Estimation and interpretation of incidence rate difference for recurrent events when the estimation model is misspecified

Recurrent events data are common in experimental and observational studies. It is often of interest to estimate the effect of an intervention on the incidence rate of the recurrent events. The incidence rate difference is a useful measure of intervention effect. A weighted least squares estimator of the incidence rate difference for recurrent events was recently proposed for an additive rate model in which both the baseline incidence rate and the covariate effects were constant over time. In this article, we relax this model assumption and examine the properties of the estimator under the additive and multiplicative rate models assumption in which the baseline incidence rate and covariate effects may vary over time. We show analytically and numerically that the estimator gives an appropriate summary measure of the time‐varying covariate effects. In particular, when the underlying covariate effects are additive and time‐varying, the estimator consistently estimates the weighted average of the covariate effects over time. When the underlying covariate effects are multiplicative and time‐varying, and if there is only one binary covariate indicating the intervention status, the estimator consistently estimates the weighted average of the underlying incidence rate difference between the intervention and control groups over time. We illustrate the method with data from a randomized vaccine trial.     

A semicontinuous culture model that links cell growth to extracellular nutrient concentration

During semicontinuous culture, a sample of fixed volume is removed at regular time intervals to make measurements and/or harvest culture components, and an equal volume of fresh medium is immediately added to the culture, thereby instantaneously enhancing nutrient concentrations and diluting cell concentration. The resulting cell concentration versus time curve (i.e., the actual cell growth curve) has a saw-toothed appearance because of the periodic dilution of cell concentration. The observed cell concentrations correspond to the peaks of the saw-toothed curve. Cell growth rates are estimated from the locus of observed cell concentrations (i.e., from the apparent growth curve obtained by connecting the peaks of the saw-toothed curve). The sole preexisting model (Fencl's mode) for estimating cell growth rate is valid only when the cells are growing exponentially at a constant rate between samplings. This model has limited validity: despite the periodic enhancement of nutrient concentration, cell growth between samplings eventually causes nutrient depletion, and the cells cease to grow exponentially. Failure to recognize the limits of validity for Fencl' model has resulted in many erroneous applications of the model and, consequently, many incorrect estimates of cell growth rates. To provide a means for correctly estimating cell growth rates, Fencl's exponential model was extended, and a new model that describes the effects of nutrient depletion on cell growth in semi-continuous culture was obtained. The new model shows that exhaustion of a single growth-limiting nutrient in semicontinuous culture causes the locus of cell concentrations observed at time intervals of Deltat to follow a logistic growth curve. The actual cell growth rate was shown to equal the apparent logistic growth rate plus the effective dilution rate -Deltat(-1) In (1 - f), where f is the ratio of sample volume to total culture volume. Moreover, the model predicts that both the apparent logistic growth rate and the apparent steady-state cell concentration should rise linearly with the concentration of growth-limiting nutrient in the input medium, but fall linearly with increases in the effective dilution rate. The new logistic model for nutrient-limited cell growth in semicontinuous culture was successfully tested using published data for Asterionella formosa, Cyclotella meneghiniana, Daucus carota, and strain L mouse cells.     

Evaluation of expected absolute error affecting the maximum specific growth rate for random relative error of cell concentration

Borzani's [(1994) World Journal of Microbiology and Biotechnology 10, 475–476] idea of evaluation of absolute error affecting the 'maximum specific growth rate' (ESGR), calculated on the basis of the first and the last time points of the entire experimental time period, is generalized to the real-life situations where the relative errors of cell concentration cannot be assumed to be constant during the experiment. Visualizing the entire experimental time period as to comprise of several successive, mutually exclusive and exhaustive time intervals, we compute specific growth rates (SGRs) for each of these time intervals. Defining maximum of these SGR values as MSGR in contrast to Borzani's ESGR our aim is to study the effect of the expected absolute error on SGRs of different intervals. This will reveal the discrepancy between the true and observed MSGRs. Assuming the relative error distribution on (0,1) to be rectangular and symmetric truncated normal with mean at 0.5 and suitable variance, the expected values of the absolute errors are evaluated and numerically tabulated using the software packages MATHEMATICA and S-PLUS. Our results thus hold for situations involving varying relative errors where Borzani's results cannot be applied. A discussion with a concrete numerical example on the misidentification of the MSGR interval due to the effect of the random relative measuremental errors reveals to an experimental biologist that ignorance of this fact may lead to his/her entire experiment being futile.     

