Comparing growth patterns of three species: Similarities and differences.

Quantitative studies of the growth of dinosaurs have made comparisons with modern animals possible. Therefore, it is meaningful to ask, if extinct dinosaurs grew faster than modern animals, e.g. birds (modern dinosaurs) and reptiles. However, past studies relied on only a few growth models. If these models were false, what about the conclusions? This paper fits growth data to a more comprehensive class of models, defined by the von Bertalanffy-Pütter (BP) differential equation. Applied to data about Tenontosaurus tilletti, Alligator mississippiensis and the Athens Canadian Random Bred strain of Gallus gallus domesticus the best fitting growth curves did barely differ, if they were rescaled for size and lifespan. A difference could be discerned, if time was rescaled for the age at the inception point (maximal growth) or if the percentual growth was compared.

Best fitting tumor growth models of the von Bertalanffy-PütterType.

BACKGROUND: Longitudinal studies of tumor volume have used certain named mathematical growth models. The Bertalanffy-Pütter differential equation unifies them: It uses five parameters, amongst them two exponents related to tumor metabolism and morphology. Each exponent-pair defines a unique three-parameter model of the Bertalanffy-Pütter type, and the above-mentioned named models correspond to specific exponent-pairs. Amongst these models we seek the best fitting one. METHOD: The best fitting model curve within the Bertalanffy-Pütter class minimizes the sum of squared errors (SSE). We investigate also near-optimal model curves; their SSE is at most a certain percentage (e.g. 1%) larger than the minimal SSE. Models with near-optimal curves are visualized by the region of their near-optimal exponent pairs. While there is barely a visible difference concerning the goodness of fit between the best fitting and the near-optimal model curves, there are differences in the prognosis, whence the near-optimal models are used to assess the uncertainty of extrapolation. RESULTS: For data about the growth of an untreated tumor we found the best fitting growth model which reduced SSE by about 30% compared to the hitherto best fit. In order to analyze the uncertainty of prognosis, we repeated the search for the optimal and near-optimal exponent-pairs for the initial segments of the data (meaning the subset of the data for the first n days) and compared the prognosis based on these models with the actual data (i.e. the data for the remaining days). The optimal exponent-pairs and the regions of near-optimal exponent-pairs depended on how many data-points were used. Further, the regions of near-optimal exponent-pairs were larger for the first initial segments, where fewer data were used. CONCLUSION: While for each near optimal exponent-pair its best fitting model curve remained close to the fitted data points, the prognosis using these model curves differed widely for the remaining data, whence e.g. the best fitting model for the first 65 days of growth was not capable to inform about tumor size for the remaining 49 days. For the present data, prognosis appeared to be feasible for a time span of ten days, at most.

Best-fitting growth curves of the von Bertalanffy-Pütter type.

INTRODUCTION: A large body of literature aims at identifying growth models that fit best to given mass-at-age data. The von Bertalanffy-Pütter differential equation is a unifying framework for the study of growth models. PROBLEM: The most common growth models used in poultry science literature fit into this framework, as these models correspond to different exponent-pairs (e.g., Brody, Gompertz, logistic, Richards, and von Bertalanffy models). Here, we search for the optimal exponent-pairs (a and b) amongst all possible exponent-pairs and expect a significantly better fit of the growth curve to concrete mass-at-age data. METHOD: Data fitting becomes more difficult, as there is a large region of nearly optimal exponent-pairs. We therefore develop a fully automated optimization method, with computation time of about 1 to 2 wk per data-set. For the proof of principle, we applied it to literature data about 217 male meat-type chickens, Athens Canadian Random Bred, that were reared under controlled conditions and weighed 28 times during a time span of 170 D. RESULTS: We compared 2 methods of data fitting, least squares using the sum of squared errors (SSE), which is common in literature, and a variant using the sum of squared log-errors SSElog. For these data, the optimal exponent-pairs were (0.43, 4.06) for SSE = 2,208.6 (31% improvement over literature values for the residual standard deviation) and (0.89, 0.93) for SSElog = 0.04599. Both optimal exponents were clearly distinct from the exponent-pairs of the common models in literature. This finding was reinforced by considering the region of nearly optimal exponents. DISCUSSION: We explain, why we recommend using SSElog for data fitting and we discuss prognosis, where data from the first 8 wk of growth would not be enough.

Optimal and near-optimal exponent-pairs for the Bertalanffy-Pütter growth model.

The Bertalanffy-Pütter growth model describes mass m at age t by means of the differential equation dm/dt = p * m a  - q * mb . The special case using the von Bertalanffy exponent-pair a = 2/3 and b = 1 is most common (it corresponds to the von Bertalanffy growth function VBGF for length in fishery literature). Fitting VBGF to size-at-age data requires the optimization of three model parameters (the constants p, q, and an initial value for the differential equation). For the general Bertalanffy-Pütter model, two more model parameters are optimized (the pair a < b of non-negative exponents). While this reduces bias in growth estimates, it increases model complexity and more advanced optimization methods are needed, such as the Nelder-Mead amoeba method, interior point methods, or simulated annealing. Is the improved performance worth these efforts? For the case, where the exponent b = 1 remains fixed, it is known that for most fish data any exponent a < 1 could be used to model growth without affecting the fit to the data significantly (when the other parameters were optimized). We hypothesized that the optimization of both exponents would result in a significantly better fit of the optimal growth function to the data and we tested this conjecture for a data set (20,166 fish) about the mass-growth of Walleye (Sander vitreus), a fish from Lake Erie, USA. To this end, we assessed the fit on a grid of 14,281 exponent-pairs (a, b) and identified the best fitting model curve on the boundary a = b of the grid (a = b = 0.686); it corresponds to the generalized Gompertz equation dm/dt = p * ma  - q * ln(m) * ma . Using the Akaike information criterion for model selection, the answer to the conjecture was no: The von Bertalanffy exponent-pair model (but not the logistic model) remained parsimonious. However, the bias reduction attained by the optimal exponent-pair may be worth the tradeoff with complexity in some situations where predictive power is solely preferred. Therefore, we recommend the use of the Bertalanffy-Pütter model (and of its limit case, the generalized Gompertz model) in natural resources management (such as in fishery stock assessments), as it relies on careful quantitative assessments to recommend policies for sustainable resource usage.

On the exponent in the Von Bertalanffy growth model.

Von Bertalanffy proposed the differential equation m'(t) = p × m(t) a  - q × m(t) for the description of the mass growth of animals as a function m(t) of time t. He suggested that the solution using the metabolic scaling exponent a = 2/3 (Von Bertalanffy growth function VBGF) would be universal for vertebrates. Several authors questioned universality, as for certain species other models would provide a better fit. This paper reconsiders this question. Based on 60 data sets from literature (37 about fish and 23 about non-fish species) it optimizes the model parameters, in particular the exponent 0 ≤ a < 1, so that the model curve achieves the best fit to the data. The main observation of the paper is the large variability in the exponent, which can vary over a very large range without affecting the fit to the data significantly, when the other parameters are also optimized. The paper explains this by differences in the data quality: variability is low for data from highly controlled experiments and high for natural data. Other deficiencies were biologically meaningless optimal parameter values or optimal parameter values attained on the boundary of the parameter region (indicating the possible need for a different model). Only 11 of the 60 data sets were free of such deficiencies and for them no universal exponent could be discerned.