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.     

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.     

A generalization of Gompertz law compatible with the Gyllenberg-Webb theory for tumour growth

We present a new extension of Gompertz law for tumour growth and anti-tumour therapy. After discussing its qualitative and analytical properties, we show, in the spirit of [16], that, like the standard Gompertz model, it is fully compatible with the two-population model of Gyllenberg and Webb, formulated in [14] in order to provide a theoretical basis to Gompertz law. Compatibility with the model proposed in [17] is also investigated. Comparisons with some experimental data confirm the practical applicability of the model. Numerical simulations about the method performance are presented.     

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 Stochastic Model of Cancer Growth Subject to an Intermittent Treatment with Combined Effects: Reduction in Tumor Size and Rise in Growth Rate

A model of cancer growth based on the Gompertz stochastic process with jumps is proposed to analyze the effect of a therapeutic program that provides intermittent suppression of cancer cells. In this context, a jump represents an application of the therapy that shifts the cancer mass to a return state and it produces an increase in the growth rate of the cancer cells. For the resulting process, consisting in a combination of different Gompertz processes characterized by different growth parameters, the first passage time problem is considered. A strategy to select the inter-jump intervals is given so that the first passage time of the process through a constant boundary is as large as possible and the cancer size remains under this control threshold during the treatment. A computational analysis is performed for different choices of involved parameters. Finally, an estimation of parameters based on the maximum likelihood method is provided and some simulations are performed to illustrate the validity of the proposed procedure.     

Bounds on life expectancy for the Rayleigh and Weibull distributions

Bounds are presented for the life expectancy or the mean residual life of an individual whose lifetime is a random variable X following a Rayleigh distribution or more generally a Weibull distribution. Simple transformations of the variables give inequalities on the Mills' ratio and the incomplete gamma functions. Some numerical computations are also reported to compare the lower and upper bounds with the exact value of the life expectancy function for several values of the parameter. When the lifetime follows a Gompertz distribution, the problem becomes complicated, and it has not been possible to construct bounds on the life expectancy function. The importance of the Gompertz distribution in the dynamics of normal and tumor growth and in the embryonic and postnatal growth of birds and mammals is demonstrated, and life expectancy is evaluated by numerical methods for a number of parameter values.     

Optimal control of tumor size used to maximize survival time when cells are resistant to chemotherapy.

The high failure rates encountered in the chemotherapy of some cancers suggest that drug resistance is a common phenomenon. In the current study, the tumor burden during therapy is used to slow the growth of the drug-resistant cells, thereby maximizing the survival time of the host. Three types of tumor growth model are investigated--Gompertz, logistic, and exponential. For each model, feedback controls are constructed that specify the optimal tumor mass as a function of the size of the resistant subpopulation. For exponential and logistic tumor growth, the tumor burden during therapy is shown to have little impact upon survival time. When the tumor is in Gompertz growth, therapies maintaining a large tumor burden double and sometimes triple the survival time under aggressive therapies. Aggressive therapies aim for a rapid reduction in the sensitive cell subpopulation. These conclusions are not dependent upon the values of the model constants that determine the mass of resistant cells. Since treatments maintaining a high tumor burden are optimal for Gompertz tumor growth and close to optimal for exponential and logistic tumor growth, it may no longer be necessary to know the growth characteristics of a tumor to schedule anticancer drugs.     

三参数增长模型拟合:以季风常绿阔叶林中两个优势乔木种群为例

Logistic、Mitscherlich、Gompertz方程是一类三参数饱和增长曲线模型,广泛地应用于许多学科领域．本文基于logistic方程饱和值K估计的三点法、四点法,推导出Mitscherlich、Gompertz方程K值的三点法、四点法估计公式,并以南亚热带季风常绿阔叶林中两种优势乔木厚壳桂、黄果厚壳桂种群为例,先用三点法或四点法估计出K值,再通过线性回归与非线性回归相结合的方法,可获得三个增长模型中三个参数的最优无偏估计．实例研究表明,两个优势种群增长数据均符合三个增长模型,但更符合增长曲线呈S形的logistic、Gompertz方程,且以logistic方程最适合于观察;黄果厚壳桂种群增长快于厚壳桂种群．     

Optimization of the cell seeding density and modeling of cell growth and metabolism using the modified Gompertz model for microencapsulated animal cell culture

Cell microencapsulation is one of the promising strategies for the in vitro production of proteins or in vivo delivery of therapeutic products. In order to design and fabricate the optimized microencapsulated cell system, the Gompertz model was applied and modified to describe the growth and metabolism of microencapsulated cell, including substrate consumption and product formation. The Gompertz model successfully described the cell growth kinetics and the modified Gompertz models fitted the substrate consumption and product formation well. It was demonstrated that the optimal initial cell seeding density was about 4-5 x 10(6) cells/mL of microcapsule, in terms of the maximum specific growth rate, the glucose consumption potential and the product formation potential calculated by the Gompertz and modified Gompertz models. Modeling of cell growth and metabolism in microcapsules provides a guideline for optimizing the culture of microencapsulated cells.     