Formulating variable carrying capacity by exploring a resource dynamics-based feedback mechanism underlying the population growth models

Most of the population growth models comprise the concept of carrying capacity presume that a stable population would have a saturation level characteristic. This indicates that the population growth models have a common implicit feature of resource-limited growth, which contributes at a later stage of population growth by forming a numerical upper bound on the population size. However, a general underlying resource dynamics of the models has not been previously explored, which is the focus of present study. In this paper, we found that there exists a conservation of energy relationship comprising the terms of available resource and population density, jointly interpreted here as total available vital energy in a confined environment. We showed that this relationship determines a density-dependent functional form of relative population growth rate and consequently the parametric equations are in the form depending upon the population density, resource concentration, and time. Thus, the derived form of relative population growth rate is essentially a feedback type, i.e., updating parametric values for the corresponding population density. This resource dynamics-based feedback approach has been implemented for formulating variable carrying capacity in a confined environment. Particularly, at a constant resource replenishment rate, a density-dependent population growth equation similar to the classic logistic equation is derived, while one of the regulating factors of the underlying resource dynamics is that the resource consumption rate is directly proportional to the resource concentration. Likewise two other population growth equations similar to two known popular growth equations are derived based on this resource dynamics-based feedback approach. Using microcosm-derived data of fungus T. virens, we fitted one derived population growth model against the datasets, and concluded that this approach is practically implementable for studying a single population growth regulation in a confined environment.     

Modeling surface growth of Escherichia coli on agar plates

Surface growth of Escherichia coli cells on a membrane filter placed on a nutrient agar plate under various conditions was studied with a mathematical model. The surface growth of bacterial cells showed a sigmoidal curve with time on a semilogarithmic plot. To describe it, a new logistic model that we presented earlier (H. Fujikawa et al., Food Microbiol. 21:501-509, 2004) was modified. Growth curves at various constant temperatures (10 to 34 degrees C) were successfully described with the modified model (model III). Model III gave better predictions of the rate constant of growth and the lag period than a modified Gompertz model and the Baranyi model. Using the parameter values of model III at the constant temperatures, surface growth at various temperatures was successfully predicted. Surface growth curves at various initial cell numbers were also sigmoidal and converged to the same maximum cell numbers at the stationary phase. Surface growth curves at various nutrient levels were also sigmoidal. The maximum cell number and the rate of growth were lower as the nutrient level decreased. The surface growth curve was the same as that in a liquid, except for the large curvature at the deceleration period. These curves were also well described with model III. The pattern of increase in the ATP content of cells grown on a surface was sigmoidal, similar to that for cell growth. We discovered several characteristics of the surface growth of bacterial cells under various growth conditions and examined the applicability of our model to describe these growth curves.     

Re-interpretation of the logistic equation for batch microbial growth in relation to Monod kinetics

Aims:  To determine the underlying substrate utilization mechanism in the logistic equation for batch microbial growth by revealing the relationship between the logistic and Monod kinetics. Also, to determine the logistic rate constant in terms of Monod kinetic constants.
Methods and Results:  The logistic equation used to describe batch microbial growth was related to the Monod kinetics and found to be first-order in terms of the substrate and biomass concentrations. The logistic equation constant was also related to the Monod kinetic constants. Similarly, the substrate utilization kinetic equations were derived by using the logistic growth equation and related to the Monod kinetics.
Conclusion:  It is revaled that the logistic growth equation is a special form of the Monod growth kinetics when substrate limitation is first-order with respect to the substrate concentration. The logistic rate constant ( k ) is directly proportional to the maximum specific growth rate constant ( μ _m) and initial substrate concentration ( S ₀) and also inversely related to the saturation constant ( K _s).
Significance and Impact of the Study:  The semi-empirical logistic equation can be used instead of Monod kinetics at low substrate concentrations to describe batch microbial growth using the relationship between the logistic rate constant and the Monod kinetic constants.     

