Page 127 - 20dynamics of cancer

112 CHAPTER 6

and outflux from that stage over the time interval [0,t]. Third, cells that
enter certain stages undergo clonal expansion. Fourth, clonal expansion
increases the number of cells at risk for making the transition to the next
stage. To account for this, outflux from a stage increases with the size
of clones in that stage.
The probabilities of being in various stages based on the influx and
outflux from each stage are

x 0 (t) = D 0 (t, 0)
t
x i (t) = u i−1 (s) x i−1 (s) D (t, s) ds i = 1,...,n − 1
0
t

x n (t) = u n−1 (s) x n−1 (s) ds,
0
where u i−1 (s)x i−1 (s) is the influx into stage i at time s, and

t
D i (t, s) = e − s u i (z)dz

is the outflux (decay) as of time t of the influx component that arrived
at time s. The integration of x i values over the time interval [0,t] means
that all influxes and outfluxes are summed over the whole time period.
The u i (t) values vary with time because the fluxes depend on clonal
expansion, so we need to express the u’s in terms of clonal expansion.
I use a logistic model to describe clonal growth. If y i (t) is the size of
the clone in the ith stage at time t, then the clone grows according to
˙ y(t) = r i y i (1 − y i /K i ), where the dot means the derivative with respect
to time, r i is the maximum rate at which the clone increases, and K i is
the maximum size to which the clone grows. Starting with a single cell,
the size of the clone after a time period s of clonal expansion follows
the well-known solution for the logistic model (Murray 1989):

K i e r i s
y i (s) = .
K i + e i s − 1
r
The subscripts describe different stages, so that the different stages may
have different rates of increase and maximum sizes.
If we assume that transitions between stages occur by somatic muta-
tion, then for each cell that makes the transition into stage i, the total
mutation capacity of that cell lineage is the mutation rate per cell, v,