Some Generalizations

Various cell cycle effects are important during tumor treatment. A considerable literature considers population dynamics approaches to such effects. Simple descriptions use a finite number of compartments, for various stages in the cell cycle, and one ODE for the time development of each compartment, e.g. [41, 42]. More difficult are biological delay systems [4].

Perhaps even more common are analyses which generalize the classic McKendrick/von-Foerster formalism. This formalism considers, as a "structure variable", chronological cell age $a$ since cell birth and deals with the number $n(t,a)$ of cells per unit age range at time t. The basic equations are

$$\partial n /\partial t +\partial n / \partial a = - \alpha (a) n , \qquad n(t, 0) = 2 \int_0^{\infty} \lambda(a) n(t,a) da$$ (12)

Here the term $\partial n / \partial a$ corresponds to the fact that a cell ages at unit rate; $\alpha (a)$ corresponds to a loss of cells to death or mitosis; and the integral represents a flux of newborn cells from mitoses at various ages. Many applications to radiobiology have been given, e.g. [43, 44]. Generalizations, including more complicated integro-differential equations or time evolution semi-group equations involving non-negative operators on a Banach space, have been applied to cell cultures and tumor growth models [45].

Reaction-diffusion PDE models (e.g [46, 47]) are commonly used to characterize spatial details of nutrient and oxygen flow to tumors, and of tumor response to the nutrients. The vascular component has been described by a number of sophisticated PDE models, as well as models more concerned with the intricate network of microvessels, including, e.g., branching process models [48 - 51].

Situations where cell number $N$ (and/or such quantities as DSBs per cell, $U$ above) must be regarded as stochastic [3] are of some importance; this is especially true in radiobiology, e.g. [43, 52 - 56]. Corresponding models showing small-number stochastic fluctuations, such as birth-death or branching models, have also been used for pre-tumor phases of carcinogenesis, e.g. [57]. In many cases such stochastic models can be regarded as models effectively involving a discretely infinite number of time-dependent unknowns, e.g. the probability $P_n(t)$ of $n$ cells for each value of $n$, with one ODE for each unknown.