Journal of Mathematical Biology

, Volume 72, Issue 6, pp 1633–1662 | Cite as

A “universal” model of metastatic cancer, its parametric forms and their identification: what can be learned from site-specific volumes of metastases

  • Leonid HaninEmail author
  • Karen Seidel
  • Dietrich Stoevesandt


We develop a methodology for estimating unobservable characteristics of the individual natural history of metastatic cancer from the volume of the primary tumor and site-specific volumes of metastases measured before, or shortly after, the start of treatment. In particular, we address the question as to what information about natural history of cancer can and cannot be gained from this type of data. Estimation of the natural history of cancer is based on parameterization of a very general mathematical model of cancer progression accounting for primary tumor growth, shedding of metastases, their selection, latency and growth in a given secondary site. This parameterization assumes Gompertz (and, as a limiting case, exponential) growth of the primary tumor, exponential growth of metastases, and exponential distribution of metastasis latency times. We find identifiable parameters of this model and give a rigorous proof of their identifiability. As an illustration, we analyze a clinical case of renal cancer patient who developed 55 lung metastases whose volumes were measured through laborious reading of CT images. The model with maximum likelihood parameters provided an excellent fit to this data. We uncovered many aspects of this patient’s cancer natural history and showed that, according to the model, onset of metastasis occurred long before primary tumor became clinically detectable.


Exponential tumor growth Gompertz law of tumor growth  Metastatic latency Model identifiability Natural history of cancer Poisson process 

Mathematics Subject Classification

60G 60E99 62M 62P 92C 


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Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • Leonid Hanin
    • 1
    Email author
  • Karen Seidel
    • 2
  • Dietrich Stoevesandt
    • 3
  1. 1.Department of Mathematics and StatisticsIdaho State UniversityPocatelloUSA
  2. 2.Halle (Saale)Germany
  3. 3.Department of Diagnostic RadiologyMartin Luther University of Halle-WittenbergHalleGermany

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