Mathematical approach for the optimal expansion of erythroid progenitors in monolayer culture

The continuous production of large numbers of red blood cells (RBCs) ex vivo is a challenging task due to process economics and complex culture conditions. In any serial passaging process, the culture conditions and operation mode are important criteria for achieving high proliferation with optimal passage lengths. The optimal inoculation concentration for serial passaging is a factor that affects both the kinetics and the total expansion performance. As part of our attempt to develop a scalable, economical and reproducible system for production of RBCs we used mathematical expressions to define the growth curves of peripheral blood derived erythroid progenitors over the course of their expansion process. We used a Gompertz function to evaluate the specific growth rate for the optimisation of inoculation concentration and passage lengths to achieve optimal expansion. This led to values of 3×10(5)cells/ml as the optimum inoculation concentration and 36h as the optimum passage length. Also the variations in growth curves confirmed the altered growth kinetics of erythroid progenitors during sequential passaging in expansion process. Cost analysis suggested a 60-h passage length at every passage, resulting in a 42.9% process-cost reduction. However, this has increased the process duration in achieving the similar expansion factor. This methodology for optimising the expansion process of peripheral derived erythroid progenitors based on optimum culture conditions could provide us with a direction and an improved performance for scale-up applications.     

Bayesian Analysis of Growth Curves Using Mixed Models Defined by Stochastic Differential Equations

Summary Growth curve data consist of repeated measurements of a continuous growth process over time in a population of individuals. These data are classically analyzed by nonlinear mixed models. However, the standard growth functions used in this context prescribe monotone increasing growth and can fail to model unexpected changes in growth rates. We propose to model these variations using stochastic differential equations (SDEs) that are deduced from the standard deterministic growth function by adding random variations to the growth dynamics. A Bayesian inference of the parameters of these SDE mixed models is developed. In the case when the SDE has an explicit solution, we describe an easily implemented Gibbs algorithm. When the conditional distribution of the diffusion process has no explicit form, we propose to approximate it using the Euler–Maruyama scheme. Finally, we suggest validating the SDE approach via criteria based on the predictive posterior distribution. We illustrate the efficiency of our method using the Gompertz function to model data on chicken growth, the modeling being improved by the SDE approach.     

Growth kinetics for shelf-life prediction: Theory and practice

Summary Predictive microbiology can be used to determine and predict the shelf-life of perishable foods under commercial distribution conditions based on microbial growth kinetics. This paper presents general microbial growth kinetics with the Monod model and the Gompertz function. Additional models are given to describe effects of food composition (e. g.a _w) and environmental conditions (e.g. temperature, gas atmosphere) as well as their interaction on the growth kinetic parameters (lag time and specific growth rate). These models can be used to predict the time to reach a critical level under any constant conditions within the range tested. A combination of microbial kinetics with an engineering accumulation approach can be used to predict the final microbial level in a food, or the loss of shelf-life, for any known time-temperature sequence, if there is no history effect or the history effect is negligible. A time-temperature indicator, could be used for predicting the remaining shelf-life of perishable foods under any distribution condition based on microbial growth kinetics.Mention of brand or firm names does not constitute an endorsement by the US Department of Agriculture over others of a similar nature not mentioned.     

Evaluation of the growth rate of MCF-7 breast cancer multicellular spheroids using three mathematical models

Abstract. Growth data on 60 multicellular spheroids of MCF-7 human breast cancer cells were fitted, on an individual basis, by the Gompertz, Bertalanffy and logistic equations. MCF-7 spheroids, initiated and grown in medium containing oestrogens, exhibited a growth rate that decreased continuously as spheroid size increased. Plots of spheroid volume v. time generated sigmoid curves that showed an early portion with an approximately exponential volume increase; a middle region or retardation phase characterized by a continuously decreasing growth rate; and, finally, a late segment or plateau phase approaching zero growth rate, that permitted an estimate of the maximum spheroid size (V_max). Growth curves generated by MCF-7 spheroids under different experimental conditions (hormones, drugs and radiation exposures) can be compared after normalization. Linearized forms of the fitted Gompertz curves provided a convenient way to express differences in growth rate.     

Combining Gompertzian growth and cell population dynamics

A two-compartment model of cancer cells population dynamics proposed by Gyllenberg and Webb includes transition rates between proliferating and quiescent cells as non-specified functions of the total population, N. We define the net inter-compartmental transition rate function: Phi(N). We assume that the total cell population follows the Gompertz growth model, as it is most often empirically found and derive Phi(N). The Gyllenberg-Webb transition functions are shown to be characteristically related through Phi(N). Effectively, this leads to a hybrid model for which we find the explicit analytical solutions for proliferating and quiescent cell populations, and the relations among model parameters. Several classes of solutions are examined. Our model predicts that the number of proliferating cells may increase along with the total number of cells, but the proliferating fraction appears to be a continuously decreasing function. The net transition rate of cells is shown to retain direction from the proliferating into the quiescent compartment. The death rate parameter for quiescent cell population is shown to be a factor in determining the proliferation level for a particular Gompertz growth curve.     

