Stochastic and deterministic models for the metastatic emission process
- 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 accessibilityAlthough 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.…
| Author details: | Christophe Gomez, Niklas HartungORCiD |
|---|---|
| DOI: | https://doi.org/10.1007/978-1-4939-7493-1_10 |
| ISBN: | 978-1-4939-7493-1 |
| ISBN: | 978-1-4939-7492-4 |
| ISSN: | 1064-3745 |
| ISSN: | 1940-6029 |
| Pubmed ID: | https://pubmed.ncbi.nlm.nih.gov/29344891 |
| Title of parent work (English): | Cancer Systems Biology |
| Subtitle (English): | Formalisms and Crosslinks |
| Publisher: | Humana Press Inc. |
| Place of publishing: | Totowa |
| Publication type: | Article |
| Language: | English |
| Date of first publication: | 2018/01/18 |
| Publication year: | 2018 |
| Release date: | 2022/03/30 |
| Tag: | Mathematical modeling; Metastasis; Poisson process; Structured population equation |
| Volume: | 1711 |
| Number of pages: | 32 |
| First page: | 193 |
| Last Page: | 224 |
| Organizational units: | Mathematisch-Naturwissenschaftliche Fakultät / Institut für Mathematik |
| DDC classification: | 5 Naturwissenschaften und Mathematik / 51 Mathematik / 510 Mathematik |
| Peer review: | Referiert |
