Accounting for individual variability in the von Bertalanffy growth model

Estimation of von Bertalanffy growth parameters has received considerable attention in fisheries research. Since Sainsbury (1980, Can. J. Fish. Aquat. Sci. 37: 241-247) much of this research effort has centered on accounting for individual variability in the growth parameters. In this paper we demonstrate that, in analysis of tagging data, Sainsbury's method and its derivatives do not, in general, satisfactorily account for individual variability in growth, leading to inconsistent parameter estimates (the bias does not tend to zero as sample size increases to infinity). The bias arises because these methods do not use appropriate conditional expectations as a basis for estimation. This bias is found to be similar to that of the Fabens method. Such methods would be appropriate only under the assumption that the individual growth parameters that generate the growth increment were independent of the growth parameters that generated the initial length. However, such an assumption would be unrealistic. The results are derived analytically, and illustrated with a simulation study. Until techniques that take full account of the appropriate conditioning have been developed, the effect of individual variability on growth has yet to be fully understood.

A simple method for estimating growth parameters from multiple length-frequency data in presence of continuous recruitment

The extended recruitment season for short-lived species such as prawns biases the estimation of growth parameters from length-frequency data when conventional methods are used. We propose a simple method for overcoming this bias given a time series of length-frequency data. The difficulties arising from extended recruitment are eliminated by predicting the growth of the succeeding samples and the length increments of the recruits in previous samples. This method requires that some maximum size at recruitment can be specified. The advantages of this multiple length-frequency method are: it is simple to use; it requires only three parameters; no specific distributions need to be assumed; and the actual seasonal recruitment pattern does not have to be specified. We illustrate the new method with length-frequency data on the tiger prawn Penaeus esculentus from the north-western Gulf of Carpentaria, Australia.

Growth curves with time-dependent explanatory variables

A class of growth models incorporating time-dependent factors and stochastic perturbations are introduced. The proposed model includes the existing growth models used in fisheries as special cases. Particular attention is given to growth of a population (in average weight or length) from which observations are taken randomly each time and the analysis of tag-recapture data. Two real data sets are used for illustration: (a) to estimate the seasonal effect and population density effect on growth of farmed prawn (Penaeus monodon) from weight data and (b) to assess the effect of tagging on growth of barramundi (Lates calcarifer)

Maximum likelihood estimation of mortality and growth with individual variability from multiple length-frequency data

We consider estimation of mortality rates and growth parameters from length-frequency data of a fish stock and derive the underlying length distribution of the population and the catch when there is individual variability in the von Bertalanffy growth parameter L-infinity. The model is flexible enough to accommodate 1) any recruitment pattern as a function of both time and length, 2) length-specific selectivity, and 3) varying fishing effort over time. The maximum likelihood method gives consistent estimates, provided the underlying distribution for individual variation in growth is correctly specified. Simulation results indicate that our method is reasonably robust to violations in the assumptions. The method is applied to tiger prawn data (Penaeus semisulcatus) to obtain estimates of natural and fishing mortality.

Population structure, mortality and growth of Pinna nobilis Linnaeus, 1758 (Mollusca, Bivalvia) at different depths in Moraira bay (Alicante, Western Mediterranean)

An investigation to characterize the causes of Pinna nobilis population structure in Moraira bay (Western Mediterranean) was developed. Individuals of two areas of the same Posidonia meadow, located at different depths (A1, -13 and A2, -6 m), were inventoried, tagged, their positions accurately recorded and monitored from July 1997 to July 2002. On each area, different aspects of population demography were studied (i.e. spatial distribution, size structure, displacement evidences, mortality, growth and shell orientation). A comparison between both groups of individuals was carried out, finding important differences between them. In A1, the individuals were more aggregated and mean and maximum size were higher (A1, 10.3 and A2, 6 individuals/100 m(2); A1, x = 47.2 +/- 9.9; A2, x = 29.8 +/- 7.4 cm, P < 0.001, respectively). In A2, growth rate and mortality were higher, the latter concentrated on the largest individuals, in contrast to A1, where the smallest individuals had the higher mortality rate [A1, L = 56.03(1 - e(-0.17t)); A2, L = 37.59(1 - e(-0.40t)), P < 0.001; mean annual mortality A1: 32 dead individuals out of 135, 23.7% and A2: 16 dead individuals out of 36, 44.4%, and total mortality coefficients (z), z(A1(-30)) = 0.28, z(A1(31-45)) = 0.05, z(A1(46-)) = 0.08; z(A2(-30)) = 0.15, z(A2(31-45)) = 0.25]. A common shell orientation N-S, coincident with the maximum shore exposure, was observed in A2. Spatial distribution in both areas showed not enough evidence to discard a random distribution of the individuals, despite the greater aggregation on the deeper area (A1) (A1, chi(2) = 0.41, df = 3, P > 0.5, A2, chi(2)= 0.98, df = 2 and 0.3 < P < 0.5). The obtained results have demonstrated that the depth-related size segregation usually shown by P. nobilis is mainly caused by differences in mortality and growth among individuals located at different depths, rather than by the active displacement of individuals previously reported in the literature. Furthermore, dwarf individuals are observed in shallower levels and as a consequence, the relationship between size and age are not comparable even among groups of individuals inhabiting the same meadow at different depths. The final causes of the differences on mortality and growth are also discussed.

