growth parameters of these bacteria. The LAB growth was evaluated in culture medium at temperatures of 4, 8, 12, 16, 20 and 30 oC. The fit of Baranyi and Roberts (BAR) and Gompertz (GO) primary models to the growth curves of LAB was compared by statistical indices, in which the BAR model showed slightly better fits to the experimental data. The BAR growth parameters were used to establish the secondary models, µmax and Nmax were established for the three LAB. The power model described the influence of temperature on the parameter λ for L. plantarum, and other bacteria showed no lag phase. The growth of LAB was strongly influenced by storage temperature and the obtained models allow predicting the growth of these bacteria within the temperature range from 4 to 30 oC.

Rumen liquor from slaughtered cattle as inoculum for feed evaluation.

Use of nonlinear mathematical models has been majorly based on in vitro gas production (GP) data generated when substrates are incubated with rumen liquor from fistulated steers. However, existing evidence suggests that rumen liquor from slaughtered cattle of unknown dietary history also generates quantifiable in vitro GP data. Fitting and description of GP data obtained from 4 diets incubated with rumen liquor from slaughtered cattle was evaluated using single-pool exponential model with discrete lag time (EXPL), logistic (LOG), Groot's (GRTS) and Gompertz (GOMP) models. Diets were formulated by varying proportions of Rhodes grass (Chloris gayana) hay and a concentrate mixed on dry matter basis to be: 1,000 g/kg Rhodes grass hay (RGH) and 0 of the concentrate (D1), 900 g/kg RGH and 100 g/kg concentrate (D2), 800 g/kg RGH and 200 g/kg concentrate (D3), 700 g/kg RGH and 300 g/kg concentrate (D4). Dietary kinetics for the models were determined by measuring GP at 2, 4, 8, 10, 18, 24, 36, 48, 72, 96 and 120 h. Model comparison was based on derived GP kinetics, graphical analysis of observed versus predicted GP profiles plus residual distribution and goodness-of-fit from analysis of root mean square error (RMSE), adjusted coefficient of determination (Adj-R2) and Akaike's information criterion (AIC). Asymptotic GP, half-life and fractional rate of GP differed (P < 0.001) among the 4 models. The RMSE, Adj-R2 and AIC ranged from 1.555 to 4.429, 0.906 to 0.984 and 2.452 to 15.874, respectively, for all diets compared across the 4 models. Based on the goodness-of-fit statistical criterion, GP profiles of D1 were more appropriately fitted and described by GRTS and GOMP than the EXPL and LOG models. The GRTS model had the lowest AIC value for D2 (2.452). Although GRTS model had the most homogenous residual dispersion for the 4 diets, all the 4 models exhibited a sigmoidal behavior. Therefore, rumen liquor from slaughtered cattle of unknown dietary history can be used to derive nutritionally important feed parameters, but choice of the most appropriate model should be made based on fitting criteria and dietary substrates incubated.

Comparative Analysis of Single-stage and Two-stage Fermentation Systems under Various Process Conditions

A comparative study of single-stage fermentation (wet or dry) and two-stage (wet-dry and dry-wet) fermentation systems was carried out under medium temperature conditions. The effect of the length of the first wet or dry fermentation stage (5-, 10-, 15-, 20-, and 25-d) in the two-stage anaerobic fermentation was investigated. The results showed that the gas production of wet fermentation and two-stage wet-dry fermentation was better than that of the dry fermentation and two-stage dry-wet fermentation. The cumulative gas production increased gradually with increased stage conversion times for the two-stage wet-dry fermentation. The gas production for the 20-d experimental group of the two-stage wet-dry fermentation system was the best. The cumulative biogas production in the anaerobic fermentation of straw correlated significantly with the changes in the degradation rates of volatile solids, cellulose, and hemicellulose (P < 0.01). The kinetic fitting analysis showed that the Reaction Curve (RC) model was more suitable for data modeling of the single-stage wet fermentation and two-stage wet-dry fermentation with straw than the Modified Gompertz (GM) and Modified Logistic (LM) models. The results of this study provided a theoretical basis for choosing a fermentation process for large-scale biogas production with straw.

