A Comparison and Catalog of Intrinsic Tumor Growth Models

Determining the mathematical dynamics and associated parameter values that should be used to accurately reflect tumor growth continues to be of interest to mathematical modelers, experimentalists and practitioners. However, while there are several competing canonical tumor growth models that are often implemented, how to determine which of the models should be used for which tumor types remains an open question. In this work, we determine the best fit growth dynamics and associated parameter ranges for ten different tumor types by fitting growth functions to at least five sets of published experimental growth data per type of tumor. These time-series tumor growth data are used to determine which of the five most common tumor growth models (exponential, power law, logistic, Gompertz, or von Bertalanffy) provides the best fit for each type of tumor.     

Modelling the growth of Japanese eel Anguilla japonica in the lower reach of the Kao-Ping River, southern Taiwan: an information theory approach

Information theory was applied to select the best model fitting total length ( L _T)-at-age data and calculate the averaged model for Japanese eel Anguilla japonica compiled from published literature and the differences in growth between sexes were examined. Five candidate growth models were the von Bertalanffy, generalized von Bertalanffy, Gompertz, logistic and power models. The von Bertalanffy growth model with sex-specific coefficients was best supported by the data and nearly overlapped the averaged growth model based on Akaike weights, indicating a similar fit to the data. The Gompertz, generalized von Bertalanffy and power growth models were also substantially supported by the data. The L _T at age of A. japonica were larger in females than in males according to the averaged growth mode, suggesting a sexual dimorphism in growth. Model inferences based on information theory, which deal with uncertainty in model selection and robust parameter estimates, are recommended for modelling the growth of A. japonica .     

Richards方程的数学属性及其在兽类生长过程中的应用

通过对Richards方程数学属性的分析表明 ,该方程具有变动的拐点值 ,因而在描绘兽类多种多样的生长过程时具有良好的可塑性。依据其方程参数n取值的不同 ,Richards方程包含了Spillman ,Logistic,Gompertz以及Bertalanffy方程。为了评估Richards方程对兽类生长过程的拟合优度 ,作者引用 1 0组哺乳动物兽类生长数据 ,将它与一些经典的生长模型如Spillman ,Logistic,Gompertz以及Bertalanffy方程共同进行了拟合比较。结果表明 ,Richards方程具有良好的拟合优度 ,适于描绘多种多样的兽类生长模式。     

Growth and annual survival estimates to examine the ecology of larval lamprey and the implications of ageing error in fitting models

This study used existing western brook lamprey Lampetra richardsoni age information to fit three different growth models (i.e. von Bertalanffy, Gompertz and logistic) with and without error in age estimates. Among these growth models, there was greater support for the logistic and Gompertz models than the von Bertalanffy model, regardless of ageing error assumptions. The von Bertalanffy model, however, appeared to fit the data well enough to permit survival estimates; using length‐based estimators, annual survival varied between 0·64 (95% credibility interval: 0·44–0·79) and 0·81 (0·79–0·83) depending on ageing and growth process error structure. These estimates are applicable to conservation and management of L. richardsoni and other western lampreys (e.g. Pacific lamprey Entosphenus tridentatus) and can potentially be used in the development of life‐cycle models for these species. These results also suggest that estimators derived from von Bertalanffy growth models should be interpreted with caution if there is high uncertainty in age estimates.     

一种获得非线性生长函数参数初始值的新方法:杂种遗传算法

本研究旨在杂种遗传算法应用于非线性生长函数的参数估计.提出了杂种遗传算法估计非线性生长函数参数的数学模型.5种非线性生长函数Gompertz、Logistic、von Bertalanffy、Richards、Brody分别拟合一个较大型的、群体类型差异大的番鸭体重生长资料,利用杂种遗传算法获得了有效初始值,在lsqcurvefit与proc nlin中获得了一致最优解的结果.表明杂种遗传算法估计非线性函数参数的实际可行性.     

