A contribution to the generalization of the Gompertz function

SCHARF (1976) discusses various growth models. For the Gompertz function the differential equation (Formula: see text) is used. In words: the difference between relative growth rate and relative growth acceleration is constant. On the other hand, according to WENK (1973), the differential equation (Formula: see text) applies to the Gompertz function. It can be shown mathematically that (Formula: see text) applies in general. From Eq. (2) one obtains without trouble (Formula: see text). Therefore, the property leading to the Gompertz function may be defined as follows; the logarithmic derivation of the relative growth rate is constant. Eq. (2) is applicable only in special cases. It can be extended by assuming that c is not constant, but a function of time. In this way, a great number of growth functions can be found, which have to be regarded as model-based extensions of the Gompertz function.     

冷却猪肉中气单胞菌生长预测模型的建立和检验

为探讨冷却猪肉中气单胞菌的生长规律,将气单胞茵接种到经过80℃无菌水灭菌的冷却猪肉中,构建Gompertz和Baranyi初级模型来描述气单胞菌在不同温度下的生长状况。结果表明：修正的Gompertz模型对气单胞菌生长曲线的拟合效果优于Baranyi模型,R2均在0．97以上。应用平方根模型和Arrhenius模型对由修正的Gompertz模型得出的最大比生长速率进行拟合,所得平方根模型拟合效果略优于Arrhenius模型。实验所建二级模型能预测0-35℃贮藏条件下气单胞菌的生长。     

The influence of local environment on the aging and mortality ofAedes aegypti (L.): Case study in Fortaleza‐CE,Brazil

It is generally assumed that the daily probability of survival of mosquitoes is independent of age. To test this assumption we have conducted a three‐year experimental fieldwork study (2005–2007) at Fortaleza‐CE in Brazil, determining daily survival rates of the dengue vector Aedes aegypti (L.). Survival rates of adult Ae. aegypti may be age‐dependent and the statistical analysis is a sensitive approach for comparing patterns of mosquito survival. The mosquito survival data were better fit by a Weibull survival function than by the more traditionally used Gompertz or logistic survival functions. Gompertz, Weibull, or logistic survival functions often fit the survival, and the tails of the survival curves usually appear to fall between the values predicted by the three functions. We corroborate that the mortality of Ae. aegypti in semi‐natural conditions may no more be considered as a constant phenomenon during the life of adult mosquitoes but varies according to the age and environmental conditions under a tropical climate. This study estimates the variability in the survival rate of Ae. aegypti and environmental factors that are related to such variability. The statistical analysis shows that the fitting ability, concerning the hazard function, was in decreasing order: Seasonal Cox, the three‐parameter Gompertz, and the three‐parameter Weibull, that was similar to the three‐parameter logistic. The advantage of using the Cox model is that it is convenient for exploring the relationship between survival and several explanatory variables. The Cox model has the advantage of preserving the variable in its original quantitative form and of using a maximum of information. The survival analyses indicate that mosquito mortality is both age‐ and environment‐dependent.     

Models for estimating empirical Gompertz mortality: With an application to evolution of the Gompertzian slope

Using data from the human mortality database (HMD), and five different modeling approaches, we estimate Gompertz mortality parameters for 7,704 life tables. To gauge model fit, we predict life expectancy at age 40 from these parameters, and compare predicted to empirical values. Across a diversity of human populations, and both sexes, the overall best way to estimate Gompertz parameters is weighted least squares, although Poisson regression performs better in 996 cases for males and 1,027 cases for females, out of 3,852 populations per sex. We recommend against using unweighted least squares unless death counts (to use as weights or to allow Poisson estimation) are unavailable. We also recommend fitting to logged death rates. Over time in human populations, the Gompertz slope parameter has increased, indicating a more severe increase in mortality rates as age goes up. However, it is well-known that the two parameters of the Gompertz model are very tightly (and negatively) correlated. When the slope goes up, the level goes down, and, overall, mortality rates are decreasing over time. An analysis of Gompertz parameters for all of the HMD countries shows a distinct pattern for males in the formerly socialist economies of Europe.     