The Logistic-Exponential Weibull Model as a Tool to Predict Natural Microflora Inactivation of Agave Mapsiaga Aguamiel (Agave Sap) by High Pressure Treatments

Agave sap (aguamiel) is a major source of nutraceutics that may be affected by natural microflora fermentation and thermal processing. High pressure processing (HPP) at 100–400 MPa/25C/1–20 min was investigated as an alternative non-thermal pasteurization by analyzing aerobic mesophiles (AM), coliforms (CL) and yeasts and molds (YM) inactivation. Standard statistics and information theory criteria determined the goodness of fit of first-order kinetics (LKM), Weibull (WBLL) and Gompertz (GMPZ) models. Non-detectable CL were found for 300 MPa/1 min, whereas 400 MPa/4 min HPP treatments reduced AM and YM by 3.86–4.72 log10 units. Statistical analysis showed WBLL best described microbial inactivation. A secondary WBLL model consisting of logistic-exponential b′(P) and exponential decay n(P) functions predicted AM (R2 = 0.835), YM (R2 = 0.867) kinetics for different pressure–time combinations. The secondary WBLL model predicted commercial HPP pasteurization conditions (5-log10 reductions of AM and YM) could be achieved at 400 MPa/5 min or 500 MPa/2 min. Practical Applications High pressure processing (HPP) is a consumer preferred alternative for food pasteurization. Aside from guacamole and some fruit and vegetable juices, there no other significant HPP foods available in México. Our research group believes that the commercialization of food products based on endemic plants such as agave sap (aguamiel), will draw the attention of national and international consumers, researchers and industrial HPP food processors to increase the number of HPP applications. Furthermore, the development of reliable pressure–time predictive models for microbial inactivation would help HPP food processors to ensure food safety and generate better budget estimates for novel products, or redefine current operating conditions since processing costs depend on the applied pressure level and pressure holding time.

Comparação de modelos não lineares para descrever o crescimento de fêmeas leiteiras mêstiças taurino x zebu no Nordeste do Brasil

Objetivou-se comparar as funções matemáticas de Brody (BD), Von Bertalanffy (VB), Logístico (LG) e Gompertz (GP), identificar a que melhor descreve o crescimento, estimar componentes da curva (peso assintótico: a; índice de maturidade: k) e caracterizar o formato (ponto de inflexão: PI; taxa de crescimento instantâneo: TCI) da curva média de crescimento de fêmeas mestiças (Bos taurus x Bos indicus) em sistema de produção semi-extensivo. Foram utilizados 537 registros de pesagens realizadas em intervalos de 30 dias durante o período de 2010 a 2012, pertencentes a três propriedades. Os valores de a e k foram, respectivamente, 1.982 kg e 0,000286 kg/dia para BD; 550,50 kg e 0,00232 kg/dia para VB; 405,20 kg e 0,006374 kg/dia para LG e 479,70 kg e 0,00353 kg/dia para GP. Baseado nos critérios estatísticos observou-se que os modelos VB, BD e LG apresentaram os maiores (&gt;0,74) coeficientes de determinação ajustados (Ra²), e menor valor do resíduo foi encontrado no modelo GP (4.918,38), indicando que todos os modelos se adequaram estatisticamente aos dados observados. Para os critérios de interpretação biológica, os modelos VB (8,48 meses) e LG (9,4 meses) apresentaram PI mais próximos do observado, o modelo VB (12,04-0,535 kg/mês) foi o único capaz de descrever a TCI, e os demais modelos subestimaram o incremento de peso corporal. O modelo de VB apresentou o melhor ajuste aos dados de crescimento de fêmeas mestiças com alta variabilidade genética em sistema semi-extensivo.

Evaluation of High Pressure Processing Kinetic Models for Microbial Inactivation Using Standard Statistical Tools and Information Theory Criteria, and the Development of Generic Time-Pressure Functions for Process Design