Modelling avian growth with the Unified‐Richards: as exemplified by wader‐chick growth

Postnatal growth in birds is traditionally modelled by fitting three‐parameter models, namely the logistic, the Gompertz, or the von Bertalanffy models. The purpose of this paper is to address the utility of the Unified‐Richards (U‐Richards) model. We draw attention to two forms of the U‐Richards and lay down a set of recommendations for the analysis of bird growth, in order to make this model and the methods more accessible. We examine the behaviour of the four parameters in each model form and the four derived measurements, and we show that all are easy to interpret, and that each parameter controls a single curve characteristic. The two parameters that control the inflection point, enable us to compare its placement in two dimensions, 1) inflection value (mass or length at inflection) and 2) inflection time (time since hatching), between data sets (e.g. between biometrics or between species). We also show how the parameter controlling growth rate directly presents us with the relative growth rate at inflection, and we demonstrate how one can compare growth rates across data sets. The three traditional models, where the inflection value is fixed (to a specific percentage of the upper asymptote), provide incompatible growth‐rate coefficients. One of the two forms of the U‐Richards model makes it possible to fix not only the upper asymptote (adult value), but also the intersection with the y‐axis (hatching value). Fitting the new model forms to data validates the usefulness of interpreting the inflection placement in addition to the growth rate. It also illustrated the advantages and limitations of constraining the upper asymptote (adult value) and the y‐axis intersection (hatching value) to fixed values. We show that the U‐Richards model can successfully replace some of the commonly used growth models, and we advocate replacing these with the U‐Richards when modelling bird growth.     

Description of the delayed microbial growth by an extended logistic equation

Logistic equations are suitable for describing microbial growth. By means of VERHULST'S logistic equation, the adaptation to sigmoid-shaped curves of growth improves with a falling ratio C_xo/C_{x, max} < 0.2, if there is no lag-phase. The known logistic equations do not take into account any lag-phase behaviour, so that noticeable deviations in the model adaptation result in this range. Therefore, an extended logistic equation of rate is proposed by which any occuring lag-time is expressed by a 1st-order lag-term. The corresponding time law allows a very good adaptation of curves of delayed growth behaviour, and changes into VERHULST'S logistic equation for a lag-time t_L = 0. Application is facilitated by instructions for the numerical determination.     

Recruitment rates in forest plots: Gf estimates using growth rates and size distributions

1 In censuses of tree populations in permanent plots, short census intervals and small population size lead to uncertainty in the observed recruitment rate of a minimum size. Increasing the census interval, however, underestimates the rate because of unrecorded 'recruit and die' events.
2 We propose a new Gf procedure for estimation of recruitment rates. Recruitment rate per area is obtained by multiplication of the density in the smallest size class (f) and the average size growth rate in that class (G) divided by the width of the class. This procedure is valid when the size distribution of the population examined is continuous with size.
3 When tree size structure is negative‐exponentially distributed, as is often the case in natural rain forest populations, the Gf estimate of the recruitment rate for a given size class was least biased close to the midpoint size of this class.
4 Gf estimates agreed well with census estimates of recruitment rate from permanent plots in rain forests. A tendency for Gf estimates to be larger than census estimates disappeared when census estimates were corrected for mortality after recruitment.
5 The effects of plot size, census interval and variation in growth rate on estimates of recruitment rate were simulated using model populations. Small plot size caused substantially more among‐plot deviation for the census count of recruitment events than for the Gf estimate. The census recruitment rate also showed larger variation among plots for shorter intervals than the Gf estimate, which was independent of census interval. The Gf estimates were therefore more accurate than census counts in many situations. More than several tens of trees were needed in a size class to allow a reliable Gf estimates.     

QTL methodology for response curves on the basis of non-linear mixed models, with an illustration to senescence in potato

The improvement of quantitative traits in plant breeding will in general benefit from a better understanding of the genetic basis underlying their development. In this paper, a QTL mapping strategy is presented for modelling the development of phenotypic traits over time. Traditionally, crop growth models are used to study development. We propose an integration of crop growth models and QTL models within the framework of non-linear mixed models. We illustrate our approach with a QTL model for leaf senescence in a diploid potato cross. Assuming a logistic progression of senescence in time, two curve parameters are modelled, slope and inflection point, as a function of QTLs. The final QTL model for our example data contained four QTLs, of which two affected the position of the inflection point, one the senescence progression-rate, and a final one both inflection point and rate.     

A method for the reconstruction of unknown non-monotonic growth functions in the chemostat

We propose an adaptive control law that allows one to identify unstable steady states of the open-loop system in the single-species chemostat model without the knowledge of the growth function. We then show how one can use this control law to trace out (reconstruct) the whole graph of the growth function. The process of tracing out the graph can be performed either continuously or step-wise. We present and compare both approaches. Even in the case of two species in competition, which is not directly accessible with our approach due to lack of controllability, feedback control improves identifiability of the non-dominant growth rate.     