鲜切生菜中沙门氏菌二级生长预测模型的建立

以鲜切生菜为研究对象,比较了修正的Gompertz、Gompertz、Logistic和MMF 4种一级模型对不同温度下鲜切生菜中沙门氏菌生长曲线的拟合情况,发现在36℃、20℃和10℃时,修正的Gompertz模型均为最佳的拟合模型,4℃时沙门氏菌生长受到抑制,对失活/存活曲线进行"镜像化"处理后发现拟合程度相对较低,相关系数为0.962 7,故未用于二级模型中;采用其他温度下的修正的Gompertz模型中的最大比生长速率作为二级模型的响应值,建立平方根二级模型;准确因子和偏差因子对二级模型的准确性验证结果表明,两者均接近1.0,说明所建立的二级模型用于预测鲜切生菜中沙门氏菌生长情况。本研究为鲜切生菜的微生物安全控制提供科学依据。     

On the 'rate of aging' in heterogeneous populations

It is well known that the rate of aging is constant for populations described by the Gompertz law of mortality. However, this is true only when a population is homogeneous. In this note, we consider the multiplicative frailty model with the baseline distribution that follows the Gompertz law and study the impact of heterogeneity on the rate of aging in this population. We show that the rate of aging in this case is a function of age and that it increases in (calendar) time when the baseline mortality rate decreases.     

Investigation of growth phases for bottlenose dolphins using a Bayesian modeling approach

The Gompertz function is the most commonly used growth function for cetacean studies. However, this function cannot represent multiple phases of growth. In this study, we present a Bayesian framework fitting parameters of a triple-logistic growth function to describe multiple phases of growth for bottlenose dolphins ( Tursiops truncatus ), simultaneously fitting and comparing all growth parameters between South Carolina (SC), Mississippi Sound (MSS), and Indian River Lagoon (IRL) cohorts. The fitted functions indicated a preliminary early, rapid growth phase, followed by a second phase of slower growth, and then a moderate growth spurt later in life. Growth parameters between geographic cohorts did not show obvious differences, although asymptotic length for SC dolphins was lower than MSS and IRL dolphins and significantly lower between females from SC and the IRL. Growth rate velocities between the sexes showed females exceed males initially (<1 yr), followed by males gaining an advantage around the ages of 3–4 yr until the age of around 15 yr when growth rates for both sexes approached zero (asymptotic length). This study demonstrates age-related changes in growth rates between bottlenose dolphin sexes and evidence of at least some differences ( i.e. , asymptotic length) across geographic cohorts.     

Forecasting the growth of multicell tumour spheroids: implications for the dynamic growth of solid tumours

The growth dynamics of multicell tumour spheroids (MTS) were analysed by means of mathematical techniques derived from signal processing theory. Volume vs. time trajectories of individual spheroids were fitted with the Gompertz growth equation and the residuals (i.e. experimental volume determinations minus calculated values by fitting) were analysed by fast fourier transform and power spectrum. Residuals were not randomly distributed around calculated growth trajectories demonstrating that the Gompertz model partially approximates the growth kinetics of three-dimensional tumour cell aggregates. Power spectra decreased with increasing frequency following a 1/f(delta) power-law. Our findings suggest the existence of a source of 'internal' variability driving the time-evolution of MTS growth. Based on these observations, a new stochastic Gompertzian-like mathematical model was developed which allowed us to forecast the growth of MTS. In this model, white noise is additively superimposed to the trend described by the Gompertz growth equation and integrated to mimic the observed intrinsic variability of MTS growth. A correlation was found between the intensity of the added noise and the particular upper limit of volume size reached by each spheroid within two MTS populations obtained with two different cell lines. The dynamic forces generating the growth variability of three-dimensional tumour cell aggregates also determine the fate of spheroid growth with a strong predictive significance. These findings suggest a new approach to measure tumour growth potential.     

Fitting and using growth curves

Summary A technique is presented for fitting and analyzing growth patterns using Gompertz, power, and exponential curves. Data collection involves measuring growth rate as a function of size. This is useful because growth rates at many different sizes can be measured at the same time, which removes the effect of environmental change from the observed growth pattern. Using size instead of age as the independent variable is important because size is usually more closely related to growth rate than is age. The particular technique presented here yields estimates of the variance of the curve parameters so that growth curves for different populations can be compared.     

A new approach of fitting biomass dynamics models to data

A non-traditional approach of fitting dynamic resource biomass models to data is developed in this paper. A variational adjoint technique is used for dynamic parameter estimation. In the variational formulation, a cost function measuring the distance between the model solution and the observations is minimized. The data assimilation method provides a novel and computationally efficient procedure for combining all available information, i.e., the data and the model in the analysis of a resource system. This technique will be used to analyze data for the North-east Arctic cod stock. Two alternative population growth models: the logistic and the Gompertz model are used for estimating parameters of simple bioeconomic models by the method of constrained least squares. Estimates of the parameters of the models dynamics are reasonable and can be accepted. The main inference from the work is that the average fishing mortality is found to be significantly above the maximum sustainable yield value.     

