TABLE II.

List of frequently physical frameworks used to model cancer cell migration and their applications.

Common modeling approaches	Applications and examples
Chemo-mechanical models based on force-dependent reaction kinetics.	Used to model sub-cellular processes such as cell-substrate bond formation, filament polymerization and gliding and mechanosensing-based changes to predict resulting cell adhesion, traction, and migration (e.g., various spring/dashpot models,51–53 active matter models,54 and molecular clutch models55).
$F\left(\mu \right)-\gamma \left(F,n\right)v=0$… Force balance between active forces driven by chemical potential, $\mu$, and dissipative forces, $\gamma v$.
$\gamma \left(F,n\right)=K{\tau }_0\mathrm{e}\mathrm{xp}(-\frac{F}{{nF}_0})$… stiffness, $K$, and number, $n$, and kinetics of molecular bonds, ${\tau }_0$ dictate drag coefficient, $\gamma$.
Agent-based models focusing on force balance between individual cells and their environment.	Used to model cell populations interacting with each other and the environment. Coarse grained to implicitly include effects of various sub-cellular processes (e.g., force-based models,56–58 energy-based models,59,60 and lattice-based/cellular Potts models61–63).
${\overrightarrow{F}}_{\mathit{active}}+{\overrightarrow{F}}_{\mathit{passive}}+{\overrightarrow{F}}_{\mathit{dissipative}}=0$…
or an energy minimization approach
$E=\sum {\lambda }_i{(A_i-A_0)}^2+\sum {\sigma }_{ij}{l}_{ij}+\sum \frac{dU}{d{\overrightarrow{r}}_i}.{\overrightarrow{r}}_i$
Thermodynamic models based on equilibrium and non-equilibrium work-free energy change relationships	Used to model both cellular and sub-cellular processes and assess the energetic states that the system can occupy (e.g., free-energy-based models64,65).
$\Delta F=\sum \Delta {\mu }_i-k_{B}T\ln \frac{\Omega }{{\Omega }_0}$…Free energy change of the system
$W=\int F_{\mathit{active}}dx$… work done by the system
Equilibrium… minimize ($\Delta F)$
Non-equilibrium … $\Delta F\gtrsim W$
Continuum phase-field models	Used to describe cell and surrounding free space as an evolving phase-field, with the moving boundary representing the cell membrane. Well suited to describe collective migration66,67 and migration of cell monolayers.68
$\frac{d{\Phi }_i}{dt}+{\overrightarrow{v}}_i·\overrightarrow{\nabla }{\Phi }_i+\frac{\delta F}{\delta {\Phi }_i}=0$… Describes the dynamics of the cell shape in response to free energy changes. The free energy functional, $F$, is chosen so that minima correspond to phases (i.e., intracellular and extracellular environment) of the system.