Probabilistic Gompertz model of irreversible growth

Characterizing organism growth within populations requires the application of well-studied individual size-at-age models, such as the deterministic Gompertz model, to populations of individuals whose characteristics, corresponding to model parameters, may be highly variable. A natural approach is to assign probability distributions to one or more model parameters. In some contexts, size-at-age data may be absent due to difficulties in ageing individuals, but size-increment data may instead be available (e.g., from tag-recapture experiments). A preliminary transformation to a size-increment model is then required. Gompertz models developed along the above lines have recently been applied to strongly heterogeneous abalone tag-recapture data. Although useful in modelling the early growth stages, these models yield size-increment distributions that allow negative growth, which is inappropriate in the case of mollusc shells and other accumulated biological structures (e.g., vertebrae) where growth is irreversible. Here we develop probabilistic Gompertz models where this difficulty is resolved by conditioning parameter distributions on size, allowing application to irreversible growth data. In the case of abalone growth, introduction of a growth-limiting biological length scale is then shown to yield realistic length-increment distributions.     

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)     

On Gompertz growth model and related difference equations

Within the context of the dynamics of populations described by first order difference equations a datailed study of the Gompertz growth model is performed. This is mainly achieved by proving several theorems for a class of difference equations generalizing the Gompertz equation. Some interesting features of the discrete Gompertz model, not exhibited by other well known growth models, are finally pointed out.     

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.     

珍稀濒危植物青钩栲种群数量特征研究

提出自适应种群增长新模型Ｓ＝ｅｘｐ（ａｌｎ＾２（１＋ｃｅ＾－ｒｔ）＋βｌｎ（１＋ｃｅ＾－ｅｔ）＋γ）,该模型包融了Ｌｏｇｉｓｔｉｃ模型、Ｓｍｉｔｈ模型、Ｇｏｍｐｅｒｔｚ模型、崔－Ｌａｗｓｏｎ模型、张－Ｌｏｇｉｓｔｉｃ模型和刘－Ｌｏｇｉｓｔｉｃ模型,运用遗传算法适应新模型进行参数估计,拟合青钩栲种群增长规律比其它种群增长模型更符合青钩栲群种的实际增长趋势,说明新模型具有一定的实用价值。     

广义Schumacher模型的改进及其应用

通过对前人提出的生长方程的具体分析,提出了一种改进的Schumacher生长方程．该模型包含了Gompenz函数、Schumacher方程及广义Schumacher方程,具有很强的自适应性和实用性．采用遗传算法。利用该模型对珍稀植物长苞铁杉和侧柏生长资料分别进行了拟合．结果表明,改进的Schumacher方程的拟合精度明显优于Schumache,方程和广义Schumacher方程,也优于经典的Logistic模型和李新运等自适应模型。可以在林木生长动态模拟及种群增长动态研究中广泛应用．     

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。     

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。     

Gompertzian re-evaluation of the growth patterns of transplantable mammary tumours in sialoadenectomized mice

Abstract
The method of the recursion formula of the Gompertz function (Bassukas & Maurer-Schultze 1988) has been applied to analyse tumour growth data taken from the literature; namely the growth perturbation of transplantable mammary tumours in sialoadenectomized mice with or without subsequent epidermal growth factor substitution (results on two mouse strains, C3H or SHN, have been reported; Inui, Tsubura & Morii 1989). The recursion formula of the Gompertz function fits growth curves to all seven sets of data well ( P > 0.05 for lack of fit test). The growth pattern of the tumours in the unperturbed hosts is Gompertzian and does not change if tumours are transplanted in sialoadenectomized mice, although the starting specific growth rate decreases in C3H mice. However, if sialoadenectomy is carried out after tumour inoculation, a complex alteration of the tumour growth evolves: tumour growth does not simply decelerate but it also shifts from the conventional Gompertzian to an exponential or even 'hyperexpo-nential' growth pattern, i.e. with an accelerating specific growth rate. Some theoretical mechanisms of this alteration, as well as the differences between the present Gompertzian analysis and a previously published Verhulstian analysis of part of the same data (Leith, Harrigan & Michelson 1991), are discussed. It is concluded that the quantitative analysis of tumour growth patterns by the method of the difference equation of the Gompertz function presently applied may substantially contribute to the improvement of the interpretation of perturbations of tumour growth-irrespective of their genesis. In contrast to the application of some a priori fixed growth function, e.g. the Verhulstian one, the present method can quantitatively interprete different growth patterns and their classification on the basis of linear regression analysis.