Will plant vigor and tolerance be genetically correlated? Effects of intrinsic growth rate and self-limitation on regrowth

Plants are known to maintain fitness despite herbivore attack by a variety of damage-induced mechanisms. These mechanisms are said to confer tolerance, which can be measured as the slope of fitness over the proportion of plant biomass removed by herbivore damage. It was recently supposed by Stowe et al. (2000) that another plant property, general vigor, has little effect on tolerance. We developed simple models of annual monocarpic plants to determine if a genetic change in components of growth vigor will also change the fitness reaction to damage. We examined the impact of intrinsic growth rate on the tolerance reaction norm slope assuming plants grow geometrically, i.e., without self-limitation. In this case an increase in intrinsic growth rate decreases tolerance (the reaction norm slope becomes more negative). A logistic growth model was used to examine the impact of self-limiting growth on the relationship between intrinsic growth rate and the tolerance reaction norm slope. With self-limitation, the relationship is sensitive to the timing of attack. When attack is early and there is time for regrowth, increasing growth rate increases tolerance (slope becomes less negative). The time limitations imposed by late attack prevent appreciable regrowth and induce a negative relationship between growth rate and tolerance. In neither of these simple cases will the correlation between vigor and tolerance constrain selection on either trait. However, a positive correlation between growth rate and self-limitation will favor fast growth/strong self-limitation in a high-damage environment, but slow growth/weak self-limitation in a low-damage environment. Thus, fundamental growth rules that determine vigor have constitutive effects on tolerance. The net costs and benefits of damage-induced tolerance mechanisms will thus be influenced by the background imposed by fundamental growth rules. This revised version was published online in July 2006 with corrections to the Cover Date.     

Theory for growth of plants derived from the nitrogen productivity concept

A theory is developed on the assumption that growth of plants is determined by the current amount of nitrogen in the plants. The nitrogen-growth relation is formalized in the nitrogen productivity concept (amount of biomass produced per amount of nitrogen in the biomass and per unit of time), which is essentially a constant for a given species under fixed environmental conditions. A number of results follow for increases in whole plant biomass: (A) The relative growth rate is a linear function of the internal nitrogen concentration. (B) The maximal relative growth rate uniquely determines the scaling of the time axis. (C) Exponential growth is consistent only with stable internal nitrogen concentration. Dose-response curves expressed in reduced variables (the ratio between a variable and the same variable for a plant growing under optimal conditions) are universal, so that all species and all environmental conditions yield the same curve. This is confirmed by experimental data. The shape (linear, exponential, etc.) of the nitrogen uptake curve is the only parameter differentiating these universal curves. The Mitscherlich curve or variations of it can be fitted very closely to the derived dose-response curves, except under exponential growth. A conclusion drawn from the analysis is that the results of nutrition experiments cannot be properly interpreted unless the variation with time of the amount of nitrogen in the plant is known. The theory can be extended to more complex situations, for example, time-varying environmental conditions.     

Growth temperature influences the underlying components of relative growth rate: an investigation using inherently fast- and slow-growing plant species

We examined the effect of growth temperature on the underlying components of growth in a range of inherently fast‐ and slow‐growing plant species. Plants were grown hydroponically at constant 18, 23 and 28 °C. Growth analysis was conducted on 16 contrasting plant species, with whole plant gas exchange being performed on six of the 16 species. Inter‐specific variations in specific leaf area (SLA) were important in determining variations in relative growth rate (RGR) amongst the species at 23 and 28 °C but were not related to variations in RGR at 18 °C. When grown at 18 °C, net assimilation rate (NAR) became more important than SLA for explaining variations in RGR. Variations in whole shoot photosynthesis and carbon concentration could not explain the importance of NAR in determining RGR at the lower temperatures. Rather, variations in the degree to which whole plant respiration per unit leaf area acclimated to the different growth temperatures were responsible. Plants grown at 28 °C used a greater proportion of their daily fixed carbon in respiration than did the 18 and 23 °C‐grown plants. It is concluded that the relative importance of the underlying components of growth are influenced by growth temperature, and the degree of acclimation of respiration is of central importance to the greater role played by NAR in determining variations in RGR at declining growth temperatures.     

