ArticleBMC cancer2019
Best fitting tumor growth models of the von Bertalanffy-PütterType.
Article in BMC cancer, 2019. The graph could read no effect estimate from its abstract, so it casts no vote on the map. Cited by 9 papers.
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Who cites it
9 citing papers in PubMed.
- Cancer interception during treatment: using growth kinetics to create a continuous variable for assessing disease response.The oncologist · 2025Article
- Proper likelihood functions for parameter estimation in S-shaped models of unperturbed tumor growth.Scientific reports · 2025Article
- Article
- Understanding patient-derived tumor organoid growth through an integrated imaging and mathematical modeling framework.PLoS computational biology · 2024Article
- A Spatially Resolved Mechanistic Growth Law for Cancer Drug Development Predicting Tumor Growing Fractions.Cancer research communications · 2022Article
- A continuum mechanical framework for modeling tumor growth and treatment in two- and three-phase systems.Archive of applied mechanics = Ingenieur-Archiv · 2022Article
- A novel interpretable machine learning algorithm to identify optimal parameter space for cancer growth.Frontiers in big data · 2022Article
- Forecasting the final disease size: comparing calibrations of Bertalanffy-Pütter models.Epidemiology and infection · 2020Article
- Estimating polymorphic growth curve sets with nonchronological data.Ecology and evolution · 2020Article
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5 authors.
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Abstract
backgroundLongitudinal 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.
methodThe 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.
resultsFor 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.
conclusionWhile 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.
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