Generic kinetic models for microbial inactivation by high pressure processing (HPP) would accelerate the development of commercial applications. The aim of this work was to develop a generic model obtained by fitting peer-reviewed microbial inactivation data (124 kinetic curves) to first-order kinetics (LKM), Weibull (WBLL), and Gompertz (GMPZ) primary and secondary models. Standard statistics (coefficient of determination (R 2), variance, residuals plots, experimental vs. predicted plots) and information theory criteria (Akaike Information Criteria, AIC; Akaike differences, ∆AIC i , Bayesian Information Criteria) determined their goodness of fit. Standard statistics showed no differences between WBLL and GMPZ, whereas information theory criteria identified WBLL as the best model (lowest AIC i value, 61.3 % of cases). LKM performed poorly according to all statistics (e.g., ∆AIC i > 10, 58.1 % of cases). The dispersion of model parameters prevented the derivation of a secondary model for the whole dataset, but clear trends and sufficient data (56 kinetic curves) were found to develop one for milk. A secondary WBLL model (b′ = 0.056–2.230, n = 0.758 − 0.403; 150–600 MPa) was the best alternative (AIC i = 183.8). A GMPZ model yielded similar predictions, but registered ∆AIC i = 19.3 reflecting its larger number of parameters (p = 8). Selecting datasets with pressure holding times of commercial interest (t ≤ 10 min) yielded different parameter estimates for the generic WBLL model (b′ = 0.079–1.859, n = 1.340–0.557; 300–600 MPa). In conclusion, information theory criteria complemented standard statistics, and the simpler WBLL secondary model (p = 4) provided a product-specific time-pressure function of industrial relevance.

Evaluation of unconstrained and constrained mathematical functions to model girth growth of rubber trees (Hevea brasiliensis) using young age measurements

No attempt has been made to date to model growth in girth of rubber tree (Hevea brasiliansis). We evaluated the few widely used growth functions to identify the most parsimonious and biologically reasonable model for describing the girth growth of young rubber trees based on an incomplete set of young age measurements. Monthly data for girth of immature trees (age 2 to 12 years) from two locations were subjected to modelling. Re-parameterized, unconstrained and constrained growth functions of Richards (RM), Gompertz (GM) and the monomolecular model (MM) were fitted to data. Duration of growth was the constraint introduced. In the first stage, we attempted a population average (PA) model to capture the trend in growth. The best PA model was fitted as a subject specific (SS) model. We used appropriate error variance-covariance structure to account for correlation due to repeated measurements over time. Unconstrained functions underestimated the asymptotic maximum that did not reflect the carrying capacity of the locations. Underestimations were attributed to the partial set of measurements made during the early growth phase of the trees. MM proved superior to RM and GM. In the random coefficient models, both Gf and G0 appeared to be influenced by tree level effects. Inclusion of diagonal definite positive matrix removed the correlation between random effects. The results were similar at both locations. In the overall assessment MM appeared as the candidate model for studying the girth-age relationships in Hevea trees. Based on the fitted model we conclude that, in Hevea trees, growth rate is maintained at maximum value at t0, then decreases until the final state at dG/dt ≥0, resulting in yield curve with no period of accelerating growth. One physiological explanation is that photosynthetic activity in Hevea trees decreases as girth increases and constructive metabolism is larger than destructive metabolism.

Modeling in vitro gas production kinetics: Derivation of Logistic–Exponential (LE) equations and comparison of models

A Logistic–Exponential (LE) model was developed by introducing a shape parameter d into the logistic function. The LE models included three mathematical forms of LE s, LE 0 and LE LAG, where LE s was a standard LE model, LE 0 was the LE s model re-parameterized with ‘ V 0 = 0’, and LE LAG was the LE 0 model with an additional discrete LAG time. The LE models generated a wide range of curve shapes, and d revealed whether rate of gas production was or was not increased during in vitro incubation. The performances of the LE 0 and LE LAG models were evaluated by comparison with published models. For example, one-pool models included exponentials with and without LAG (EXP LAG and EXP 0), Gompertz (GOM), Logistic (LOG), generalization of the Mitscherlich (GM) and Michaelis–Menten (MM); while two-pool models included diphase Michaelis–Menten (DGMM), diphase Exponential with LAG (DEXP LAG), diphase Gompertz (DGOM) and diphase Logistic (DLOG) models. Gas production data consisting of 23 curves, obtained from a wide range of feeds, were used to investigate the fit performance of the models. Statistical criteria used were analysis of residuals ( i.e., run test and serial correlation) and goodness-of-fit ( i.e., residual mean squares, root mean squared prediction error, akaike's information criterion, observed versus predicted regression). Gas production attributes of final asymptotic gas volume ( V F ), gas volume at 24 h ( V 24 h ), half-life ( t 0.5) and fractional rate of gas production at half-life ( μ 0.5) were compared among one-pool models. The LE 0 was better than EXP LAG and EXP 0 models based upon overall statistical criteria, and better than GM and LOG models according to akaike's information criterion. The MM model had higher goodness-of-fit than the LE models, and had the highest V F and t 0.5, and lowest V 24 h and μ 0.5. The GOM and LOG had higher goodness-of-fit than the LE models, but positive gas volumes at ‘ t = 0’ were biologically meaningless. Two-pool models could improve the fit to curve, but led to declined robustness. There were no differences of predicted gas production attributes between the LE and GM models. Results indicate that the LE models can be alternatives to describe in vitro gas production kinetics.