Analysis of G.F. Gause experimental time-series by means of continuous-time models

Four population dynamics models, namely Verhulst, Gompertz, Rosenzweig, and Svirezhev ones, have been used to approximate two well-known time-series of Paramecia aurelia and P. caudatum population size (Gause, 1934). The parameters are estimated for each of the models by the least-square method (with global fitting) in two different ways: with and without an additional upper bound for a parameter value. In the latter (traditional) case, when the deviations of theoretical (model) trajectories from experimental time-series have been tested for normality (Kolmogorov-Smirnov test, Shapiro-Wilk test) with zero average, and for the presence/absence of serial correlations (Durbin-Watson criteria), the best results are obtained for the Gompertz and Verhulst models. In the former, more realistic, case (when we impose an additional constraint that the parameter meaning the carrying capacity of the environment has to be greater than any element in the sample), the best results are observed for the Gompertz model. Under this constraint, the canonical technique for deviation analysis can be applied in a restricted version only.     

A curvilinear approach to the kinetic analysis of linoleate peroxidation in aqueous liposomes by 2,2'azobis(2-amidoinopropane) dihydrochloride

Lipid peroxidation is a common feature of many chemical and biological processes, and is governed by a complex kinetic scheme. A fundamental stage in kinetic investigations of lipid peroxidation is the accurate determination of the rate of peroxidation, which in many instances is heavily reliant on the method of finite differences. Such numerical approximations of the first derivative are commonly employed in commercially available software, despite suffering from considerable inaccuracy due to rounding and truncation errors. As a simple solution to this, we applied three empirical sigmoid functions (viz. the Prout-Tompkins, Richards & Gompertz functions) to data obtained from the AAPH-mediated peroxidation of aqueous linoleate liposomes in the presence of increasing concentrations of Trolox, evaluating the curve fitting parameters using the widely available Microsoft Excel Solver add-in. We have demonstrated that the five-parameter Richards' function provides an excellent model for this peroxidation, and when applied to the determination of fundamental rate constants, produces results in keeping with those available in the literature. Overall, we present a series of equations, derived from the Richards' function, which enables direct evaluation of the kinetic measures of peroxidation. This procedure has applicability not only to investigations of lipid peroxidation, but to any system exhibiting sigmoid kinetics.     

Kinetics and modelling of cell growth and substrate uptake inCentella asiatica cell culture

In this study, we have conducted kinetics and modelling studies ofCentella asiatica cell growth and substrate uptake, in an attempt to evaluate cell growth for a better understanding and control of the process. In our bioreactor cultivation experiment, we observed a growth rate of 0.18/day, a value only 20% higher than was seen in the shake flask cultivation trial. However, the observed maximum cell dry weight in the shake flask, 10.5 g/L, was 14% higher than was achieved in the bioreactor. Ninety seven percentage confidence was achieved via the fitting of three unstructured growth models; the Monod, Logistic, and Gompertz equations, to the cell growth data. The Monod equation adequately described cell growth in both cultures. The specific growth rate, however, was not effectively predicted with the Logistic and Gompertz equations, which resulted in deviations of up to 73 and 393%, respectively. These deviations in the Logistic and Gompertz models may be attributable to the fact that these models were developed for substrate-independent growth and fungi growth, respectively.     

Dietary restriction of rodents decreases aging rate without affecting initial mortality rate – a meta‐analysis

Dietary restriction (DR) extends lifespan in multiple species from various taxa. This effect can arise via two distinct but not mutually exclusive ways: a change in aging rate and/or vulnerability to the aging process (i.e. initial mortality rate). When DR affects vulnerability, this lowers mortality instantly, whereas a change in aging rate will gradually lower mortality risk over time. Unraveling how DR extends lifespan is of interest because it may guide toward understanding the mechanism(s) mediating lifespan extension and also has practical implications for the application of DR. We reanalyzed published survival data from 82 pairs of survival curves from DR experiments in rats and mice by fitting Gompertz and also Gompertz–Makeham models. The addition of the Makeham parameter has been reported to improve the estimation of Gompertz parameters. Both models separate initial mortality rate (vulnerability) from an age‐dependent increase in mortality (aging rate). We subjected the obtained Gompertz parameters to a meta‐analysis. We find that DR reduced aging rate without affecting vulnerability. The latter contrasts with the conclusion of a recent analysis of a largely overlapping data set, and we show how the earlier finding is due to a statistical artifact. Our analysis indicates that the biology underlying the life‐extending effect of DR in rodents likely involves attenuated accumulation of damage, which contrasts with the acute effect of DR on mortality reported for Drosophila. Moreover, our findings show that the often‐reported correlation between aging rate and vulnerability does not constrain changing aging rate without affecting vulnerability simultaneously.     