Shapes and functions of bird-growth models: how to characterise chick postnatal growth

We compare four candidate models (logistic, Gompertz, von Bertalanffy, and extreme value function) for modelling the growth of birds. We fitted the models to two empirical data sets of chick growth (six biometric measurements) of African black oystercatchers Haematopus moquini from South Africa and little stints Calidris minuta from Russia, and identified the best-fitting growth curves by Akaike's information criterion. We also determine fitted and derived parameters, including the relative value (size) at hatching, the placement of inflection, the (normalised) growth rate constant, and the adult value (upper asymptote). The preferred model together with these factors describes how fast (or abruptly) the curves asymptote, and illustrates why growth is poorly characterised by the growth rate constant alone. Though the extreme value function model has not (as far as we know) been applied to chick growth data before, it appears to return the best fit for some parameters in our data sets. For example, we found that in African black oystercatchers two very different models best characterise two of the measurements: the extreme value function model and the Bertalanffy model for tarsus growth and body mass growth, respectively. In addition, we discuss the usefulness of fixing the upper asymptote to the adult value (e.g., adult body mass) and recommend a fixed upper asymptote in most cases.     

Growth curves of body weight and their relationship to sexual maturity in laboratory-bred male African green monkeys (Cercopithecus aethiops)

Nonlinear growth models having a three- or four-parameter family were applied to individual body weight data of 5 male African green monkeys for estimating their growth patterns. Body weight was measured from birth to six years of age and 58 to 114 data items per monkey were collected. The average body weight at birth was 360g with the standard deviation of +/- 25g, 4.54 +/- 0.29 kg at five years of age, and 4.50 +/- 0.12 kg at six years of age at which point body weight was judged to have reached a plateau. Five growth models (Gompertz, Logistic, Richards, Bertalanffy and Brody) were applied to the growth data in this study. As a result, two (Gompertz and Logistic) of the five models were found applicable to all data from the five monkeys. However, the coefficient of determination (R2) obtained by application of the two models were not so large (0.919 +/- 0.05 in Gompertz, 0.889 +/- 0.01 in Logistic). Therefore the data were divided into two groups according to monkey age: the first group being from monkeys between birth and 2 years 10 months of age and the second group was from monkeys older than 2 years 10 months of age. The Gompertz model fitted best the data of the first group in four of the five animals (R2 = 0.982 +/- 0.011). The age at the inflexion point in the Gompertz model nearly corresponded to the age of weaning. The Logistic model was most suitable for the date of the second group in all five animals (R2 = 0.955 +/- 0.038).(ABSTRACT TRUNCATED AT 250 WORDS)     

Post-natal growth of elephants Loxodonta africana in Etosha National Park, Namibia

Post-natal growth in the African elephant ( Loxodonta africana ) was described using three alternative mathematical models, and two age estimation schedules. Von Bertalanffy, Gompertz and Logistic equations all provided adequate models of post-natal growth in a species for which age estimation methods are largely unsubstantiated. Gompertz and Logistic models overestimated pre-weaning growth and underestimated adult size. Self-accelerating growth is of short duration (one and three years in females and males, respectively), and we found no evidence of a secondary growth spurt in males. Males, nevertheless, continue to grow throughout their lifespan, while females reach asymptotic size at the age of 35–40 years. We found no evidence of differences in growth rate of males and females up to 10 years, and there does not seem to be differential investment in male and female offspring. Growth rates of captive elephants differ substantially from all wild populations studied and may not serve as adequate references for the revision of existing age estimation schedules.     

Age and growth of the European catfish (Silurus glanis) in a Turkish Reservoir and comparison with introduced populations

The European catfish, Silurus glanis, is native to eastern Europe and western Asia and is among the largest freshwater fish in the world. Despite its increasing economic importance and its frequent introductions, the ecology and life-history of this species is poorly known due to the difficulty of sampling such a large species in large rivers and standing waters. Our study provides the first data on age and growth of this species in Turkish waters, where it is native. We report the length-weight relationships and age and size structure of this population, which were significantly different between females and males. A marginal increment analysis indicated that annulus formation occurred between May and June. The estimates of three growth functions (von Bertalanffy, logistic and Gompertz) are reported, with the von Bertalanffy growth providing a better fit and more realistic parameter estimates. Growth rates were significantly higher in males than in females and were overall higher compared to other native populations but similar to introduced populations of similar latitude.     

