Transitions and Transport Among Cancer-Cell Migration Strategies

Definitions and mathematical statement

The source distinguishes three named cancer-cell movement strategies: amoeboid, mesenchymal, and collective. These are descriptive labels in the supplied record; no mathematical definitions of the strategies or rules for their interaction are supplied.

The stated research questions are:

1. Which strategy is advantageous for invasion?
2. What triggers transitions between strategies?
3. What dynamics occurs when all three strategies are present?
4. What are their transport properties, including invasion speeds?

These questions constitute a research programme rather than a uniquely specified mathematical proposition. The record does not prescribe a dynamical model, state variables, spatial domain, environmental parameters, initial or boundary conditions, transition rules, or quantifiers over admissible models and environments. It also does not define the criterion for invasion advantage, the transport observables, or the meaning of invasion speed. In particular, it does not specify whether speed means a transient front velocity, an asymptotic spreading speed, or another observable. No such choice is imposed here. The record also does not specify whether the three labels are mutually exclusive states or how collective movement relates to the other two labels. A precise model-dependent statement therefore requires additional hypotheses and definitions.

The supplied registry labels the problem partially solved. Its supplied literature assessment reports that experiments and switching models identify environmental triggers for amoeboid--mesenchymal transitions, and that hybrid models compare collective and solitary invasion and speeds, while optimal strategy and fully coupled three-mode dynamics remain context-dependent and unresolved. This is supplied dated curation, not an independently verified mathematical conclusion.

Short English statement

Determine how amoeboid, mesenchymal, and collective cancer-cell migration compare for invasion, what induces transitions, how the strategies interact when all are present, and what transport behaviour and invasion speeds result. The source leaves the mathematical model, comparison criterion, conditions, and speed definitions unspecified, so this remains a research programme requiring further formulation and human review.

Sources

• Local UnsolvedMath registry, supplied complete record for permanent ID $30002575$, source problem code catalogue entry, title: Transitions and Transport Among Cancer-Cell Migration Strategies. The collection is Oberwolfach Reports Open Problems. No individual proposer is supplied. The record gives proposed year $2014$, labels the statement exact, and labels the status partially_solved; these are supplied registry annotations.
• 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 supplied background describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions. The citation's year $2015$ and the registry's proposed year $2014$ are retained without reconciliation.
• Supplied literature source, labelled "Migration-plasticity model": https://doi.org/10.1038/s41598-017-09300-3.
• Supplied literature source, labelled "Confinement transition model": https://doi.org/10.1021/acs.nanolett.9b01597.
• Supplied literature source, labelled "Hybrid multiscale invasion model": https://doi.org/10.1007/s11538-024-01286-0.
• Literature assessment checked at 2026-08-21, according to the supplied record. This date identifies supplied dated curation derived from the OWR material, not independent verification. The supplied assessment attributes progress to experiments, switching models, and hybrid models, but describes optimal strategy and fully coupled three-mode dynamics as context-dependent and unresolved. Time-sensitive status and cited claims require independent verification.