Figure
10.1 corresponds to the following system of massbalance ordinary differential equations (using deterministic equations for the means of the underlying stochastic variables and focusing on expected values and rates of increase in model quantities
I,
M, and
T, rather than on first passage times):

N = constant (with a value that may depend on smoking),

dP/dt = a ^{*} N – (r + f)P(t),

dI/dt = f ^{*} P(t) – (c + d – b)I(t),

dM/dt = c ^{*} I(t) – (h + e – g)M(t),

dT/dt = h ^{*} M(t).
The first of these equations sets the size of the normal lung stem cell compartment to a constant, N, that is assumed to be homeostatically maintained and that may depend on smoking behavior. Thus, if transients are ignored, normal stem cell numbers are assumed to remain steady at some level, N, during smoking (reflecting assumed homeostasis). While there is good evidence that smoking increases lung cellularity, it is not clear exactly how this affects the numbers of lung stem cells at risk of carcinogenic transformations. Hence, in Table 10.4, this entry ranges from 1 (no smokinginduced amplification) to 3 [the value of an early increase in cellularity noted by Mancini et al. (1993), although not specifically for stem cells].
The remaining equations express the identity that the rate of growth in each compartment at each moment is the difference between the total inflow from all sources (typically, cell births in the compartment plus new immigrations from the preceding compartment) and the outflows to all destinations (typically cell death, differentiation, or transition to a subsequent compartment). Arguably, the rate parameter h is excessively simplified, as the acquisition of fully tumorigenic properties may require over 30 events (both clonal genetic alterations and epigenetic lesions such as promoter hypermethylations) that can occur in different ways and orders and that can lead to different specific histological types of lung cancer (Minna et al., 2002; Wistuba et al., 2001). The simplistic final equation, dT/dt = h ^{*} M(t), emphasizes that the formation of such fully tumorigenic cells, however complex, is driven and limited by the number of earlystage malignant cells from M entering the process.
Rather than solving the above dynamic system for timevarying values of
P(
t),
I(
t),
M(
t), and
T(
t), we consider steadystate solutions for
P,
I, and
M by setting their time derivatives equal to zero and solving the resulting algebraic system, yielding

P = a ^{*} N/(r + f),

I = [f ^{*} P(t)/(c + d – b)] = (p ^{*} a ^{*} N)/(c + d – b) [using the definition p = f/(f + r)],

M = [c ^{*} I(t)/(h + e – g)] = (c ^{*} p ^{*} a ^{*} N)/[(c + d – b)^{*}(h + e – g)]

= (p ^{*} a ^{*} N)/{[1 + (d – b)/c]^{*}[(h + e – g)]}.
These solutions are physically meaningful if the initiated and early malignant cell populations do not spontaneously grow without bound (as fully tumorigenic cells would do) but instead tend to become extinct, i.e., if
c +
d >
b and
h +
e >
g. (This differs from the assumptions of most TSCE models, in which the net birth rates are assumed to remain positive. For lung cancer, premalignant clonal expansion and
in situ carcinoma both appear to have selflimited growth.) However, the preceding formulas provide key components of the transient solutions even if the net birth rates are positive (Hazelton et al., 2005). If the net birth rates are negative, so that
c +
d >
b and
h +
e >
g, then the formulas are well motivated; otherwise, they provide only a heuristic guide to the contributions of different mechanisms to cancer risks, and a timevarying analysis is required for more accurate results. Subject to these caveats, the expected steadystate production of new tumorigenic cells per unit time can be written as a product:
$$\begin{aligned}&\hbox{production rate of tumorigenic cells} =
dT/dt = h^\ast M (t)\\
&\quad= (p^\ast a^\ast N)^\ast [c/(c + d  b)]^\ast [h/(h + e  g)] \\
&\quad= (p^\ast a^\ast N)/ \{[1 + (d  b)/ c]^\ast [1 + (e  g)/ h]\}\end{aligned}.$$
An increase in any of the components p, a, N, [c/(c + d – b)], or [h/(h + e – g)] by a factor of (1 + x) will multiply the production rate of malignant cells by the same factor, motivating the use of relative risk factors in Table 10.4. Interpretively, (p ^{*} a ^{*} N) is just the flux of new initiated cells created per unit time, while [c/(c + d – b)]^{*}[h/(h + e – g)] reflects the expected fraction of tumor cells eventually exiting the promotionprogression pipeline for each initiated cell entering it.