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A new method to estimate parameters of the growth model for metastatic tumours

Esmaeil Mehrara14*, Eva Forssell-Aronsson14, Viktor Johanson2, Lars Kölby2, Ragnar Hultborn3 and Peter Bernhardt14

Author Affiliations

1 Department of Radiation Physics, University of Gothenburg, Sahlgrenska University Hospital, Göteborg, SE - 413 45, Sweden

2 Department of Surgery, University of Gothenburg, Göteborg, Sweden

3 Department of Oncology, University of Gothenburg, Göteborg, Sweden

4 Department of Medical physics and Biomedical Engineering, Sahlgrenska University Hospital, Göteborg, Sweden

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Theoretical Biology and Medical Modelling 2013, 10:31  doi:10.1186/1742-4682-10-31

Published: 9 May 2013



Knowledge of natural tumour growth is valuable for understanding tumour biology, optimising screening programs, prognostication, optimal scheduling of chemotherapy, and assessing tumour spread. However, mathematical modelling in individuals is hampered by the limited data available. We aimed to develop a method to estimate parameters of the growth model and formation rate of metastases in individual patients.

Materials and methods

Data from one patient with liver metastases from a primary ileum carcinoid and one patient with lung metastases from a primary renal cell carcinoma were used to demonstrate this new method. Metastatic growth models were estimated by direct curve fitting, as well as with the new proposed method based on the relationship between tumour growth rate and tumour volume. The new model was derived from the Gompertzian growth model by eliminating the time factor (age of metastases), which made it possible to perform the calculations using data from all metastases in each patient. Finally, the formation time of each metastasis and, consecutively, the formation rate of metastases in each patient were estimated.


With limited measurements in clinical studies, fitting different growth curves was insufficient to estimate true tumour growth, even if patients were followed for several years. Growth of liver metastases was well described with a general growth model for all metastases. However, the lung metastases from renal cell carcinoma were better described by heterogeneous exponential growth with various growth rates.


Analysis of the regression of tumour growth rate with the logarithm of tumour volume can be used to estimate parameters of the tumour growth model and metastasis formation rates, and therefore the number and size distribution of metastases in individuals.

Modelling tumour growth; Metastasis; Dissemination; Gompertzian