Growth curves of body weight changes in laboratory-bred female African green monkeys (Cercopithecus aethiops)

Nonlinear growth models having three or four parameter family were applied to individual weight data of female African green monkeys for estimating their growth pattern. The body weight was measured continuously from birth to six years of age with five female laboratory-bred monkeys. A total of 95 weight data were collected from each monkey. The average body weight was 330 g with the standard deviation of +/- 15 g at birth, and 2.71 +/- 0.33 kg at four years of age. The body weight of female African green monkeys was judged to reach a plateau after about four years of age. Five growth models (Gompertz, Logistic, Richards, Bertalanffy, Brody) were applied to these weight to age data. The most suitable coefficient of determination between growth data and growth model was obtained by the application of Gompertz equation. Three parameters of Gompertz equation, mature size (A), rate of maturing (K) and inflexion point (e-1 A) were analyzed in relation to age of menarche. Strong correlations between age of menarche and maturing rate, as well as between age of menarche and inflexion point were observed.     

Modelling Biolog profiles' evolution for yeast growth monitoring in alcoholic fermentation

Aims: A research was undertaken to explore the possibility to express with suitable mathematical models Biolog metabolic curves obtained for oenological yeasts and to use such models for monitoring yeast growth in alcoholic fermentation. Methods and Results: Experimental curves of metabolic activity in Biolog YT microplates, obtained in a previous work for various oenological yeast strains in pure cultures and mixed populations, at various cell concentrations, have been modelled with Gompertz’s, Gompertz’s modified and Lindstrom’s mathematical equations. Lindstrom’s model proved to be the most suitable to fit the curves of the oenological yeasts under study, providing the highest correlation coefficients between experimental and calculated data. The model made it possible to recognize, in mixed yeast populations, the presence of active dry yeasts used for guided fermentations. Model’s constant parameters were used for a numerical characterization of yeast curves. Conclusions: The application of the model to the experimental data resulted to be suitable for an early prediction of the successive evolution of yeast growth. Significance and Impact of the Study: The results obtained indicate the possibility to develop protocols for monitoring yeast presence during alcoholic fermentation, with an early assessment of the correct evolution of their growth, especially when active dry yeasts are employed.     

Development of sexual dimorphism in two sympatric skinks with different growth rates

Sexual size dimorphism (SSD) is widespread in animals, especially in lizards (Reptilia: Squamata), and is driven by fecundity selection, male–male competition, or other adaptive hypotheses. However, these selective pressures may vary through different life history periods; thus, it is essential to assess the relationship between growth and SSD. In this study, we tracked SSD dynamics between a “fading‐tail color skink” (blue tail skink whose tail is only blue during its juvenile stage: Plestiodon elegans) and a “nonfade color” tail skink (retains a blue tail throughout life: Plestiodon quadrilineatus) under a controlled experimental environment. We fitted growth curves of morphological traits (body mass, SVL, and TL) using three growth models (Logistic, Gompertz, and von Bertalanffy). We found that both skinks have male‐biased SSD as adults. Body mass has a higher goodness of fit (as represented by very high R² values) using the von Bertalanffy model than the other two models. In contrast, SVL and TL for both skinks had higher goodness of fit when using the Gompertz model. Two lizards displayed divergent life history tactics: P. elegans grows faster, matures earlier (at 65 weeks), and presents an allometric growth rate, whereas P. quadrilineatus grows slower, matures later (at 106 weeks), and presents an isometric growth rate. Our findings imply that species‐ and sex‐specific trade‐offs in the allocation of energy to growth and reproduction may cause the growth patterns to diverge, ultimately resulting in the dissimilar patterns of SSD.     

Development of mathematical models (Logistic, Gompertz and Richards models) describing the growth pattern of Pseudomonas putida (NICM 2174)

Bacterial growth curve, which is asymptotic after a certain period, is described using three different mathematical models, namely, Logistic model, Gompertz model and Richards model. The equations for these three models are fitted by evaluating the mathematical parameters involved in these models. This is done by applying the method of partial sums to the data in Table 1 containing the optical density values for different cell mass at different time intervals. The sum of square of residuals between the expected optical density values and the experimental values is calculated for each of these models. In the cases tested, the Logistic model was found to be the best fit for the growth curve of Pseudomonas putida (NICM 2174) and was found to be easy to use. These results fit the data very well at 5% level for more than 70% of the readings.     