Nematode growth patterns and the moulting cycle: the population growth profile

Population growth profiles of Caenorhabditis elegans and Panagrellus redivivus constructed from length frequencies have a number of steps in them coinciding with the number of extrauterine moults. Each step has a constant size relationship with that of one of the midmoults measured directly.
The profiles could only have the shape they do if there are corresponding steps in the true growth curve of individual worms: the fact that previous workers have been unable to detect these steps being due to the limitations of techniques available for the study of synchronous and individual growth curves. Nevertheless, a synchronous system with Trichostrongylus retortaeformis gives qualitative support to the findings from population profiles.
The population growth profile is a new tool in the study of environmental effects on moulting, though there are theoretical reasons why the true growth curve cannot be derived from it.
Abandonment of the "continuous growth" model for post-embryonic development simplifies the framing of hypotheses to explain ecdysis in nematodes.     

Nonlinear estimation of specific growth rate for aerobic fermentation processes

The specific growth rate of the biomass, a very important parameter of almost every fermentation process, cannot be measured directly or estimated from related variables, as the concentrations of biomass, substrates, or products, due to the lack of reliable and cheap sensors. In this article a stable adaptive estimator of the specific growth rate is designed for those aerobic processes where the measurement of the oxygen uptake rate is available on-line. This particular approach can be applied also for other reaction rates if the model of the process satisfies some very general assumptions, which make the dynamics of the measured reaction rate a nonlinear function only of two unknown parameters, the specific growth rate and its time derivative. With respect to a previous similar approach, the new estimator has one additional parameter and a different nonlinear structure. From the analysis of the dynamics of the estimation error, a tuning criterion is derived, by which the two different algorithms can be compared under similar conditions. Simulation results show a good performance of both estimators for various kind of processes and disturbances. (c) 1995 John Wiley & Sons, Inc.     

A 450 million years long latitudinal gradient in age‐dependent extinction

Leigh Van Valen famously stated that under constant conditions extinction probability is independent of species age. To test this 'law of constant extinction', we developed a new method using deep learning to infer age‐dependent extinction and analysed 450 myr of marine life across 21 invertebrate clades. We show that extinction rate significantly decreases with age in > 90% of the cases, indicating that most species died out soon after their appearance while those which survived experienced ever decreasing extinction risk. This age‐dependent extinction pattern is stronger towards the Equator and holds true when the potential effects of mass extinctions and taxonomic inflation are accounted for. These results suggest that the effect of biological interactions on age‐dependent extinction rate is more intense towards the tropics. We propose that the latitudinal diversity gradient and selection at the species level account for this exceptional, yet little recognised, macroevolutionary and macroecological pattern.     

Ecological and mathematical considerations on self-thinning in even-aged pure stands

It is emphasized in growth analysis of self-thinning populations that relative mortality rate pertains to the difference between relative growth rates and net assimilation rates, each of which are definable on a mean plant size basis or on a biomass basis. The time trends of the ratio of relative mortality rate to relative growth rates to be expected according to Tadaki's, Shinozaki's and Hozumi's models are compared with that of the eastern white pine population, and a good agreement is exhibited. As an alternative to Hozumi's model, a new model is constructed to unite the logistic theory of plant growth and the 3/2 power law concerning self-thinning, which so far have usually been applied independently to growth analysis. To construct the model the following assumptions are made: the fundamental equation to relate mean plant weight with density in self-thinning population proposed by Shinozaki, and a special population with a specific initial density which follows thew-p trajectory of the 3/2 power law type and has an exponential decrease in its density with biological time. Properties of the model are examined from ecological and mathematical viewpoints.     

The kinetics of mycelial growth.

Based on the assumption that mycelial growth follows the logistic growth law, formulae have been developed to express the growth of fungal colonies under a variety of geometric constraints. Analysis was done of Deppe's (1973) results on surface colony growth, where the mass of the colony grew exponentially during most colonial growth, and of Trinci's (1970) results on submerged "pellet" growth, where the mass of the colony increased as the cube of time during most colony growth. In both cases, the linear dimensions of the colony were increasing linearly while the mass was changing in these quantitatively different manners. It is concluded that these disparate growth behaviours result from different habits of growth; in two-dimensional colony growth a new region of space if invaded by an amount of mycelium small in proportion to the final "carrying capacity" of the region, and in three-dimensional colony growth a region is invaded with an amount of mycelium almost equal to the region's final limiting mycelial mass. Thus, the types of growth law for colony mass which are applicable for a particular organism in a particular physical environment depend critically on the degree to which the invading hyphae initially occupy the space.