An algorithm to fit the Gompertz function to growth curves

An algorithm to fit the Gompertz growth Junction is presented.This algorithm is easy to program on a microcomputer. The algorithmis based on employing a searching technique to solve a set ofequations derived from the Gompertz function. Its applicationmay prove valuable when access to a computer mainframe is difficult.Such a method may be useful in construction of a specific growthcurve in biology, or as a managerial tool in livestock enterprise,as well as in the clinical treatment of tumors. Demonstrationof the successful application of this algorithm in experimentallivestock growth data are presented.     

Classical mathematical models for prediction of response to chemotherapy and immunotherapy

Classical mathematical models of tumor growth have shaped our understanding of cancer and have broad practical implications for treatment scheduling and dosage. However, even the simplest textbook models have been barely validated in real world-data of human patients. In this study, we fitted a range of differential equation models to tumor volume measurements of patients undergoing chemotherapy or cancer immunotherapy for solid tumors. We used a large dataset of 1472 patients with three or more measurements per target lesion, of which 652 patients had six or more data points. We show that the early treatment response shows only moderate correlation with the final treatment response, demonstrating the need for nuanced models. We then perform a head-to-head comparison of six classical models which are widely used in the field: the Exponential, Logistic, Classic Bertalanffy, General Bertalanffy, Classic Gompertz and General Gompertz model. Several models provide a good fit to tumor volume measurements, with the Gompertz model providing the best balance between goodness of fit and number of parameters. Similarly, when fitting to early treatment data, the general Bertalanffy and Gompertz models yield the lowest mean absolute error to forecasted data, indicating that these models could potentially be effective at predicting treatment outcome. In summary, we provide a quantitative benchmark for classical textbook models and state-of-the art models of human tumor growth. We publicly release an anonymized version of our original data, providing the first benchmark set of human tumor growth data for evaluation of mathematical models.     

A principle of fractal-stochastic dualism and Gompertzian dynamics of growth and self-organization

The emergence of Gompertzian dynamics at the macroscopic, tissue level during growth and self-organization is determined by the existence of fractal-stochastic dualism at the microscopic level of supramolecular, cellular system. On one hand, Gompertzian dynamics results from the complex coupling of at least two antagonistic, stochastic processes at the molecular cellular level. It is shown that the Gompertz function is a probability function, its derivative is a probability density function, and the Gompertzian distribution of probability is of non-Gaussian type. On the other hand, the Gompertz function is a contraction mapping and defines fractal dynamics in time-space; a prerequisite condition for the coupling of processes. Furthermore, the Gompertz function is a solution of the operator differential equation with the Morse-like anharmonic potential. This relationship indicates that distribution of intrasystemic forces is both non-linear and asymmetric. The anharmonic potential is a measure of the intrasystemic interactions. It attains a point of the minimum (U(0), t(0)) along with a change of both complexity and connectivity during growth and self-organization. It can also be modified by certain factors, such as retinoids.     

Life equations for the senescence process

The Gompertz law of mortality quantitatively describes the mortality rate of humans and almost all multicellular animals. However, its underlying kinetic mechanism is unclear. The Gompertz law cannot explain the mortality plateau at advanced ages and cannot give an explicit relationship between temperature and mortality. In this study a reaction kinetics model with a time dependent rate coefficient is proposed to describe the survival and senescence processes. A temperature-dependent mortality function was derived. The new mortality function becomes the Gompertz mortality function with the same relationship of parameters prescribed by the Strehler–Mildvan correlation when age is smaller than a characteristic value δ, and reaches the mortality plateau when age is greater than δ. A closed-form analytical expression for describing the relationship of average lifespan with temperature and other equations are derived from the new mortality function. The derived equations can be used to estimate the limit of average lifespan, predict the maximal longevity, calculate the temperature coefficient of lifespan, and explain the tendency of the survival curve. This prediction is consistent with the most recently reported mortality trajectories for single-year birth cohorts. This study suggests that the senescence process results from the imbalance between damaging energy and protecting energy for the critical chemical substance in the body. The rate of senescence of the organism increases while the protecting energy decreases. The mortality plateau is reached when the protecting energy decreases to its minimal levels. The decreasing rate of the protecting energy is temperature-dependent. This study is exploring the connection between the biochemical mechanism and demography.