Minimal Mechanistic Models of Cancer-Cell Invasion

Definitions and mathematical statement

The source asks for the identification of subcellular processes essential for cancer-cell invasion and for a suitable level of mechanistic detail in a model. Here a mechanistic model means a mathematical description representing biological processes involved in invasion; the source specifies neither a class of such models nor a mathematical definition of invasion. The terms "essential", "realistic", and "sufficiently simple" are criteria to be specified, rather than defined mathematical predicates in the supplied record.

The supplied research question is:

Which subcellular processes are essential for cancer-cell invasive behavior, and what level of mechanistic detail is needed for a model that is realistic yet sufficiently simple?

A precise mathematical formulation remains unresolved. The source does not specify the cancer-cell population or biological setting, the candidate subcellular processes, the invasion observables, a model class, validation data, an adequacy criterion, or a measure of mechanistic complexity. It also does not state whether "essential" means necessary for invasion itself or necessary for a model to reproduce specified observations, or whether the requested conclusions should hold across all cancer types and environments or only in a particular setting. No equations, domains, initial or boundary conditions, numerical thresholds, or mathematical quantifiers are supplied. Consequently, the question cannot be converted into a uniquely specified theorem, existence problem, or minimization problem without additional assumptions.

The supplied literature assessment describes models incorporating cytoskeletal, extracellular-matrix, confinement, and phenotype mechanisms, while reporting no validated consensus on the essential subcellular processes or the minimal adequate detail. These are contextual mechanisms mentioned in the curation, not a prescribed list of essential subcellular processes. The record labels the problem open; this label and the assessment are supplied dated curation, not independently verified mathematical conclusions.

Short English statement

Determine which subcellular processes must be represented to explain cancer-cell invasion, and how much mechanistic detail a realistic, sufficiently simple model needs. The source leaves the biological setting, meaning of necessity, validation criteria, and measure of simplicity unspecified, so a unique mathematical formulation requires further choices. This is a statement-only draft requiring human mathematical review.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30002570$, title "Minimal Mechanistic Models of Cancer-Cell Invasion", source problem code catalogue entry, in the collection "Oberwolfach Reports Open Problems". No individual proposer is identified. The record gives the proposed year as 2014.
• Supplied report citation: Mini-Workshop: Mathematical Models for Cancer Cell Migration (2015), Oberwolfach Reports, DOI 10.4171/owr/2014/19, pp. 1075--1109. https://doi.org/10.4171/owr/2014/19. The background describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions. The citation's 2015 designation and the record's 2014 proposed year are retained without resolving their relationship.
• Supplied literature source, "Multiscale invasion model": https://pmc.ncbi.nlm.nih.gov/articles/PMC10293590/.
• Supplied literature source, "Molecular-pathway invasion model": https://doi.org/10.1371/journal.pcbi.1004571.
• Literature assessment checked on 2026-08-21, according to the supplied record: status "open"; multiscale models incorporate cytoskeletal, extracellular-matrix, confinement, and phenotype mechanisms, but the assessment reports no validated consensus identifying essential subcellular processes or a minimal adequate level of detail. This is supplied dated literature triage derived from the OWR curation, not independently verified here. Time-sensitive status and cited claims require independent verification.
• The registry marks the statement status "exact" and supplies a verification note describing it as clear, self-contained, publication-ready, and retained byte-for-byte. These are source-record assertions, not an independent verification of the report wording or of mathematical completeness.