Age and growth of an Iberian cyprinodont, Aphanius iberus (Cuv. & Val.), in its most northerly population

Age and growth of an Iberian cyprinodont, Aphanius iberus (Cuv. & Val.), in its most northerly population, La Rubina lagoon (Alt Empordà wetlands, NE Spain), was studied. The age groups for this Mediterranean population (O +, 1 + and 2 +) agree with those reported for the Atlantic population (R. Guadalquivir marshes), in the southern limit of the species distribution in the Iberian peninsula. Differences in age structure between these populations and the one previously studied in La Tancada lagoon (R. Ebro delta marshes) can be attributed to age estimation bias. Male A. iberus data appear to fit similarly to a Gompertz growth function and to a von Bertalanffy growth function, but L _x, was clearly underestimated. A Gompertz growth function was not a good fit for females, perhaps because female growth is clearly faster and less attenuated than male growth. The life history and growth pattern ofthis species is similar to that in other cyprinodontiform fishes, typified by a short longevity. The growth model of this species does not seem to fit types I and II of Sebens's classification, which are characteristic of fish.     

Evaluation of mathematical models to describe testicular growth in Blackbelly ram lambs

The primary objective was to compare various mathematical models to describe scrotal circumference (SC) and paired testis volume development in Blackbelly ram lambs. The study was conducted in the state of Querétaro, México (20° 43' N, 100° 15' W). Spring-born Blackbelly ram lambs (n = 41) were housed outdoors and fed alfalfa hay and concentrate. Body weight, SC, and testis length, diameter, and volume were recorded every 2 wk from 24 to 172 d of age (June 18 to November 3). The following mathematical functions were used to model SC-age and testis volume-age relationship: Von Bertalanffy, Brody, Gompertz, Logistic, and Richards. The suitability of the models was evaluated based on parameter values and standard errors, residual mean square, the coefficient of determination (R²), and the average prediction error (APE). All models, except for Brody's, had good fit to SC (R² > 0.98) and testis volume (R² > 0.95), and produced similar growth curves in the range of ages studied. The logistic model predicted SC at maturity quite well, 33.6 ± 0.6 cm as compared with 33.9 ± 0.5 cm observed in adult animals; all models had APE's smaller than ±7% between 56 and 168 d of age. The Bertalanffy model predicted testis volume at maturity quite well, 513 ± 22 cm³ as compared with 488 ± 20 cm³ calculated for adult animals. The logistic model had a good fit to testis volume during the period of study, but underestimated the volume at maturity by 28%. All models, except for Brody's, had APE's smaller than ±14% between 98 and 168 d of age. The logistic and Bertalanffy models predicted the inflection point for SC at 83 and 59 d of age, and testis volume at 116 and 109 d of age, respectively. In conclusion, all models, except for Brody's, had good fit to actual SC and testis volume data in the range of age evaluated, whereas the logistic and Bertalanffy's models made the best predictions for adult SC and testis volume, respectively.     

Characterization of commercial and biological growth curves in the Segureña sheep breed

Non-linear models were analysed to describe both the biological and commercial growth curves of the Segureña sheep, one of the most important Spanish breeds. We evaluated Brody, von Bertalanffy, Verhulst, logistic and Gompertz models, using historical data from the National Association of Segureña Sheep Breeders (ANCOS). These records were collected between 2000 and 2013, from a total of 129 610 weight observations ranging from birth to adulthood. The aim of this research was to establish the mathematical behaviour of body development throughout this breed’s commercial life (birth to slaughter) and biological life (birth to adulthood); comparison between both slopes gives important information regarding the best time for slaughter, informs dietary advice according to animals’ needs, permits economical predictions of productions and, by using the curve parameters as selection criteria, enables improvements in growth characteristics of the breed. Models were fitted according to the non-linear regression procedure of statistical package SPSS version19. Model parameters were estimated using the Levenberg–Marquardt algorithm. Candidate models were compared using the determinative coefficient, mean square error, number of iterations, Akaike information coefficient and biological coherence of the estimated parameters. The von Bertalanffy and logistic models were found to be best suited to the biological and commercial growth curves, respectively, for both sexes. The Brody equation was found to be unsuitable for studying the commercial growth curve. Differences between the parameters in both sexes indicate a strong impact of sexual dimorphism on growth. This can emphasize the value of the highest growth rate for females, indicating that they reach maturity earlier.