Fig. 1: Growth, state regulation and stochastic state transition combine to give the cell density evolution in the proposed Cell Population Balance model.

In each panel, the cell density u is represented at time t in black, and at a subsequent time t + Δt in colour. The number of cells with cell state in the interval [a, b] is represented by the hashed (black or coloured) area. Growth (which includes cell division and cell death) results in an upward or downward shift of the cell density. In this toy example (top left panel), cell division takes place faster than cell death, so the cell density at time t + Δt is shifted upward compared to the cell density at time t. Consequently, the number of cells with cell state in [a, b] increases between t and t + Δt. State regulation results in a horizontal displacement of the cell density. In this example (center left panel), a concentration phenomenon is taking place towards the center of the state space. Here, the number of cells with cell state in [a, b] increases between t and t + Δt. Stochastic state transition leads to a mixing of the cell population. The evolution of the number of cells with cell state in [a, b] depends on the cell density at every other point of the state space. This results in a diversification of the population, that is in a flattening of the cell density curve (bottom left panel). In this example, the number of cells with cell state in [a, b] decreases between t and t + Δt. All three mechanisms combine in the cell population balance model to give the evolution of the cell density (right panel).