With the exception of the last section we have assumed that the tumour cell population is homogeneous involving only one cell type. In spite of the good quantitative comparison with data, such as regards life expectancy after resection, in the last section we saw that variable sensitivity to chemotherapeutic drugs exhibited by gliomas necessitated a model with tumour cell heterogeneity. Gliomas are known to be heterogeneous (polyclonal) with heterogeneity generally increasing with grade. The more malignant cells are believed to have a stronger propensity to mutate thus increasing heterogeneity. Therefore, we expect to see different types of cells within the tumour (for example, Pilkington 1992). The basic model can be extended to consider the case of a polyclonal tumour as we saw in the last section by simply creating two (or more) cell populations within the tumour that may have different diffusivity and growth rates. In the model equation (11.3) (the homogeneous tissue situation) the cell density C becomes the vector of cell densities c and the diffusion coefficient D and the growth rate p now become diagonal matrices of diffusion coefficients and growth rates, respectively. The total tumour population, at a given point in space and time (x, t), is the sum of the components of the c vector. In the chemotherapy model in the last section we took the two cell populations to be independent. Often, however, the subpopulations are not independent but are connected by mutational events transferring cancer cells from cell population i to cell population j . A fairly general but basic model to account for population polyclon-ality can be written in the dimensional form dc
d t where T is a matrix representing transfer between subpopulations. To complete the mathematical problem we use initial and boundary conditions n ■ DVc = 0 for x on dB (the boundary of the brain) c(x, 0) = f(x) for x in B (the brain domain).
We expect that the introduction of multiple cell populations in the tumour introduces more heterogeneity in the growth patterns of the simulated tumour. Clinical and experimental results have shown fingering and branching of the visible tumour.
In this section we consider, by way of example, the existence of two clonal subpopulations within a tumour with mutational transfer. To be specific we assume one population has a high growth and low diffusion coefficient while the second population has a moderate growth rate and high diffusion coefficient; there are various other scenarios we could take. Let c=d=(D D)•»=cs:)• =o;
where the D's, p's and k's are constant parameters, and (11.60) becomes d U
d t with given initial conditions u (x, 0) = f (x) and v(x, 0) = g(x). With u the more rapidly proliferating population and v the more rapidly diffusing population, D2 > Di and p1 > p2. We further assume u cells are the only tumour cells initially present (f (x) > 0 and g(x) = 0). With some small probability, k ^ p1, u cells mutate to form v cells. Although we do not include it here, each of the cell populations could retain the ability to diffuse faster in white matter regions.
Initially, let us suppose there is a source of u tumour cells that have mutated from healthy cells and can proliferate faster then the neighboring normal cells thereby starting to form a tumour. We can think of k as a measure of the probability of u tumour cells mutating to become the rapidly diffusing tumour cell population v. Introduce the nondimensional variables x = x, t = p1t, 0 = — < 1, a = k, v = ■ -, (11.62)
D1 p1 p1
u(x, t) = —^^ /-P1 x,p1t ), v(x, t) = v ( /—x,p1t ), (11.63) p1u0 D1 p1u0 D1
where u0 = / f (x)dx, the total original cancer cell population. Growth is measured on the timescale of the u population proliferation and diffusion is on the spatial scale of the u cell diffusion.
Equations (11.61) now become d u o
The parameter a = k/p1 ^ 1 is the proportion of the first subpopulation's growth lost to mutation.
In one space dimension on an infinite domain we can write down the analytical solutions from which some interesting and highly relevant conclusions can be deduced as we shall see. In one space dimension d u d 2u
dv d 2v
Let us take the initial source of u tumour cells to be u(x, 0) = S(x) and take v(x, 0) = 0. The v population diffuses faster than u so v > 1. The growth rate of the u population is larger than v so fi < 1. The growth rate of u is much higher than the probability of mutation so a ^ 1.
The solution of (11.66) can be solved separately from the v equation and has the solution u(x, t) = J_exp | (1 — a)t — , (11.68)
\ 4t j which on substituting this result into the v equation (11.67) gives dv d2 v 1 / x2 \
dt = vd!2+fiv +a V4n7 exp((1 — a)t — «). (1L69)
We use a Fourier transform in the spatial variable x to solve this equation. With the transform and its inverse defined by
F— 1 [ F (t ; m)](x, t) = —\ F (t ; m)eimx dm 2n J—œ
the transformed equation for v is then d V 2
— = v(iw)2V + fiV + ae(l—a—m2)t with V(t = 0; w) = 0, d t where V(t; w) = F[v(x, t)]. The solution for V is then given by a e(1—a—w2)t — e(fi — vw2)t
Taking the inverse transform gives, after some algebra, v(x, t) as v(x, t) = F-1[V(t; w)]
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