Effect of low inoculation density in the scale-up of insect cell cultures

The scale-up of insect cell cultures and the production of baculovirus with these cultures is dependent on the inoculation density applied. The effect of applying a low inoculation on the specific growth rate and on the duration of the lag phase was tested. Three different cell lines, HzAm1, Ha2302, and Sf21 were tested in a total of five cell line/medium combinations. Growth in suspension culture was examined, and data obtained were fitted with the Gompertz equation. A significant decline in specific growth rate with decreasing inoculation density was observed in all cell line/medium combinations, except for HzAm1. No critical inoculation density, below which no growth would occur, was found. In suspension culture in shake flasks, an inoculation density of 5 x 10(4) cells/mL is achievable, without severely influencing the overall growth rate. A lower inoculation density in suspension culture results in less steps in the scale-up process and might be a tool in bypassing the viral passage effect.     

Characteristic Species Dependent Growth Patterns of Mammalian Neoplasms

Evidence is presented, arising from an analysis of published data on tumour growth in three species of laboratory animals and in human multiple myeloma supporting a species specific relation between two supposedly independent parameters in the Gompertz equations frequently used to quantify tumour growth curves. This evidence supports the conjecture of Norton et al. (1976), based on their observations of the growth kinetics of a murine melanoma and a rat mammary carcinoma, that such a relation may be a general feature of tumour growth. Published data on the growth of xenografts of human colorectal tumours in immune-deprived mice suggests that the observed growth relation reflects the ability of a particular species to support a tumour of a certain maximum size. the existence of this relation greatly simplifies the task of predicting complete patterns of undisturbed neoplastic growth in these species.     

A stochastic model in tumor growth

A stochastic model of solid tumor growth based on deterministic Gompertz law is presented. Tumor cells evolution is described by a one-dimensional diffusion process limited by two absorbing boundaries representing healing threshold and patient death (carrying capacity), respectively. Via a numerical approach the first exit time problem is analysed for the process inside the region restricted by the boundaries. The proposed model is also implemented to simulate the effects of a time-dependent therapy. Finally, some numerical results are obtained for the specific case of a parathyroid tumor.     

Growth with regulation in random environment

The diffusion model for a population subject to Malthusian growth is generalized to include regulation effects. This is done by incorporating a logarithmic term in the regulation function in a way to obtain, in the absence of noise, an S-shaped growth law retaining the qualitative features of the logistic growth curve. The growth phenomenon is modeled as a diffusion process whose transition p.d.f. is obtained in closed form. Its steady state behavior turns out to be described by the lognormal distribution. The expected values and the mode of the transition p.d.f. are calculated, and it is proved that their time course is also represented by monotonically increasing functions asymptotically approaching saturation values. The first passage time problem is then considered. The Laplace transform of the first passage time p.d.f. is obtained for arbitrary thresholds and is used to calculate the expected value of the first passage time. The inverse Laplace transform is then determined for a threshold equal to the saturation value attained by the population size in the absence of random components. The probability of absorption for an arbitrary barrier is finally calculated as the limit of the absorption probability in a two-barrier problem.     

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．     

Growth with regulation in fluctuating environments

In this paper deterministic growth laws of a logistic-like type are initially introduced. The growth equations are expressed by first order differential equations containing a third order nonlinear term. Such equations are then parameterized in a way to allow for random fluctuations of the intrinsic fertility and of the environmental carrying capacity, thus leading to diffusion processes of new types. Their transition p.d.f. and asymptotic moments are then obtained and a detailed study of the extinction problem is performed within the framework of the first passage time problem through arbitrarily fixed threshold values. Some statistically significant quantities, such as the mean time necessary for the process to attain an assigned state, are obtained in closed form. The behavior of the diffusion processes here derived is finally compared with that of the well known diffusion processes obtained by parameterizing logistic and Gompertz growth equations.Work supported in part by the Group for Mathematical Information Sciences (GNIM) of the National Research Council and by Progetto Finalizzato Sofmat, Contract No. 82.00845.97     

A generalized Luria-Delbrück model

We develop extensions of the Luria-Delbrück model that explicitly consider non-exponential growth of normal cells and a birth-death process with mean exponential or Gompertz growth of mutants. Death of mutant cells can be important in clones arising during cancer progression. The use of a birth-death process for growth of mutant cells, as opposed to a pure birth process as in previous work on the Luria-Delbrück model, leads to a large increase in the extra Poisson variation in the size of the mutant cell populations, which needs to be addressed in statistical analyses. We also discuss connections with previous work on carcinogenesis models.     

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.     

Leaf-area Growth in Pelargonium zonale

The curve of growth in area of Pelargonium leaves against timeis strongly asymmetrical, and cannot be fitted by the usuallogistic curve. It is shown that the Gompertz curve providesan excellent fit, and the significance of its parameters isdiscussed.     

Stochastic Gompertz model of tumour cell growth

In this communication, based upon the deterministic Gompertz law of cell growth, a stochastic model in tumour growth is proposed. This model takes account of both cell fission and mortality too. The corresponding density function of the size of the tumour cells obeys a functional Fokker--Planck equation which can be solved analytically. It is found that the density function exhibits an interesting "multi-peak" structure generated by cell fission as time evolves. Within this framework the action of therapy is also examined by simply incorporating a therapy term into the deterministic cell growth term.     

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.     

On the therapy effect for a stochastic growth Gompertz-type model

We consider a diffusion model based on a generalized Gompertz deterministic growth in which carrying capacity depends on the initial size of the population. The drift of the resulting process is then modified by introducing a time-dependent function, called "therapy", in order to model the effect of an exogenous factor. The transition probability density function and the related moments for the proposed process are obtained. A study of the influence of the therapy on several characteristics of the model is performed. The first-passage-time problem through time-dependent boundaries is also analyzed. Finally, an application to real data concerning a rabbit population subject to particular therapies is presented.     

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。