Effect of sub-lethal heating and growth temperature on expression of the ribosomal RNA rrnB P2 promoter during the lag phase of Pseudomonas fluorescens

Current models for the lag phase of food-borne pathogens are limited by our poor understanding of the physiological changes taking place as bacterial cells prepare for exponential growth. In a previous paper in this series, a strain of Pseudomonas fluorescens containing the Tn 7-luxCDABE gene cassette regulated by the rRNA promoter rrnB P 2 was used to measure the influence of starvation on the lag phase duration (LPD OD) and growth rate ( R OD). rrnB P 2 promoter activity increased exponentially during the lag phase, and was characterized by lag (LPD Exp) and rate ( R Exp) parameters. In the present study, this work was expanded to include the influence of growth temperature (10 to 30 °C) and exposure to sub-lethal heating at 47 °C. With these additional datasets, the LPD Exp was often more pronounced than had been noted with starvation, so the original exponential association model (EXP) was compared to logistic and Gompertz (GOM) models. Based on root mean square error, the GOM model gave the better fit for some of the sub-lethal heating and growth temperature datasets; however, the EXP model was assessed as best overall. Increased growth temperature and decreased time of sub-lethal heating produced significant decreases in LPD OD and LPD Exp and increases in R OD and R Exp. The results suggest that different stressors have differential effects on gene expression and subsequent growth.

Comparison of nonlinear functions for describing the growth curve of three genotypes of dairy cattle

The Richards (RCD), the Gompertz (GMP), the monomolecular (MON) and the logistic (LOG) functions were compared for describing the growth curve of 343 female dairy cattle for which 12–18 body-weight measures were available. The animals were divided into four groups, and each function was assigned to one of the groups. Analyses of variance of growth traits estimated by the four functions and of the residual mean squares (RMS) indicated that all differences between functions in estimated mature weight (A) were statistically significant (P &lt; 0.01 to P &lt; 0.0001) except for RCD vs. GMP (P &lt; 0.27). Differences between functions in RMS were significant for RCD vs. LOG (P &lt; 0.0026), MON vs. LOG (P &lt; 0.0044) and GMP vs. LOG (P &lt; 0.0584). RCD differed from all other functions in average lifetime growth rate (P &lt; 0.1 to P &lt; 0.0001) and from MON (P &lt; 0.0004) and GMP (P &lt; 0.0003) in average lifetime maturing rate. Analysis of residuals indicated that both GMP and LOG overestimated birth weight. MON overestimated mature weight, whereas LOG tended to underestimate this parameter. The Richards function was chosen to describe the actual growth curves of individual females of three different genotypes. Predictive equations for the three breed groups were derived. Key words: Dairy cattle, growth curves, nonlinear functions

Growth delay and tumour relapse after adjuvant chemotherapy; the validity of models

(b) the logari thm of the factor, by which the number of cells is reduced (log kill) by a drug, is proport ional to the dose of the drug used. This model was assumed valid also for solid tumours with a number of restrictions which were not quant i ta t ively formulated: resistance to t rea tment could occur either on a cell kinetic basis, by development of a resistant subline of cells, or due to the tumour being located in a sanctuary where it is not reached by the drug. More recently, Norton and Simon have attempted to quant i fy the predictions of cell kill when cell kinetic mechanisms are taken into account. In the first place they expanded the use of the Gomper tz ian tumour growth model as proposed by Laird [2, 3] and demonstra ted that it is in better agreement with experimental da ta than the exponential growth model [4, 5] and they followed Lloyd's [3] assumption that the growth rate of a treated tumour is similar to that of an untreated tumour with a similar number of living tumour cells. Their main contention is, that cell kill is at each instant proport ional to GF, the relative tumour growth rate [6]. On this basis they built a model which predicts an inversion point below which tumour cell kill becomes