Stochastic and Deterministic Models for the Metastatic Emission Process: Formalisms and Crosslinks

Part of the Methods in Molecular Biology book series (MIMB, volume 1711)


Although the detection of metastases radically changes prognosis of and treatment decisions for a cancer patient, clinically undetectable micrometastases hamper a consistent classification into localized or metastatic disease. This chapter discusses mathematical modeling efforts that could help to estimate the metastatic risk in such a situation. We focus on two approaches: (1) a stochastic framework describing metastatic emission events at random times, formalized via Poisson processes, and (2) a deterministic framework describing the micrometastatic state through a size-structured density function in a partial differential equation model. Three aspects are addressed in this chapter. First, a motivation for the Poisson process framework is presented and modeling hypotheses and mechanisms are introduced. Second, we extend the Poisson model to account for secondary metastatic emission. Third, we highlight an inherent crosslink between the stochastic and deterministic frameworks and discuss its implications. For increased accessibility the chapter is split into an informal presentation of the results using a minimum of mathematical formalism and a rigorous mathematical treatment for more theoretically interested readers.

Key words

Poisson process Structured population equation Metastasis Mathematical modeling 



The authors thank Florence Hubert, Charlotte Kloft, and Andrea Henrich for suggestions and critical reading of the manuscript. NH gratefully acknowledges financial support by the Agence Nationale de la Recherche under grant ANR-09-BLAN-0217-01.


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Copyright information

© Springer Science+Business Media LLC 2018

Authors and Affiliations

  1. 1.Aix Marseille UniversitéCNRS, Centrale Marseille, I2M UMR 7373MarseilleFrance
  2. 2.Department of Clinical Pharmacy and BiochemistryFreie Universität BerlinBerlinGermany
  3. 3.Present address: Institute of MathematicsUniversität PotsdamPotsdamGermany

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