Progress in modeling malignant brain tumors
Len Sander & David Mobley
February 19, 2004 [pdf version]

In this brief report I will say what we have been doing in modeling invasion and tumor growth using continuum methods. Our intention is to get an overall picture of the development of the in vitro tumors that were observed by Deisboeck, et. al [1]. We are not, at the moment, trying to see branch formation; this will probably need discrete simulations, as in [2]. Our hope is that this report will evoke comment. Our ideas about what triggers invasion and what limits growth are
based, as closely as we can do, on the biology as we understand it. That’s what we need help on, and probably further experiments.

The general picture is that we have two ‘species’ of tumor cells: proliferative, which we call s, and invasive, a. The notation is taken from bacteria: there are bacteria colonies whose members can be ‘social’ or ‘adventurous’, and this gives rise to characteristic patterns. Our specific model is close to that of the Murray group [3], but generalized to two species (they only have one).

Here are the equations. We work in three dimensions (it is just as easy as 2 because we are assuming everything is spherically symmetric.) For the central tumor, we want the s cells to proliferate. Thus:

(1)

What we have done is say that there is proliferation with rate K = 0.831/day, which we get from the doubling time. The term cuts othe growth when the cells are crowded, i.e., when the density is larger than so.

The Heaviside function, , is given by . What it is trying to do is represent the conversion from s to a. If the density of s is small enough, then the s cells lack gap junctions (or something) and stop proliferating. The second term in brackets converts them to a-type cells with the same doubling time. In words, if the density is higher than , s-cells proliferate. If the density is lower than , they convert. We guess that is the percolation density: below the tumor is no longer connected.

Now for the invasive cells we have motion and no proliferation. The experiment [1] seems to say that nutrient governs, at least in part, the motion of the a-cells, since artificial introduction of nutrient causes invasive cells to gather. Here is our model (see [2] for more details):

(2)

Figure 1: Invasive velocities referred to a fixed center.

There is diffusion, chemotaxis (the second term) with coefficient, and which follows the gradient of nutrient, . The last term is the production of a-type cells by the tumor.

Finally, we need to model the nutrient. We suppose that it is uniformly distributed initially, but that the central spheroid consumes so much that n goes to 0 very quickly inside it. Also, the invasive cells consume nutrient. All these rates are known [2], and are put in.

The first thing we need tried to fit is the invasive velocities in [1]. This is shown in Figure 1. We can get the first peak in the invasive speed, but we don’t have a clue why there is a second peak in the experiment. Perhaps there is secretion of waste by dying cells in the MTS which moves the a cells away. Any ideas??

We can also look at the distribution of the various species. In Figure 2 we have the proliferative cells, the invasive cells, and the nutrient after 10 days, doing the best we can for the parameters. This figure is interesting: we find that the invasive cell density peaks outside, but near the tumor, as must be the case.

Now this can be refined in lots of ways, but what we would chiefly like to hear is whether the basic ideas make sense. We take our ideas about the conversion from the work of Berens [4], but it certainly is an interpretation which might be wrong.

Figure 2: Densities after 10 days. The important curves are the numerical solutions.

References:

[1] Deisboeck TS, Berens ME, Kansal AR, Torquato S, Stemmer-Rachamimov AO, Chiocca EA.
Pattern of self-org Cell Prolif. 2001 Apr;34(2):115-34. PMID: 11348426.

[2] Sander LM, Deisboeck TS. Growth patterns of microscopic brain tumors. Phys Rev E
66:051901. (2002)

[3] Burgess P. K., P. M. Kulesa, J. D. Murray, and E. C. Alvord. The interaction of growth rates
and diusion coecients in a three-dimensional mathematical model of gliomas. J. Neuropath.
Exp. Neurol. 56,704-713 (1997).

[4] Gap junction intercellular communication in gliomas is inversely related to cell motility Mc-
Donough W.S., Johansson A, Joee H, Giese A, Berens M.E. , Gap junction intercellular
communication in gliomas is inversely related to cell motility , Int. J. Devel. Neurosci., 17:
601-611 (1999)