Evaluation of growth and changes in body composition following neonatal diagnosis of cystic fibrosis

Early deficits in nutritional status that might require specific treatment and early response to nutritional therapy were studied longitudinally in 25 infants with cystic fibrosis (CF) diagnosed by neonatal screening, using anthropometric and research body composition methodology, and evaluation of pancreatic function. At the time of confirmed diagnosis (mean 5.4 weeks), body mass, length, total body fat (TBF), and total body potassium (TBK) were all significantly reduced. Following diagnosis and commencement of therapy there was a normalization of weight, length, and TBK by 6-12 months of age, indicating catch-up growth. But in some individuals the response was incomplete, and as a group, mean total body fat remained significantly lower than normal at 1 year of age. Seven of 25 (28%) were pancreatic sufficient at diagnosis, and all but one had evidence of declining pancreatic function requiring the institution of pancreatic enzyme therapy during the next 1-9 months. The median age of commencement of enzyme therapy was 10 weeks (range 5 weeks to 11 months). These longitudinal assessments emphasize the dynamic changes occurring in absorptive function, body composition, and nutritional status following neonatal diagnosis of cystic fibrosis and may reflect previously described abnormalities of energy metabolism in this age group. Abnormal body composition is evident in most CF infants following diagnosis by neonatal screening but pancreatic damage may still be evolving. We suggest that early active nutritional therapy and surveillance for changes in pancreatic function are warranted in CF infants diagnosed by neonatal screening.

Survival, growth and quality of life in children after orthotopic liver transplantation: A 5 year experience

The aims of this study were to investigate outcome and to evaluate areas of potential ongoing concern after orthotopic liver transplantation (OLT) in children. Actuarial survival in relation to age and degree of undernutrition at the time of OLT was evaluated in 53 children (age 0.58-14.2 years) undergoing OLT for endstage liver disease. Follow-up studies of growth and quality of life were undertaken in those with a minimum follow-up period of 12 months (n = 26). The overall 3 year actuarial survival was 70%. Survival rates did not differ between age groups (actuarial 2 year survival for ages <1, 1-5 and >5 years were 70, 70 and 69% respectively) but did differ according to nutritional status at OLT (actuarial 2 year survival for children with Z scores for weight <-1 was 57%, >-1 was 95%; P = 0.004). Significant catch-up weight gain was observed by 18 months post-transplant, while height improved less rapidly. Quality of life (assessed by Vineland Adaptive Behaviour Scales incorporating socialization, daily living skills, communication and motor skills) was good (mean composite score 91 ± 19). All school-aged children except one were attending normal school. Two children had mild to moderate intellectual handicap related to post-operative intracerebral complications. Satisfactory long-term survival can be achieved after OLT in children regardless of age but the importance of pre-operative nutrition is emphasized. Survivors have an excellent chance of a good quality of life and of satisfactory catch-up weight gain and growth.

Estimation of growth parameters from multiple-recapture data

This article develops a method for analysis of growth data with multiple recaptures when the initial ages for all individuals are unknown. The existing approaches either impute the initial ages or model them as random effects. Assumptions about the initial age are not verifiable because all the initial ages are unknown. We present an alternative approach that treats all the lengths including the length at first capture as correlated repeated measures for each individual. Optimal estimating equations are developed using the generalized estimating equations approach that only requires the first two moment assumptions. Explicit expressions for estimation of both mean growth parameters and variance components are given to minimize the computational complexity. Simulation studies indicate that the proposed method works well. Two real data sets are analyzed for illustration, one from whelks (Dicathais aegaota) and the other from southern rock lobster (Jasus edwardsii) in South Australia.

