Figure 8.

Hybrid microscopic-macroscopic modeling. (a) Kim, Stolarska and Othmer (92-93) coupled a discrete model of proliferating cells (red circles) with a viscoelastic continuum model of the necrotic core (cyan solid), by coupling the triangular computational mesh through the quiescent region (blue lines). The model could successfully transfer forces and masses between the two representations. Reproduced by permission from (92). (b-d) Wise, Lowengrub, Cristini, Frieboes and colleagues (10, 62-63, 65) used a discrete model of motile glioma cells (blue dots in (b) and (c)) to model the EMT in glioma, and a continuum mixture model (gray regions in (b) and (d)) to model regions of tumor cell aggregation. The model was successful in dynamically choosing between discrete and continuum representation according to localized biophysical criteria (in this case, a threshold density, coupled with hypoxic considerations), while maintaining mathematical consistency between the models and conserving mass and momentum during any switch. Palisading cells were observed to move up oxygen gradients (blue dots; see (c)), and once cells aggregated in normoxic regions in sufficient numbers, they were converted back to the continuum representation, thus creating a satellite tumor (outer gray regions in (d)). Adapted with permission from (63, 65).