Groucho homologue Grg5 interacts with the transcription factor Runx2-Cbfa1 and modulates its activity during postnatal growth in mice

Runx2-Cbfal, a Runt transcription factor, plays important roles during skeletal development. It is required for differentiation and function of osteoblasts. In its absence, chondrocyte hypertrophy is severely impaired and there is no vascularization of cartilage templates during skeletal development. These tissue-specific functions of Runx2 are likely to be dependent on its interaction with other proteins. We have therefore searched for proteins that may modulate the activity of Runx2. The yeast two-hybrid system was used to identify a groucho homologue, Grg5, as a Runx2-interacting protein. Grg5 enhances Runx2 activity in a cell culture-based assay and by analyses of postnatal growth in mice we demonstrate that Grg5 and Runx2 interact genetically. We also show that Runx2 haploinsufficiency in the absence of Grg5 results in a more severe delay in ossification of cranial sutures and fontanels than occurs with Runx2 haploinsufficiency on a wild-type background. Finally, we find shortening of the proliferative and hypertrophic zones, and expansion of the resting zone in the growth plates of Runx2(+/-)Grg5(-/-) mice that are associated with reduced Ihh expression and Indian hedgehog (Ihh) signaling. We therefore conclude that Grg5 enhances Runx2 activity in vivo.

Growth defect in Grg5 null mice is associated with reduced Ihh signaling in growth plates

Gene-targeted disruption of Grg5, a mouse homologue of Drosophila groucho (gro), results in postnatal growth retardation in mice. The growth defect, most striking in approximately half of the Grg5 null mice, occurs during the first 4-5 weeks of age, but most mice recover retarded growth later. We used the nonlinear mixed-effects model to fit the growth data of wild-type, heterozygous, and Grg5 null mice. On the basis of preliminary evidence suggesting an interaction between Grg5 and the transcription factor Cbfa1/Runx2, critical for skeletal development, we further investigated the skeleton in the mice. A long bone growth plate defect was identified, which included shorter zones of proliferative and hypertrophic chondrocytes and decreased trabecular bone formation. This decreased trabecular bone formation is likely caused by a reduced recruitment of osteoblasts into the growth plate region of Grg5 null mice. Like the growth defect, the growth plate and trabecular bone abnormality improved as the mice grew older. The growth plate defect was associated with reduced Indian hedgehog expression and signaling. We suggest that Grg5, a transcriptional coregulator, modulates the activities of transcription factors, such as Cbfa1/Runx2 in vivo to affect Ihh expression and the function of long bone growth plates.

Estimating equations for parameters in stochastic growth models from tag-recapture data

James (1991, Biometrics 47, 1519-1530) constructed unbiased estimating functions for estimating the two parameters in the von Bertalanffy growth curve from tag-recapture data. This paper provides unbiased estimating functions for a class of growth models that incorporate stochastic components and explanatory variables. a simulation study using seasonal growth models indicates that the proposed method works well while the least-squares methods that are commonly used in the literature may produce substantially biased estimates. The proposed model and method are also applied to real data from tagged rack lobsters to assess the possible seasonal effect on growth.

An improved Fabens method for estimation of growth parameters in the vonBertalanffy model with individual asymptotes

In the analysis of tagging data, it has been found that the least-squares method, based on the increment function known as the Fabens method, produces biased estimates because individual variability in growth is not allowed for. This paper modifies the Fabens method to account for individual variability in the length asymptote. Significance tests using t-statistics or log-likelihood ratio statistics may be applied to show the level of individual variability. Simulation results indicate that the modified method reduces the biases in the estimates to negligible proportions. Tagging data from tiger prawns (Penaeus esculentus and Penaeus semisulcatus) and rock lobster (Panulirus ornatus) are analysed as an illustration.

Growth curves with explanatory variables and estimation of the effect of tagging

The von Bertalanffy growth model is extended to incorporate explanatory variables. The generalized model includes the switched growth model and the seasonal growth model as special cases, and can also be used to assess the tagging effect on growth. Distribution-free and consistent estimating functions are constructed for estimation of growth parameters from tag-recapture data in which age at release is unknown. This generalizes the work of James (1991, Biometrics 47 1519-1530) who considered the classical model and allowed for individual variability in growth. A real dataset from barramundi (Lates calcarifer) is analysed to estimate the growth parameters and possible effect of tagging on growth.

Modelling growth rate of Penaeus monodon Fabricius in intensively managed ponds: effects of temperature, pond age and stocking density

Records of shrimp growth and water quality made during 12 crops from each of 48 ponds, over a period of 6.5 years, were provided by a Queensland, Australia, commercial shrimp farm, These data were analysed with a new growth model derived from the Gompertz model. The results indicate that water temperature, mortality and pond age significantly affect growth rates. After 180 days, shrimp reach 34 g at constant 30 degrees C, but only 15 g after the same amount of time at 20 degrees C. Mortality, through thinning the density of shrimp in the ponds, increased the growth rate, but the effect is small. With continual production, growth rates at first remained steady, then appeared to decrease for the sixth and seventh crop, after which they have increased steadily with each crop. It appears that conservative pond management, together with a gradual improvement in husbandry techniques, particularly feed management, brought about this change. This has encouraging implications for the long-term sustainability of the farming methods used. The growth model can be used to predict productivity, and hence, profitability, of new aquaculture locations or new production strategies.